CA2091920A1 - Apparatus and method for picture representation by data compression - Google Patents

Abstract

A method and all apparatus for representing picture by data compression are provided. The method comprises dividing the picture into regions, registering the value of a brightness function, fixing a characteristic scale for each region, dividing the region into cells having a linear dimension in the order of the characteristic scale, identifying basic structures in each cell, representing said structures by models, and storing and/or transmitting data defining the models, which data represent the primary compression of the picture and can be further compressed or otherwise processed. The apparatus comprises means for carrying out the several steps of the method. The basic structures may be chosen from among predetermined lists and such structures are described. In a preferred aspect of the invention they are identified and represented by constructing geometric models representing them, associating to each geometrical model a mathematical model, condensing said geometrical or mathematical models to define a global model for each cell and encoding and quantizing the data defining the global model. Methods are described for constructing the geometric models, based on the distribution of the values of the brightness function by determining a grid, determining an approximating function, at each point of the grid, and comparing the values of the derivatives and gradients of the approximating function with a predetermined set of thresholds. When representing color pictures, the method is applied successively to each of the basic colors or to a luminance signal and two reduced bandwidth colour-information carrying signals obtained by known transformation of colour data. An image of the original picture can then be produced from the compressed data, and it may be a faithful image or a desired transformation thereof, if the compressed data have been correspondingly transformed. Various application of the invention are described, comprising the application to video sequences.

Description

- 1 - 2 ~ ?, ~
~097/2~24~9~
APPARATU~ AND lV~OD FOR PI~TI1~2E
~2EP~?ESENIATIONB~DATACOMP~E~SION

Field of the invention This invention relates to apparatus and methods for representing pictures by data compression, particularly, but not exclusively, for the purpose of storing and/or transmitting the compressed data and subsequently reconstructing the picture in a manner as faithful as possible.

The representation of various objects by data compression is a problem with which the art has been increasingly occupied in recent times. The problem is encountered in many cases, e.g.
when a picture, or a succession of pictures, for example constituting a television broadcast, has to be registered in a magnetic memory, such as a video tape, or is to be transmitted over a distance by electromagnetic waves or by cable. On the one hand, it is of considerable economic importance to increase as much as possible the amount of optical and acoustic information that can be registered on a given memory, whereby to reduce the size and cost of magnetic tapes or other information storage means. On the other hand, the available wave bands are increasingly crowded, and so are the cables, and it is increasingly necessary to compress the transmitted data, so that as great a number of them as possible may be transmitted over a given frequency or by a given cable. Data compression problems, - 2 - 2 ~ 2 20Y712~ I/g2 therefore, are increasingly acute, both in data storage and in data transmission.

In particular, the art has dealt with the problem of compressing the data which represent an object, e.g. a picture. A process for the production of images of objects is disclosed in EPA 0 465 852 A2, which process comprises the steps of: (1) approximating the object by a model comprising at least one differentiable component; (2) establishing the maximum allowable error and the degree of the polynomials by which the differentiable components of the model are to be approximated; (3) constructing a grid of a suitable pitch; (4) computing the coefficients of the Taylor polynomials of the aforesaid differentiable components at selected points of said grid.

However, none of the method and apparatus of the prior art are wholly satisfactory. Either the degree of compression is too small, or the picture cannot be faithfully reconstructed - vi~.
"decompressed" - from the compressed data, or both. There is another important requirement, not satisfied by known compression methods: application of image processing operations on compressed data, and natural extendibility of the compression scheme to video sequences compression.

In describing this invention, two-dimensional pictures, in particular color pictures, such as those created on a television screen, are considered, but three- or more than three-dimensional 20~1~20 2097/24291~V~2 objects could be represented by the apparatus and rnetho~l provided by the invention, e.g. by defining them by means of views or cross-sections in differ~nt planes.

The efficiency of a compression method depends on the one hand on the degree of compression, which should be as high as possible, but on the other on the faithfulness with which the picture reconstrcuted from the compressed data reproduces the original one. Perfect reproduction is obtained when the two pictures are visually undistinguishable. Two pictures are considered to be "visually undistinguishable", as de~ned by the MPEG (Motion Picture Expert Group of the International Standard Organization), when any ordinary viewer cannot distinguish between them when viewing them from a distance equal to six times the picture height. Different requirements for visual undistinguishability may be de~lned for different applications, such as: high end computer imaging, PC computer imaging, PC or video games, multimedia, pre-press applications, fax, colour video conferencing, ~Tideophone, archiving, medical imaging, aerial picture analysis, etc. However, the invention does not always require that the picture representation and the original be visually undistinguishable, though this is generally preferred: the degree of similarity may depend on the particular application and on the degree of faithfulness that is required of the representation in each case.

4 2~91~
20Y71~2~/H/g2 SlmLmarv of the inventiQn Broadly, The method of picture representation by data c~mpression according to the invention comprises the steps of:
1 - subdividing the picture into regions;
2 - registering the values of a brightness function (or "grey levels") preferably at each pixel of the picture;

3 - fixing for each region a characteristic scale (hereinafter indicated by L) preferably def~med in terms of a number of pixels;
~ - dividing each region into cells, each comprising a number of points (pixels) del'ined by two variables (coordinates), said cells having linear dimension in the order of L, and preferably equal approximately to L;
5 - identifying in each cell the "basic elements" or "structures", as hereinafter defined;
6 - in each cell, representing the basic elements by models (or submodels), as hereinafter defined; and 7 - storing and/or transmitting, for each cell the data defining each model, said data together representing the "primary compression" of the picture.
- Optionally, said data may be further compressed by any suitable methods, or otherwise processed, as will be explained hereinafter.

The regions into which the picture is subdivided are chosen in such a way that the data to be stored and/or transmitted for each of them will not be too numerous, and thus will not create files that are too cumbersome, particular with regard to the hardware that is available and to its capacity. Therefore in some cases the ~ ' .

2091~2~
2~ 2 whole picture may be considered as a single region, or conversely, in other cases the regions will only be small fractions of the whole picture. Thus a suitable subdivision into regions will offer no difficulties to skilled persons.

The picture to be represented is defined by the brightness values of the basic colours tusually three) for each pixel, or by equivalent data. Said values may be available in the form of computer files, or may be transmitted by a picture generating apparatus, e.g. a TV
camera, or may be read by means of scanners. In any case, when a colour picture is to be compressed, the method according to the invention can be applied separately to each of the three (or two or four) basic colours, and corresponding monochrome picture images are obtained. Alternatively, transformation of colour data, by methods known in the art (see e.g. R.J. Clarke, Transform Coding of Images, Academic Press, 1985, from page 248) may be carried out, and the three original monochrome signals, corresponding to the RGB system, can be transformed into one (monochrome) luminance signal and two reduced bandwidth colour-information carrying signals (sometimes referred to collectively, hereinafter, as "colour information signals).

Therefore, hereinafter the expression "brightness function" (or "grey levels") should be understood as meaning the function defined by the array of the brightness values of any basic color or of the values of any of the luminance and/or colour information carrying signals.

-6- 2~919~
~0~71~241~/92 By decompressing the compressed and stored and/or transmitted data relative to the various cells, which contain all the chromatic information required, a "picture image", viz. an image which closely approximates the picture, can be reconstructed. Said data include a brightness value for each pixel and for each basic colour, or equivalent information deriving from the transformation hereinbefore mentioned, and this information permits any apparatus capable of creating an image, be it e.g. a computer which has stored the said information in its memory, or a printer, a still camera, a l'V camera, and so on, which receive the information from a computer, to create the picture image.
Such apparatus and their operation are well known to persons skilled in the art.

The information def~lning, in any chosen way, the brightness distribution of the various colors or signals, may be a function of time. This will occur e.g. whenever a motion or a television picture is compressed and reconstructed. In such a case, the method steps according to the invention should be carried out in a very short time, e.g. in the order of 30 frames per second.

The apparatus according to the invention comprises:

A - means for def~lning the brightness values of the basic colours, or equivalent information, preferably at each pixel of the original picture;

.

20Y7/2~ If92 2 0 9 ~ 9 2 0 B - means for registeri~g the said brightness Yalues or equivalent information, as scts of values associated with the pixels of a number of cells, of prcdetermined size, of each region of the picture;

C - means for determining the parameters of any one of set of basic models, in particular by minimizing the square deviation of the values of said basic model from the values of a brightness function at the pixels of the cell; and D - means for storing and/or transmitting information defining the types of basic models chosen and the said parameters thereof.

The means A- for determining the brightness values of the points of the cell may be different depending on the particular embodiment OI the invention. They may consist, e.g., merely in means for relaying to khe apparatus values which are de~ned by means that are not part of the apparatus, in particular by the apparatus which creates or transmits the original picture. Thus, if the invention is applied to the compression of a television movie, for the purpose of registering it on a video tape, the brightness values relative to each point of the television receiver screen are transmitted, as functions of time, from a broadcasting station via electromagnetic waves or via cable, and these same values can be relayed directly to the registering means B-. The brightness value determining means will then essentially be a part of the television -8- 2~91920 20Y7/2424J~1~92 receiver: said values will be registered in the apparatus accorcling of ths invention concurrently with their appearing as optical values on the receiver screen. In this case, one may say that the picture is being compressed in real time. A similar situation will prevail if the picture to be compressed is not being transmitted, but has been registered on a magnetic tape: the reading of the tape, that would be carried out in order to screen the registered picture in a normal way, will directly provide the brightness values. In other embodiments, the invention may be used to compress a picture that is already optically defined. Then the brightness value determining means will be normally constituted by a scanner.

The storing and/or transmitting means D- may be conventional in themselves, and may be constituted, e.g. by magnetic tapes, such as video tapes, by optical or magnetic disks, or by television broadcasting apparatus, and the like.

The stored and/or transmitted, compressed picture must be reconstructed by decompression from the compressed data, so that it may be viewed. Therefore, there must be additionally provided E - means for reconstructing the picture by producing at each point of each cell a color brightness, for each color, the value of which is defined by the value at said point of said basic model having said parameters.

2~97/2424/ED1~2 2~ 2~

In some embodiments, decompressing means E- are part of the apparatus according to the invention. Thus, if the invention is used to compress data for recording television pictures on video tapes, the apparatus will comprise means for actuating the television screen to screen pictures defined by the compressed data. This will generally occur when the reconstructed picture must be seen at the location at which it has been compressed.
However, if the point at which the picture is to be seen, is dif~erent from the one at which the apparatus comprising components A-to D- is located, means E- will not be physically a part of sad apparatus. In general, mean~ E- are functionally, but not necessarily or even usually structurally connected with means A-to D-. These latter, while usually connected with one another, need not necessarily be structurally sombined.

In a preferred aspect of the invention, the basic structures comprise background areas, edges, ridges, positive and negative hills and, optionally, saddles as hereinafter defined.

In a preferred aspect of the invention, the identification of the basic structures and their representation by models are carried out by the following steps:

I. constructing geomet~c models representing said structures;

-lO- 209~2~
~7l2424~g2 II. associating to each of said geometric models a mathematical model representing it;

III. condensing said models to deffne a global model for each cell of the region; and subsequently IV. encoding and quantizing the data defining said global model;
wherein steps II and III may be partly concurrent.

It should be understood that, since, as has been said before, the process of the invention is calTied out separately on each basic color or on each luminance or color information carrying signal3 the "objects" which the geometric models are intended to represent may not be, and generally are not, actual physical objects which the eye would discern in the picture, but represent characteristics of the distribution of the basic color brightness or the value of the color signal under consideration. Therefore, in principle, the objects and the corresponding models could be quite dif~erent ~or dif~erent basic colors or color information signals.
Thus, in principle, the arnount of compressed data required to represent a color picture would be three or four times as large as the amount required ~or each basic color or color in~ormation ~ignal. However, it has been surp~isingly discovered that it is pos~ible to identify in the models two different kinds of parameters, which will be called respectively "geometric parameter~" and "brightness parameter~", such that the geometric parameters are, in most practical cases, the same for - -~ 1- 2~92~
2~ 424/~92 all the basic colors or color information signals. Thus, in most cases it is enough to process the monocromatic luminance signal and add the color data later, which color data, when operating according to the invention, may require as little as an additional 3% to 10% appro~imately of data. This is an important feature of this invention, particularly in its preferred aspect.

It is therefore another preferred aspect of the invention a method of compressing color pictures, which comprise3 carrying out the compression method hereinbefore defmed with reference to one basic color or monochrome signal and successively repeating it for the remaining basic colors or color information carrying signals, by using the same m~dels with the same geometric parameters and determining the appropriate brightness parameters, as will be defiIled hereinafter.

In the process according to the invention, geometric andlor mathematical models are considered as "representing" picture elements or other geometric and/or mathematical models whenever they approximate these latter to a degree dete~mined by absolute or relative parameters or thresholds, the determination of which is part of the invention. The ab~olute thresholds have a ~ed value. The relative thresholds depend on the values of the quantity considered over a certain area, or over the entire region.
Usually, but not necessarily, the relative thresholds have the form of kM, where k is a coefficient and M is the average value of the quantity over the area that has been chosen for averaging.

-12- 209192~
~097~429~92 The various thresholds will be defined as they come into play during various stages of the process, for specific embodiments of the invention.

D~finitiQn Q~e ~ic el~ or ~trU~ture~

By "basic elements" is meant, in the broad definition of the invention, a number of 8iDlple structures such that in combination they approximate any actual structure or "object"
that can be found in the picture. In carrying out the invention, a list of such basic elements iB prepared for each application.
Usually the same list is adequate for all applications of the same nature, e.g. for all TV pictures.

As used in this specifilcation, the term "submodel" is to be construed as meaning: a) an array of grey levels or RGB values for a certain part of a picture (e.g. grey leYel z = ~ab(x,y), wherein is an expression depending on parameters a and b); or b) the geometry of certain objects on a picture (e.g. the form of a certain curve can be represented as Y = ~cd(x), such as e.g. y = cx~d~2).

The term "model" means an expression consisting of one or more submodels, and allowing for computing for any given ~,y a grey le~el z = ~(~,y).

2~97/~4~2 - 1 3 2 0 ~ 0 The parameters of submodels, representing the geometry of objects, are called "geometric parameters", and the parameters of submodels, representing grey levels, are called "brightness parameters". Some of the model~ explicitly contain submodels responsible for the position and the geometry of the described objects, as shown in the following example:

In the model z = ~(~,y), wherein ~ is equal to alx+bly+cl, if y2ax2+~x+X, and is equal to a2x+b2y+c2, if y<ax2+,~x~%, the geometric submod01 is the curve y = ocx2+,Bx+x and a"B,% are the geometric parameter~. z = alx~bly~cl and z = a2x+b2y~cz are two other submodels of this model.

Polynomial model~ or submodels are those given by polynomials of low degree (usually ~ 4), with coeffilcients assuming a limited number of values tusually ~ 2~6).

To "represent" a picture, or a part thereof, or a certain object that is in the picture, by a model, means to replace the original grey or RGB levels z = f~x,y) by the grey level model values z = ~(~9y).

In all the definition~ of picture obje~ts (not of models and submodels), hereinaf~er, reference is made to a given part of a picture of a size approximately equal to the scale L. Therefore said definitions are scale-dependent. A list of basic elements, particularly suitable for representing 1~ pictures, but also for other applications, will now be described. EIowever, persons 209~ ~2Q

2097124241~92 skilled in the art may modify it and add other basic elements, when dealing with other applications or with particular cases of the same application.

Smooth objects and smQot~ç ons - The word "smooth"~ is used and will always be used hereinafter to define those brightness distributions (brightness surfaces) that can be represented by a polynomial P(x,y) of a low degree, e.g. a degree generally not higher than 4, in a visually undistinguishable way. Thus smooth regions are those in which the brightness surface z = f(x,y) defining the distribution of a colour in the picture, can be so represented; and smooth objects are those any part of which inside any cell can be ~o represented by a polynomial model or submodel of a low degree. Analogously, a "smooth curve" is a curve in the neighbourhood of whose intersection with any cell the picture can be represented by a model, within which the curve is defined by a polynomial of low degree.

Simple models or submodels are those containing a small total number of parameters (usually ~ 6), each of these parameters assuming a limited number of values (usually ~ 2~6).

~QQ~j~ - An object, part of a picture, is said to be "simple", if ~or any cell the part of the object inside the cell can be represented in a visually undistinguishable way (as hereinbefore def;ned) by a simple model or subrnodel.

-1S- 209~920 20971~4241~192 (~ur~dlinear structures - Are those in which the brightness distribution can be represented by a surface z = f(~,y), generated by a simple (as the word is defined hereinbefore) pro~le, a point of which follows a simple curve, the parameters defining the profile being simple functions of the position of said point on said curve.
Curvilinear structures can be unbounded (in the cell under consideration), or bounded at one end, or at both ends to constitute a segment. They can also form nets, when several curvilinear structures ars joined at some points or portions, that will be called "crossings" .

Local simple elementa ~ An element is "local" if its diameter iB
comparable with L, at most 2 to 3 L. Local simple elements are those that are local and simple, as the latter word is defined hereinbefore.

In practice, the aforesaid types of brightness distribution6 are never found in their pure form, but types that are sufficiently similar to be treated as such are generally found.

The determination of the characteristic scale L is a fundamental step. If L is too small, the structure found in the cells - the "object"
- can easily be represented by hasic element models, but the amount of data that will be involved in the compression i8 too high for the compression to be successful. On the other hand, if L is too large, it is impossible to represent the objects in a visually undistinguishable way by means of basic element models.

, -16- 20~192~

Therefore the choice of L will depend on each specific application, and L will be chosen as the largest scale that permits to approximate the actual objects by means of basic element models in such a way as to achieve a visually undistinguishable representation, or at least as faithful a representation as desired for the specific application; and it will also depend, of course, on the quality and resolution of the original picture.

L is expressed in terms of pixels. For instance, when applying the inYention to the compressioll of teleYision images, it is found that L should be comprised between 10 and 16 pixels, e.g. about 12 pi~els. For most applications, L may be comprised between 6-8 and 48 pixels, but these values are not a limitation. A frequent value is L=16. It is appreciated that each cell, if square, contains L~L pixels, 80 that if L iB 16, the cell will contain 256 pixels. A
square cell having the dimensions LxL will be called the "standard cell". Since most hardware is designed to operate with ASCII symbols, such a size of cell or a smaller one i8 COnYeIlieIlt.
If a certain object i8 simple or smooth with respect to a given scale L, it is simple or smooth with respect to any smaller scale.

It has been found that, for representing television pictures, if the picture is divided into cells of 4x4 pixels, and in each cell the grey levels are appro~imated by second degree polynomials, an essentially visually undistinguishable picture representation is obtained. ~efore in a television picture any object is smooth and simple on a characteristic æcale of 4. Furthermore, for such an -17- 2~9~2~
2097~'M~2 application, the array of basic elements hereinbefore described i8 adequate and sufficient for picture representation on any scale between 8 and 16, preferably of 12.

In the preferred aspect of the invention, the basic structures are defined as follows:

background (also called sometimes "smooth';~ areas are those wherein the values of the brightness function may be considered to be changed slowly;

edges are curvilinear structures on one side of which the values of the brightness function undergo a sharp change;

ridges are curvilinear structures defined by a center line, the cross-sections of which in planes perpendicular to the center line are bell like curves, and they are positive or "white" (ridges proper) or negative or "black" (valleys), according to whether at the point of each cross-section located on the center line the brightness function value is a maximum or a minimum;

hills are points or srnall areaS at which the brightness ~unction value is a maximum (positive - white - hills or hills proper) or a minimum (negative - black - hills or hollows) and decrease6 or increases, respectively, in all directions from said point or small area.

/H192 - 1 8 - 2 ~ 9 1 ~ 2 ~
saddles are curvilinear structures which comprise a central smooth region bounded by two edges, wherein the brightness function values increase at one edge and decrease at the other.
They may be treated as basic structures, for convenience purpose, or they may be separated into their aforesaid three components:
two edges and an intermediate smooth area.

The said basic structure~ represent a particular case of the basic elements hereinbeIore defined. In particular, (white and black) ridges and edges are specific cases of curvilinear structur~s.
Hills and hollows are specific cases of local simple structures.

Preferably they are identified through the derivatives of the brightness function z = frx,y), x and y being the Cartesian (or other) coordinates of a coordinate system of the region considered, or through the derivatives of an approximating function, as will be explained hereina~ter.

13e~i~tis~of 1 he llh~awi~
In the drawings:

Figs. 1 to 50 - geometrically illustrate va~ious stages or particular cases of the processes described as emhodiments of the invention, and each of them will be described, in order that its content may be clearly understood, at the appropriate stage of the description of preferred embodiments;

-' 9- 2~ 2~
~OY7~429~
Fig. 51 is an original RGB still picture of the size approximately 480x720 pixels;

Fig. 52 represents the same picture after compression by the process according to the invention to 1/60 of the initial data volume and decompression;

Fig. 53 represents the same picture after compression to 1/40 by the standard JPEG method and decompression;

Figs. ~4, ~5, ~6 show the stages of analysis of a small area of a 48x48 pixels region of Fig. 51, shown at a greatly enlarged scale, wherein;

Fig. 56 represents the result of the edge-line analysis;

Fig. ~7 represents the original of the 48x48 pixels region, its representation by the global model before quantiæation and after quantization (small pictures from left to right). Big pictures represent the ~ame images in a 1:5 zoorned form;

Fig. 58 represents the result of a picture processing operation, performed on a compressed fo~n of the above picture;

Figs. ~9, ~0, 61 represent three color separations (R,G,B) of the picture of Figure ~2 compressed and decompressed according to the invention;

-20- 2~9192~
2097/2424~

Figures 62-67 represent the video sequences compression;

Figure 6~ represents 10 frames (as hereinafter defined) of a videosequence (on a 48x48 region), the upper line being the original sequence and the bottom line the sequence after compression (1:150);

Figures 63 and 64 ~how the results of analysis of the frames 4 and 7 (control frames) as still pictures;

Figure 65 shows (by yellow and blue components) the results of the motion predicted for the components from frame 4 to frame 7;

Figures 66 and 67 show the interpolated positions of the components (blue lines) against the real position (yellow lines) on the intermediate frames 5 and 6;

Fig. 68 is a picture to be represented by compression;

Fig. 69 is its representation obtained by decompressing data that had been compressed at a ratio of 1:35;

Fig. 70 is another picture to be represented lby compression; and Fig. 71 iB its representation obtained by decompressing data that had been compressed at a ratio of 1:50.

20Y71~L29~2 ~ o 9 ~

aile51 ~il~ti~n of Pre~e~l Eml~imenl~
In the process of the invention in its preferred aspect, the following steps are carried out:

Preferably, a grid is fixed in the region considered.

Preferably, each point of the grid considered is assigned to one of three domains indicated as Al, A2 and A3, according to the values of the derivatives of f(x,y) at that point and to the following criteria.

Domain Al contains all the points where all the said derivat*es of orders 1 and 2 are small and do not exceed a certain threshold Tl. A convenient way of expressing that condition, to which reference will be made hereinafter, is to say that the domain Al is the set of the points of the grid for which I Vf 12 < GabS~ ~l2 + ~22 < Sabs In the above formula, GabS and Sabs are thresholds, the values of which will be set forth hereinafter. ~1 and ~2 are the eigenvalues of the matrix $fi'dx2 V2~dxdy l/2~2~dxdy $~dy2 which will be designated hereinafter as W.

2097/~4241H/92 -2~ - 2 0 ~ 1 9 2 ~
Domain A2 includes the grid points where the gradient of khe brightness function f is large This condition can be expressed by saying the set A2 is defined by 1 Vf 1 2 2 max [Gabs. Grel]
where GabS is the same as the above and Grel is a relative threshold equal to Kgrad Ml wherein Ml is the mean value of .
I Vf 12 over the region considered. Typical values of the parameter Kgrad are between 0.2 and 0.~, and ~0 ~ GabS~ Sabs ~ 2~0.

Preferred values of these coefficients are Gabs = Sabs = 100, Kgrad =
0.3.

Domain A3 includes the points in which the second order derivatives are bigger than a threshold T3, while I grad(f) I does not exceed another threshold T2. T3 and T2 are relat*e thresholds, viz. they are defined as a certain percentage of the average value of the second order derivatives and of I grad(f) I respectively.

In a more preferred form of the invention, the domain A3 is actually the sum total of five sub-domains defined by the following conditions:
(1) 1 ~2/~l l < M ratio~ 2 <
(2) 1 ~2/~l 1 < M rat;o~ ~1 + ~2 >
(3) 1 ~2/~11 2 ~ ratio~ ~1 < , ~2 ~

(4) 1 ~2/~l l > M ratio~ ~1 > ~ ~2 >

(5) 1 ~2/~l l > M ratio~ 2 ~
assuming 1~1 12 1~2 -23- 2~920 20Y712~2~/Hf92 wherein Mratjo is another threshold (typical values whereof will be given later).

Preferably the brightness function is approximated by an approximating function p(x,y). When this is done, the function p and its derivatives and gradient should be substituted for the function f and its derivatives and gradient in all the formulae in these appear, e.g. as hereinbefore.

The approximating function is any convenient function, preferably involving a limited number of parameters, the values and derivatives of which are close enough to those of the brightness function in the area considered for it to be representative of the brightness function. In particular, it may conveniently be a polynomial of the second degree having the form p(X, y) = aoo + aloX + aoly + a2ox2 + allxy + ao2y2, though a dif~erent function, e.g. a polynomial of a higher degree or a function other than a polynomial, might be chosen in particular cases. The coefficients of p(x,y) are determined by minimizing the expression constituted by the sum of the square of the differences between the values of z and p at the grid points over the entire window considered, or, in other words, by minimizing the square deviation of the function p from the function æ over the area considered. More complex functions, including polynomials of degree 3 and higher, can also be used whenever the use is required for a better implementation of the process according to -2~- 20~1~20 2097/2424nV92 the invention, as well as more complicated measures of approximation than the square deviation.

In this manner, the approximating function and its derivatives will have a given value at each grid point. Preferably, the derivatives used in the following steps of the process of the invention are those up to the order 2. Within the degree of accuracy desired for the process, the derivatives and the gradient of the approximating function p(x,y) can be regarded as being the same as those of the brightness function z = f~x,y).

The matrix W becomes then a20 1/2a 1/2all ao2 In the aforesaid preferred embodiment of the invention, the same comprises therefore the following steps:

1) The values o~ a b~ightness funstio~ z = f(2~, y) alr(e registered preferably at each pixel of the picture or of a region thereof, if it is chosen to operate successively on a number of regions in order to contain the number of data to be handled at each time. The regions are preferably squares of side R (hereinafter "the region scale").

O l~he values of the der~vatives of the b~i~atn~s ~unc1ion z = ~
preferably in order up to 2, e.g; are appro2~imately compu~d by ~e following steps. A "window" and a grid related thereto are determined. The window is a square having a side "1" which is ordinarily of a few pixels, preferably between 3 and 6, and more commonly of 4. The grid is constituted by the pixels themselves, if "1" is an uneven number of pixels, and iB constituted by the central points between the pixels, viz. the points located halfway between adjacent pixels both in the x and in the y direction, if "1"
is an even number of pixels. These alternative definitions of the grid are adopted for reasons of symmetry.

3) For each g~id point (pi~el or central point between pi~els), 1~e brightness function z(x,y) is appro2~imated by the approximating function p, preferably a polynomial, as hereinhefore explained.

4) A number of thresholds is fi2~:ed. The various thresholds may be fixed or not at different stages of the process, or may vary in different parts of it for specific applications of the invention. But, for convenience of exposition the thresholds will be considered as being all fixed during each step of the process.

5) Each ~id point is as~gned to one of 1 bree dom~ins indicated as , A2 and A3, aecordi~g to the values of the del ivatives of the appro2~imating function at that point and to the criteria heneinbefore e2~plained.

-26- 2~91920 20Y7~42~
Domain Al contains all the points where all the derivatives of the approximating function p (from now on considered in place of the brightness function f) of orders 1 and 2 are small and do not exceed a certain threshold Tl. A convenient way of expressing that condition, as noted, is to say that the domain Al is the set of the points of the glid for which I Vf l 2 < GabS~ ~12 + ~22 < Sabs-In the above formula, GabS ~nd Sab~ are the aforementioned thresholds, the values of which are related to the value of 1. ~1 and ~2 are the eigenvalues of the rnatrix W

a~o l/2a~

l/2all ao2 Domain A2 includes the grid points where the gradient of the averaging function p is large This condition can be expressed, as noted, by saying the set A2 is defined by I Vf 1 2 2 max [Gabs~ Grel]
where GabS is the same as the above and Grel is a relative threshold equal to Kgrad M1 wherein M1 is the mean value of I Vfl 2 over the entire region or its parts. Typical and preferred values of the parameters or thresholds will be given hereinafter.

Domain A~ includes the points in which the seoond order derivatives are bigger than a threshold T3, while I grad(p) I does not exceed another threshold T2. T3 and T2 are relative thresholds, -27- 209192~
2~712424~V92 viz. they are defined as a certain percentage of the average value of the second order derivatives and of I grad(p) I respectively.

In a preferred form of the invention, as has been said, the domain A~ is actually the sum total of five sub-domains defined by the following conditions:

(1) 1 ~2/~l 1 < M ratio~ 2 <
l 1 < M ratio~ ~1 + ~2 >
(3) 1 ~ l l > M ratio, ~1 < , ~2 <
(4) 1~ l1 >Mratio, ~1>, ~2>
(5) 1~ l >Mratio~ 2<
assuming 1~ 2 l wherein Mratio is another threshold (typical values whereof will be given later ).

6) The basic sl~ructure~ are identif;ed in the domains. Preferably, these structures include, as noted, smooth regions, edges, ridges, and hills, and optionally saddles, wherein ridges and hills may be positive or negative, or black or white, the negative (black) ridges being "valleys" and the negative (hlack) hills being depressions or "hollows", as they will be sometimes called hereinafter. Other elements may be associated to the said ones, and in some cases, some of them may be omitted.

- : . --28 - 2 ~ 2 ~
20~7/242~/E~2 The background (or smooth) regions are those defined by the domain ~1, and in them the brightness function and the approximating function have only moderate variations.

The edges essentially indicate the sharp passage from two different values of the brightness and they may constitute the passage from one smooth region to another~ characterized by different average values of the brightness function. They are defined in the domain ~2. They are the first example of a curvilinear structure.

(Positive and negative) ~idges and hills are defined in various sub-domains into which the domain A3 is preferably divided.

Ridges are curvilinear structures defined by a center line, the cross-sections or profiles of which in planes perpendicular to the center line are bell-like curves that can generally be approximated, in the simplest manner, by parabola.
Mathematically, the center lines of the regions are determined by condition that the f~lrst derivative of the brightness function vanishes in the direction of the bigger eigenvalue of the second differential of the brightness function.

Hills are structures in which the brightness ~unction has a maximum or a minimum and respectively decreases or increases in all directions from said maximum or minimum poin$. They may in general be approximated in the simplest manner by 2~9~2~

2~g7/24~9JlY92 paraboloid. Mathematically, the central points of the hills and the hollows are determined by the condition of the vanishing of the first differential of the brightness function.

The detailed identif~lcation of the structures, particularly in the domain A~, will be described hereinafter for a preferred embodiment of the invention.

7) The cllrvilinear ba~ic ~tructures are appro~imated by lines related to their center lines and parameters related to their profiles. This step can be considered as the construction of geometric models.

In this connection, it is important to note that although the basic structures are represented geometrically, essentially by surfaces, those surfaces define, with some approximation, a physical quantity, viz. a brightness or a colour information signal, as a function of geometric variables, viz. two coordinates in the plane of the picture to be compressed. Those structures l;herefore describe in geometric terms a brightness or signal distribution over the picture surface.

Sometimes the profiles of the curvilinear structures must be defined only in the immediate vicinity of the center line, as will be better explained hereinafter: in fact it is essentially sufficient to define the curvature of the structure at the center line in a plane perpendicular to the center line.

20Y71~2~92 8) Preferably, superiluous structur~s and/or por~ions thereof are eliminated. Structures and their portions are considered "superfluous" if their elimination does not substantially affect the quality of the compression, viz. does not cause the reproduced picture obtained by decompressing the compressed data to be unacceptably different from the original. This will be generally true if they are superimposed or almost superimposed to other structures or portions thereof or do not differ from them to a degree defined by appropriate thresholds fixed as provided for in Step 4. Specif1c criteria will be æet forth in describing an embodiment of the invention, but in general can be determined by skilled persons for each type OI application, taking into account the quality required for the picture's reproduction. Omission of this step or reduction thereof to a minimum will not damage the quality of the picture reproduction? but may affect the degree of compression and thus render the process less economical. This step, further, may be partially concurrent with Step 7 and will be completed in Step 9, hereinafter.

9) The basic ~truct~res are ~epreæentedby mathematical models.
This step consists in the approximate representation of the basic structures by mathematical models according to two criteria: a) that the approximation is sufficiently close; b) that the number of data required to define the models be as small as possible. It will be understood that each basic structure could be represented in an exact or almost exact manner by mathernatical expressions, but _31_ 2~91920 2097/~429~g2 these would in general require an excessive number of parametsrs for their definition, so that the degree of compression would be relatively low without any significant gain as to the accuracy of the reproduction. Conversely, oversimplified models could be chosen, whereby a very high compression would be obtained but the quality of the reproduction would not be satisfactory. A compromise between the two exigencies of a high compression and a good reproduction quality must be achieved by an appropriate choice of the mathematical models. Examples of such models will be given hereinafter, but it should be understood that it is within the skill of the expert person to construct the appropriate ones for each basic structure and each particular application of the invention, by approximation methods as described herein.

10) The moslels ~ nsh ucted are inte~relabed, preferably by 1~he omission of part of 1~hem, ~o consh~uct a ~lobal model ~or ea~h ~ell of the region. The model parts that are omitted are the ones that overlap or that do not contribute anything significant to the quality of the picture reproduction. The criteria to be followed are essentially the same as in Step 8. A number of standard models will be defined for each type of application of the invention, and in any case, to each type of model will be associated a code identifying it, so that each specific model will be identified by its code and a set of parameters specific to it. At this point the compression process has reached the stage of primary compression, the primary -32- 209~ ~20 2~ /H/92 compressed data obtained consisting of codes identifying the type of each model and the parameters of each model.

11) Preferably, 'Yilte~in~' is carried out. This stage consists of the elimination of information that is considered excessive because it has only a little psycho-visual significance. The simplification involves dropping some models completely and eliminating some excessive parameters of the models that are retained. The evaluation of the parameters to be dropped is carried out on the global model, since parameters that are significant on the local basis, viz. if we consider each specific model, lose significance in the global model. In part this operation has been carried out in the construction of the global model, during which parts of the local models are dropped. However, it is verified and completed in this stage.

It should be understood that the steps so far described need not necessarily be completely separate from one another and need not necessarily take place in the order in which they are listed, but variations may be made in said order, and further, one or more steps may be carried out in separate stages and/or in part or in the whole concurrently with another step or steps.

12. Quantization. This step consists in approximating the parameter values of the primary compressed data on the basis of a predetermined set of values, which are limited in number. Each parameter is substituted by the closest value of this set. The 2097/~42~J92 2 ~ 9 1 ~ ~ 0 criteria for quantization depends on several considerations, such as the degree of compression that is desired, tlle degree of accuracy or resolution of the compressed picture that is required, the geometric and visual interpretation of the models, and so forth. Some examples of quantization will be given for a specific embodiment of the invention.

13) Encoding. The quantized data are represented in the form of a binary file. This operation is lossless, since when the binary file is decoded, the quantized values of -the parameters that have been used to form the binary file are entirely recovered. Certain criteria to be followed in this operation will be described in describing an embodiment of the invention. Huffman-type coding can be used in some cases.

14) Decompression. This consists in reproducing from the binary file the quantized values of the parameters and those values are substituted into a global model representing the brightness function for each region, which will direct a computer to attrihute to each point of the compressed picture its appropriate brightness levels.

All the above operations can be repeated severally for the various basic colors, thus obtaining monochrome picture compressions which are combined. Alternatively, transform coding of color data, by methods known in the art, may be carried out and the original monochrome signals corresponding to the RGB system -3~ 9~92~
20~7/242~If~2 can be completed by transform coding into one monochrome luminance signal and two re~uced bandwidth color-information carrying signals.

In a preferred embodiment of the invention, the process steps are carried out as follows.

Sbep 1 -The values of the brightness function z(x,y) are registered.

step 2 -The side "1" of the window is assumed to be 4 pixels. Consequently the grid is constituted by the central points between the pixels.
However, it may be preferable to use other or additional values of "1", as will be explained hereinaPter.

step 3-The brightness function z = f(x,y) is approximated by the approximating function p(~,y) which, in this embodiment, is a polynomial of the second degree having the form p(x, y) = aoo +
alox + aoly + a2ox2 + allxy + ao2y2. The coefficients of the function p are determined by minimizing the square deviation of said function from the function z. The values OI the derivatives of z are approximated by the coefficients of p(x,y). All the computations are performed wit~ the machine accuracy of the hardware on which the process is implemented. The accuracy of twelve bits and less is sufficient.

20Y~1242~192 ~35~ 2 St~p 4 The following absolute and relative ~hreshold values are preferably used in the embodiment described herein. For each one, three numbers are given. The first two numbers give the bounds, between which the described parameter usually is to be fixed. The third number in parentheses gives the preferred value of the parameter.

Gabs~ Sabs (grey levels): 50-250 (100);
Kgrad: 0.1-0.5 (0.2);
Mratio 0.2- 0.5 (0.3);
Kj: 0.5,1.2 (0.8);
DCenter 0.5~1 pixel (0.7 pixel);
TSlope (grey values): 20-40 (35);
DSegment 2-4 pixels (3 pixels);
D1Segm 1-3 pixels (2 pixels);
D2Segrn 0.3-0.8 pixel (0.5 pixel);
TSegm 20-40 (30);
BSegm (grey levels): 10-30 (20);
K'j=1.2Kj;
C1= 0.8;
DComp 0.3-1 pixel (0.5 pixel);
r: 1-~ pixels (3 pixels);
Dgcenter = Dcenter;
Kcurv = 1-3 (2);
Wgrad = 2-6 (3.5);

,, -3 6 - 2 ~ 9 1 ~ 2 0 Mrjdge (grey levels): 20-60 (30);
Dlgrad: :1-5 pixels (3 pixels);
D2grad: û.2-lpixel (0.5 pixel);
~grad: 6-10 pixels (8 pixels);
~grad: 20%-~0% (30%);
o: 0.05-0.2 (0.1);
K: 0.0~-0.2 (0.1);
d: 2-4 pixels (3 pixels);
F1, F2, F3, F4 (grey levels): 5-20 (10);
Q1, Q2, Q3, Q4, ~5, Q6: 0.3-1.2 (0.6);
Sr (grey levels): 10-30 (20):
T8: 0.3-1 pixel (0.5 pixel);
Tg: 0.1-0.3 (0.2);
T1o (grey values): 5-20 (10).

Step 5-At each point of the grid, gr;d constructed as described hereinbefore, the quadratic form g(x,y) = a2ox2 + a11xy + ao2y2~
which constitutes the second degree part of the approximating function p, is reduced to the principal axes by means of an orthogonal transformation P. P is represented by 132l 1~22 In the new coordinates x', y', the polynomial p(x,y) takes the form p(x',y') = a'oo + a'1ox'+a'o1y' + ~1x~2 + ~2y~2. ~1 and ~2 are the eigenvalues of the matrix W hereinbefore defined, and ~
~21), ~2 = (~12, ~22) are the eigenvectors of the same matrix W

20g712~2~1JH/92 ~ 2 ~ 9 1 9~ ~
defined by: W~ = ~,B or of the second differential d2p, and a'oo =
aoo.

The points are assigned to the different domains as follows. The domains Al, A2, and A3 are defined in the following manner.

Given p(x,y) - ~ aij xiyj (for i+j~2), the matrix P, the coefEicients a ij, ~ 2 and ~t and ,B2 can be computed by various well known computation procedures. For ~ approximately equal to ~2 the computation of P and ,B may become insufficiently accurate. In this case the matrix P is not computed and is merely defined to be equal to I (identity matrix).

Denote by Vf the vector (alo, aol) of the gradient of p(x,y) at each point of the grid I Vp I = \ ¦ alo2 + aol2 = ~ a~lo2 + a ol2 since the transformation P is orthogonal. Denote by Ml = the mean value of the I Vf 1 2 over all the grid points in the region.

The set Al is defined as the set of those points of the grid, for which: I Vf l 2 < Gabs~ ~12 ~ ~22 < ~abs The thresholds GabS~ Sabs usually take values between 50 and 250.
In this example we assume GabS = 100, Sabs = 100.

2~Y712~24J~192 -38- 2~ 20 The set A2 is defined as the set of those points of the grid for which:
I Vfl 2 > max [GabS~ Grel]~ where Gabs is as above, and Grel =
Kgrad-Ml. Typical values of the parameter Kgrad are between 0.1 and 0.5. In this example we assume Kgrad = 0.2.

The thresholds GabS~ Sabs and Kgrad above, as well as Kj, j=1, .......
5, Mratjo which will be defined below, are the parameters of the detection of the basic structures, which must be fixed for each region of the picture. Their usual and preferred values, for the present application, will be indicated.

The set A1 will serve as the basis for constructing the smooth regions and the background. The set A2 will be used for construction of the edges.

The set A3 is then constructed, wherein the (white and black) ridges, (white and black) hills and, in this embodiment, saddles will be identif;ed. At the ridge points one eigenvalue is big, and the second small and the derivative in the direction of the biggest eigenvalue is zero. At hills and hollows, dp = 0, and both eigenvalues are big and have the same sign. At the saddle points, dp - O and both eigenvalues are big and have opposite signs.The values of dp and of the eigenvalues identify the said basic structures as far as the invention is concerned.

-39- 2~g~920 20Y7/24~92 The eigenvalues are always ordered in such a way that l ~1 1 >
1~2l The conditions that divide the domain A3 into five subdomains A3;, j being 1 to 5, have already been set forth. The points of the various subdomains form the basic structures of this domain as follows:
(1) l ~2/~l l < M ratio, ~1 + ~2 < O - white ridges -(2) l ~2/~l l < M ratio, ~ 2 > O - black ridges -(3) l ~1 l > M ratio~ ~1 < O, a~,2 < O - white hills -(4) 1 ~2t~l 1 > M ratio, ~1 > ~ ~2 ~ O - black hills or "hollows" -(5) l ~2/~1 1 2 M ratio~ 2 < 0 - saddle points -The threshold Mratio usually takes value~ between 0.2 and 0.5 andtypically 0.3.

Denote as M2j = the mean value of ~,l2 + ~,22 over the grid points in the region where ~1 and ~2 satisfy one of the above five conditions, viz. the condition (j) wherein j = 1, ..,.,5.

Let A3j be the set of gridpoints in the region where ~1 and ~2 satisfy the condition ~j) and I Vfl 2 < max [Gabs~ Grel], ~l2 ~ ~22 >
max [Sabs, Sjre~]. Here the thresholds Ga~,S, Grel, Sabs are as def~ned above, Sjrel = Kj M2j. The typical values of the parameters Kj, j = 1, ..., 5, are between 0.5 and 1.2, or typically K
=0.8.

': '~

, 2097/2429J~/g2 An easy alteration (instead of ~ 2, ~ 2 ~ ~2/~1 is used) makes all the quantities computed till now homogeneous and symmetrical expressions in ~1 and ~2. Hence they can be expressed in terms of the initial coef~lcient a;j, without performing the reduction to the principal axes. Thus, the computations can be arranged in such a way that the reduction will be performed only at the points of the (usually small) set A3.

The identification of the ridges, hills and saddles will now be described exemplified for an embodiment of the invention.

In a new coordinates system (x',y') the polynomial p(x',y') is given by p(x',y') = a'oo + a'lo x' + a'oly' + 3llx'2 + ~2y~2.

On the sets A31 and A32 we perform an additional transformation x" = x' + a~lo/2~1~ Y = Y-We get:
p(x~y~)=a~oo+a~oly~+~lx~l2~l2yll2 (Note that by the definition of A~ 1 1 is big).

We say that the considered gridpoint is part of a white (black) ridge if it belongs to A3l (A32, respectively) and I a~1oi2~1 1 <
Dcenter, I a 01 1 ~ MSlope- Here the thresholds Dcenter and MSlope usually take the values 0.5 pixel < Dcenter < 1 pixel, and 20 < TSlope 40. In this embodiment we assume Dcent~r = 0.7 pixel, Tslope =
3~.

20~92~

~0~ /E~32 We associate tn such a gridpoint a white (black) segment (of unit length), centered at the point x" = 0, y" = 0 (in the new coordinates x", y") and directed along the y" axis in this new coordinate system. We also associate to this gridpoint the values ~1 and a"oo.
The segments thus constructed approximate the ridge lines, since at them the derivative of f and of its approximating function p in the direction of the biggest eigenvalue is zero.

The condition I a'1o/2~1 1 < DCenter is caused by the fact that p approximates f in a reliable way only near the center of the square having the size "1" and therefore, the ridge is detected in a reliable way only if it is close enough to the center of the square, and I a'1o/2~ 1 is the distance of the ridge from said center.

On the gridpoints, belonging to the sets A~3, ~4, A3s, we perform an additional transformation: -x"' = x", y"' ~ y" ~ a'ol/2~2 (On these sets, both ~1 and ~2 are big).
We get: p(x"',y"') = a~oo + ~ )2 ~ ~2 (y )2-We say that the considered gridpoint represents a positive (white) hill, a negative (black) hill (hollow) or a saddle point, if it belongs to A33, A34 or A3s, respectively, and I a'o1 /2~11 < Dcenter ..
. ~:

-~2- 20919~0 20g7/24241~K

I a'ol/2~2 1 < Dcenter Here Dcenter iS the same parameter as above. We associate to such a point the center of the new coordinate system x"', y"', the value a"'oo and the values ~ 2, and the direction of the eigenvectors ~l"B2 as defined above.

The center of the new coordinate system is the point where dp = 0.
Once more, this center should be close enough to the center of the cell for the detection to be reliable.

To each positive or negative hill or saddle point, we also associate a unit segment, centered at the center of the new coordinate system x"' y"', and directed along the y"' axis.

All the segments obtained are separated into two sets: "white"
segments, for which ~1 < 0, and "black" ones, for which ~1 > 0.

The segments constructed by now correspond to the positive and negative ridges, the positive and negative hills and to the saddle points. Although by definition only ridges represent curvilinear structures, both hills and saddle points, if prolonged can visually represent a part of a curvilinear structure and be processed as such. Since curvilinear structures form the most coherent and compact elements in picture description, it is desirable to include as many details as possible in simple curvilinear structures, instead of representing them as separated local elements.

43 2~91920 20Y7/24~41~g2 Therefore, on this stage we consider all the segments obtained as potential representation of the "ridge lines" and attempt to construct from them maximally extended geometrically simple compounds.

step 6 ~
Structure components are constructed in this step. Each component consists of an orderly array of closely related segments of the same "color". Herein, we consider white segments. The construction of the black ones is completely similar.

Each segment constructed is characterized by its central point (which was defined in local coordinates for each 1 x 1 square; this point is not necessarily one of the gridpoints), and by its direction.
Also the approximate grey level value a"oo at this central point and the biggest eisenvalue ~1 are associated to each se~nent. We now carry out the following steps.

1. Regardless of the origin of each segment in a certain 1 x 1 -square, we represent now all the segments by the coordinates of their central points in the global coordinate system, viz. the coordinate system originally associated with the RxR region on which we operate, and by the angles with respect to the global coordinate system. The sel; of s0~nents is ordered, iEor example, by the natural ordering of 1 x 1 - squares of their o~igin.

-44- 20~ 32 ~097/~ 92 2. We consider the first segment sl and associate to it all the "neighbouring" segments sj,viz. those segmentswhose centers are closer to the center of sl than Dsegm~ The threshold Dgegm usually takes values between 2 and 4 pixels. Here we assume DSegm = 3 pixels.

For each segment among the neighboring segments sj, we check the following conditions (see explanatory Fig. 1):

i. The distance dl between the center of sl and the projection of the center of sj onto the sl, is less than Dlsegm ii. The distance d2 between the center of sj to its projection onto s is less than D2segm.

iii. The angle between the directions of sl and sj is less than Tsegm -iiii. The difference between the grey levels aoo associated to sl andsj does not exceed (in absolute value) Bsegm-3. The thresholds DlSegm~ D2segm~ Tsegm~ Bsegm~ haveappro~imately the following usual and preferable values:

2097/2~2~92 ~45 ~ 2 0 ~ 1 9 2 ~

Dlsegm = 1-3 pix., preferably 2, D2~egm = û.3 - 0.8, preferably 0.5 pix., TSegm = 30, BSegm = 10-30, preferably 20 grey levels.

4. Those among the sj, which satisfy all the conditions i, ii, iii, iiii, are included into the component constructed.

On s1 one of two possible directions is fixed. Then the included segments are ordered according to the order of the projections of their centers on s1, and on each of them one of two possible directions is chosen in a way coherent with sl.

5. We take the first and the last segments with respect to the order introduced, and for each of them we repeat the operations 1-4, adding segments which are still free (i.e., do not belong to the component under construction or to other components).

6. We order the segments, added to the first one, and the segments already in the component, according to the order of the projection of their centers on the first component. The same for the last component. In this way we order all the segments of the component under consideration.

7. We take the first and the last segment and repeat the operations 1-5.

~0~7~24~2 2 ~ 2 ~
8. We repeat the aforesaid operations until no free segments can be added to the component. This completes construction of the component, which is represented by an ordered array of segments of the same colour.

9. We take the first among the remaining free segments and repeat operations 1-8, constructing the next component.

As it was explained above, we try to construct a small number of long components instead of the big number of short ones. As a result, it is desirable to have more segments available. Therefore the values of the thresholds Kj and D~ent, given above, are rather "liberal". As a result, some components can be constructed which represent visually insignificant structure. To remove these components, the following procedure is applied:

1. We construct an array of "strong segments", repeating all the operations described, but with the thresholds K'J greater than Kj.
By construction, each strong segment is also a regular segment, and thus the array of strong segments is a subarray of all the regular segments.

2. We construct the components from the regular segments, but then we drop each component which does not contain at least one strong segment.

The preferable values of K'J are about 1.2 times bigger than Kj.

20Y7/2424~ 2 0 ~ 1 9 2 0 Components, constructed above, represent the ridges detected.
The hills and hollows, found as described above, represent the corresponding simple local structures. In principle hills and hollows are not curvilinear structures and are detected separately from the ridges, as described above. However, by construction, some part of hills and hollows can appear exactly on the components constructed. If their width is approximately the same as the width of the corresponding ridge, they can be omitted.

Thus, the hill (hollow) is omitted if the corresponding segment belongs to some component, and the bigger eigenvalue ~1 at this point satisfy 1~1 1 > C1~, where A is the mean value of ~1 over the segments of the component, and Cl is a threshold. Usually C1 =
0.8.

This operation, however, essentially belongs to Step 11 (filtering).

St~p7 -The operations described in this stage are essentially approximation operations. It is to he understood that they need not necessarily be performed at this stage, but may in part or on the whole be performed at an earlier or at a later stage.

1. The components that have been constructed are subdivided into the parts which satisfy the following condition: the projection of all the segments of the component onto the interval, joining the -48- 2~ 20 central points of the first and the last segments, should be one-to~
one and in the order deflned above. ~7Ve check whether the condition is satisf~led starting from the second segment and continue with the following segments until we find one which does not satisfy the condition. The beginning of this segment is the point which mark the passage from the conmponent part considered to another one. The process is then repeated starting from said point. Fig. 2 shows an example of the construction of a component part.

2. Now we approximate components, up to required accuracy, by polygonal lines. This is done as follows, with reference to Fig. 3:

a) We construct the interval, joining the centers A and B of the first and the last segments of the component part constructed as under 1.

b) We find the segment on the component part for which the distance d of its center C to AB is maximal. If d < DComp~ the procedure is completed, and the interval AB provides the required approximation. If d ~ DComp~ we consider the intervals AC and (:~B and repeat the procedure with each part AC and CB of the component separated.

c) This subdivision process is continued until the component is approximated by a polygonal line, joining the centers of some of its segments, with the following property (see Fig. 4): for each -49~ 20~192Q
segment of the component, the distance of its center to the corresponding interval of the polygonal line, is less than DComp. In this embodiment, the threshold DComp is usually equal to approximately 0.3 to 1 pixel, preferably 0.5.

By now, each component is approximated by the polygonal line, joining the centers of some of its segments. This polygonal line is a geometric submodel, representing the central line of the ridge.

In order to def~me the profile, the following two numbers are associated with each interval Ii of the approximating polygonal line~ Ii, which is the mean value of ~1 over all the segments between those whose centers the interval Ii joins, including the end segments (remember that a~l the segments of each CDIPponent are ordered); ai = aIi~ which is the mean value of the brightness a"oo over the same segments as above.

Finally, each ridge componen~ in A~ is represented by:
- the ordered array of the vertices Pi of the polygonal line (i.e., the endpoints of the intervals of this line). These points are given hy their global coordinates in the region.
- the ordered arrays of the numbers ~i and ai, defined as above.

The said numbers define the profile of the ridge, specifically its level coordinates (brightness values = ai) and its curvature in the vicinity of $he center line (~i). Theoretically, the curvature is defined in an infinitesimal interval about the point of the center 2097/24241~I/92 ~50~ 2 0 9 1 ~ 2 0 line considered and perpendicularly thereto, but here we deal with finite, though small, intervals and therefore the curvature defines the profile in an interval in the order of the distance between adjacent pixels, which, for the purposes of these operations, is the smallest significant interval.

The identification and approximation of edges will now be described. As stated, the set A2 has been defined by I Vp 12 ~ max [GabS~ Grel], and thus consist of the gridpoints, where the gradient Vp is "big".

However, the set A2 is spread around the regions of a high slope of z = f(x,y), and does not by itself represent the central lines of the edges. Therefore, we perform first a certain filtering of the points in A~, trying to screen out the points far away from the edge's central lines.

Consider a point in A2 and let p(x,y~ = aoo ~ alox + aoly + a2oX2 +
allxy + ao2y2 be the approximating polynomial computed at this point. Vf by definition is the vector (alo, aol). We consider the s0t of values which p assumes on the line Q, parallel to Vf passing through said point, and f;nd the verte~ of the parabola, del'ined by said set of values. If the distance of this vertex from the said point is smaller than DgCenter, the gridpoint is excluded from the set A2, and we pass to the next gridpoint in A2. Here, the threshold Dgcenter iS usually equal to DCenter defined above. Thus, we exclude from consideration the points which are too close to the -S 1 - 2 ~ ~ ~ 9 2 0 sides of the edges. In this way, DgCenter determines the minimal width of the edges which can be recognized.

For each of the remaining points we compute the number 2 1 a2o(aol)2 a11 alo aol ~ ao2(a10)2 1 K =
((ao1)2 + (alo)2 )3/2 Mathematically, lc is the curvature of the level curve of p(x,y) at (O,O), viz. the curve de~med by p(x,y) = p(O,O). It approximates the curvature of the center line of the edge at this point. We drop the points where K > KCUI'V
where KCurv = 1-3 or typically 2.

Thus, we omit the highly curved edges which usually appear around small local patches, which we have earlier identified as hills or hollows. The remaining gridpoints form a smaller set A2', at which the m~re detailed analysis is performed.

At each gridpoint of A2' we construct a polynomial q(~,y) of degree 3, providing the minimum of the square deviation ~[f(x,y) -q(x~y)]2 over all the pixels of the l'xl'-cell, centered at this gridpoint. The parameter l' is usually slightly bigger than l. In this embodiment l' = 6.

209192~

~097~24~2 This polynomial is restricted to the straight line Q passing through the origin (the gridpoint) in the direction of Vf = (alo~ ~ol), viz. the values which the polynomial assumes on the line Q are considered as a function of the distance of their projections on the line Q from the origin (variable t). We get a polynomial q of degree three of the variable t. These operations are illustrated in Fig. 5.
We find the points of extrema (maximum and minimum points) of q(t), tl and t2, by solving the quadratic equation q'(t) = O, and the values al = q(tl), a2 = q(t2)-The following condition i8 checked: tl and t2 exist, tl < O, t2 ~ ,and I tl I and I t2 I do not e~ceed Wgrad. Eere the threshold Wgrad determines essentially the maximal width of the edges that can be detected in a reliable way. Usually, Wg,ad is approximately equal to l/2 1'. In this embodiment Wgrad = 2-6, preferably 3.5.
(Alternatively, one can check the condition that both the points t and t2 on the line Q belong to the l'xl'-cell considered.) If the condition set forth above is not satisfied, the gridpoint is e~cluded from the set A2', and we pass to the next gridpoint.

If the condition is satisf~led, we associate to the grid point considered the following object (called an "edge element"):
1. The unit segment Ie centered at the point C in Fig. 6, which is the central point of the interval [tl, t2~ on the line Q, and directed in an orthogonal direction to Vf (i.e., in the direction of the edge).

~53~ 20~19~0 2. The segment [tl, t2] of the line Q (by construction, this segment is centered at the same point C, and is directed along Vf~. We also associate to each edge element two values al and a2. a1 = q(tl) and a2 = q(t2) step 8 ~
The following operations are designed to detect geometric connections between the ridges, found before, and the edges under construction, to eliminate super~luous data.

Consider the example, illustrated in Fig. 7, a and b. Therein the broken lines denote the ridge components, detected earlier, and the marks Y denote the segment Ie, detected as described hereinbefore and shown in Fig. 6. The lines 1l and 12 in Fig. 7b usually appear on sides of edges, because of the typical profile of the edge, illustrated in Fig. 8. They can be omitted from the data, since the required profile is detected as described in Step 7, and can be reconstructed from the data thus obtained.

In the example of Fig. 7c and d, the edges E1 and E2 are detected on both sides of the ridge 12. The lines 1l and 13 have the same nature as the lines in the first example. Here the lines 11 and 13 and the edges E1 and E2 can be omitted.

Ridges and edges that are particularly closely related will be said to have an "adherency" relationship. To detect such relationships, we proceed as follows:

5q 209~
2097~4~2 1. The ridge components, detected earlier, are stored in a certain order. We start with the first component. It is represented by an ordered array of segments.

2 For each segment in this array, those edge elements are considered whose centers are closer to the center of the segment than Dlgrad. Dlgrad = 1-5, preferably 3 pixels. This procedure is illustrated in Fig. 9.

3. Then, the following condition is checked (see Fig. 9): if the segment is a white (black) one, the distance p is computed between its center and those from the points tl and t2 of the edge element which is the maximum (minimum) of the corresponding profile.
If p 2 D2g,ad, we pass to the next from the edge elements above. If p < D2grad, the edge element iB marked as "adherent" to the segment considered, and the segrnent is marked as "adherent" to edge elements ~rom the side, where the center of the edge element lies. Here I~2g~ad = 0.~-1, preferably 0.~ (It should be remembered that the segments are oriented.). Then we pass to the ne~t edge element and repeat the procedure. (If the segment is already marked, this mark is not changed.) When all thc edge elements, chosen as described in 1 are checked, we pass to the next segment of the component.

4. After all the segInents have been processed, the following data are obtained:

-55- 20~920 209'7/~4~

a. 33ach segment of the component considered is marked as "adherent" or "unadherent" on each of its sides.

b. Some of the edge elements are marked as "adherent" at one of the points t1 or t2.

5. Now all the segments of the component under consideration are proc0ssed once more in the following way:

Starting from the first seg~ent (in their standard order), the segments are checked until the first segment marked as adherent from the left appears (simultaneously, exactly the same procedure is performed on the right side). The number of this segment is memorized. If the distance of its center from the center of the first segment in the component is greater than the ~grad, the center of this segment is marked as the partition point. ~grad is a threshold, the value of which is usually between 6 and 10 pixels.
In this embodiment ~grad = 8.

Then the next segment marked as adherent from the left is found.
If the distance of its center iErom the c0nter of the previous segment~ marked from the left, is greater than ~grad, both its center and the center of the previous segment are marked as the partition points. If not, the next segrnent marked from the left is found and the procedure is repeated.

2 ~ 0 -~6-209712~241~92 Exactly the same procedure is simultaneously performed on the right side of the component.

6. As a result, the component is divided into subcomponents by the partition points found as described. For each subcomponent on each side, by construction the adherent segments are either uni~ormly denser than ~grad, or uniformly sparser than ~grad.
See Fig. 10.

7. For each subcomponent the proportion of the marked segments to all the segrnents is computed on each side.

Now several cases are distinguished:

a) If the marked segments are ~grad - dense on both sides, the subcomponent is preserved, and the marking of the edge elements, adherent to its segments, is preserved.

b) If the marked segments are ~grad - dense on one side, and on the second side the proportion of the marked segments is less than grad when grad is a parameter of values between 20% and 50%, preferably 30%, the subcomponent is omitted, and the marking of the edge elements adherent to it is canceled.
If the proportion is larger than or equal to grad, we proceed as in a).

' 57 2~9~.92~
20~7/24~
c) If the marked segrnents are Agrad - sparse on both sides of the subcomponent, we proceed as in a).

8. For each subcomponent which has not been omitted, the following numbers are computed:

leftvalue, equal to the mean value of al or a2 on all the edge elements adherent from the left (from al or a2 take the value corresponding to the free end OI the edge element).

left~idth - the mean value of the length of the intervals rtl, t2] for the edge elements, adherent from the left.

and similarly for the right hand side of the subcomponent.

9. Now we pass to the next ridge component and repeat all the operations 1-8, and 80 on, until all the components have been processed.

As a result each component has been subdivided into subcomponents, a part of these subcoInponents has been omitted, and for the rest the data as in 8 has been computed. All the edge elements are still preserved and some of their extreme points tl, t2 are marked as adherent.

At thi~ stage all the edge elements are omitted which are marked at at least one of their extreme points. The surviving "free" edge 2~71~424~
elements are used now to construct "edge components".
Remember that each edge element consists of the interval [t1, t2]
and the segment Ie - see Fig. 11 - centered at the center C of [t1,t2]
and orthogonal to it (thus, I~ i8 directed along the edge, which is orthogonal to the direction of [t1, t2]-Now the array, consisting of all the edge segments Ie, is formed.All the operations, described hereinbefore for the construction of components (Step 6) above are applied to the segments Ie. Only the condition iiii in operation 2 for said construction is omitted. The D1segm, D2segm, Tsegm~ are replaced by the parameters D^~1Segm~
D~2segm~ T~segm, whose values are usually the same.

As a result, a number of edge components is constructed.

10. At this stage the procedure described in Step 7 for the division of ridge components into parts and their appro~imation by polygonal lines is applied to the edge components. The parameter DComp is replaced by Dedge, whose typical values are between 0.3 and 1, typically 0.5.

As a result, each edge component is approximated by the polygonal lines joining the centers of some of its segments. This polygonal line is a submodel, representing the central line OI the edge.

20~920 2097~D92 In order to construct the pro~lle, the following three numbers are associated to each interval Jj of the approximate polygonal line:

minj, which is the average value of all the minima of the edge elements, constituting the part of the component between the ends of Jj(i.e. the average of the minima of oc1, a2).
~, which is the average value of all the maxima, respectively (i.e. the average of the maxima of a1, a2).
widtllj, which is the average value of all the lengths I t2-tl 1 over the same edge elements.

Finally, each edge component is represented by:
- the ordered array of the vertices Pi of the polygonal line (i.e., the end points of the intervals Jj). These points are given by their global coordinates on the region.
- the ordered array of the numbers 3~, }a~, width3.

The operations, described above, can be implemented in two variants:

a) After construction of the ridge components, they are subdivided and approximated by polygonal lines (approximation procedure).
Then the procedure for the construction of edge lines is applied to each subcomponent obtained in this subdivision, separately. It can cause the further subdivision of the subcomponents and their corresponding approximation by polygonal lines.

20919~0 2ûg7/2~2~92 b) The procedure for the construction of edge lines is applied to the entire ridge components; it produces a certain subdivision of each component into subcomponents. Then the approximation procedure is applied to each subcomponent separately.

In each case, in addition to the numbers ~i and a;, associated to each interval Ii of the approximating polygonal line, we associate to each interval also the numbers leftvaluel lQftwidth;~
ri~htvalue;, t~1eidthi, determined as described. The averaging here is done over all the edge elements, adherent to the ridge segments in the interval Ii-The approxirnation procedure can be modified to give a bettervisual approximation to the components which are curved - see Fig. 12.

The subdivision points can be shifted into the convexity direction o the component, thus diminishing the distance from the component to the approximating polygonal line roughly twice.

In the model construction stage hereinafter we describe the models and methods which utihze the information obtained in the detection stage~ only in a partial way. In particular, the following alterations can be made:

Each ridge component (or an interval in its polygonal approximation) has the adherent edge elements on the righ~ and ~61- 2091920 on the left of it. The "widths" I t2-tl 1 of these edge elements can be used to approximate the corresponding ridges by the models with the variable width (see Fig. 13):

Each edge component (or an interval in its polygonal approximation) is constructed from the edge elements. Their width allow for approximation of the edges by the variable width models (see Fig. 14):

The edge profile analysis as described hereinbefore neglects an information obtained by approximation by polynomial q, viz. its maximum and minimum values, unless b~th the profile extrema are within the distance Wgr~,d (or 1') from the gridpoint under consideration. This restricts the width of the edges detected to approximately Wg,ad, or l', which in the present realization are of the order of ~-6 pixels.

Eowever, the proiles of the form shown in Fig, 15 a, b, and c, can be used to detect wider edges, by analyzing the adherency relations among the corresponding edge elements illustrated in Fig.16aandb.

The numbers leftvalue, rightvalue, etc., computed for each component (or an interval of the polygonal line)9 together with the marking of the edge elements, allow for a construction of curvilinear structures, more complicated than those used above.
For example, the profile like that shown in Fig. 17, which is -62- 2~1920 2ûg~,'12~2 typical for many pictures, can be easily identified and approximated, using the "end adherency" marking of the edge elements, their vvidths and the numbers leftvalue, rightvalue etc., providing the grey levels of the profile at the extrema.

In the implementation described here - see Fig. 18 - we omit the edge components adherent to the surviving ridge components.
This operation increases the compression ratio, but in some cases it can cause quality problems. The part A'M"DC - Fig. 18 - is represented by a ridge on the ridge component AB, while CE and DF are represented by edge components. This can cause a visually appreciable discontinuity in the shape of the lines A'CE
and A"DF on the picture.

An appropria$e choice of the parameters reduces this effect to a completely acceptable level.

However, another realization is possible where the gradient components CA' and DA" are memorized and used to correct the representation of the element CA'A". The decrease of the compression ratio, caused by storing this additional information, is eliminated almost completely by the fact it is correlated with the rest of the data.

step 9 -63- 2~91920 20Y7/242~J~2 This involves representing basic structures by mathematical models.

By now the following output has been produced by the preceding process steps:

1) Ridge polygonal lille3. To each interval Ii on such a polygonal line, the numbers ~;, ai, leftvalue;, leftwidthi, rightvaluei, rightwidthi are associated.

2) Edge polygonal lines. To each interval Ji on such a line, the numbers minj, maxj (together with the indication of what side of the component max (min) are achieved), widthj are associated.

O Hills (hollows). Each one is given by the point (represented by its global coordinates on the region), ~1 and ~2, l ~l l > 1~2 l (For hills, ~1 < ~2 < 0, for hollows ~1 > ~2 > 0), and the angle ~ (0 < ~ <
180), between the direction of ~2 and the first axis of the global coordinate system on the region.

4) The set Al of the "smooth" or the '7background" point3 on the g~id; at each gridpoint belonging to Al, the value aoo (the degree zero coefficient) ofthe approximating polynomial p(x,y) is stored.

Basic models that can conveniently be used will now be described.
The following model is used to represent a ridge over one interval of the corresponding polygonal line - see Fig. 19.

-64- 209192~
2097/~424/~92 Assume that the interval I is given by the equation (in the global coordinate system u, v on the region):
au+~v+c=O,a2+,B2=l, and the two lines 11 and 12, orthogonal to I and passing through its ends are given by -,Bu + av + cl = 0, -,Bu + av + c2 = O (notice that c2-cl is the length of I, assuming that c2 > cl).
Then the model ~Iridge (u,v) is defined by ~ Yr (au~ ~v + c), for q~Iridge (u, v) = cl - ~(c2-Cl) ~ ,Bu - av 5 c2 + o(c2 - cl) O, for ,Bu - o~ < cl - O(C2-cl)~
or ,Bu - av > c2 + ~(c2-cl).

There, ~ is a parameter between 0.05-0.2, preferably 0.1.

The function yrr(s) (s = au + ,Bv + c) can be chosen in various ways.
In the most accurate approximate mode it must satisfy the following properties - ses Fig. 20:
~r ( - leftwidth) = leftvalue ~r ( rightwidtO = ~ghtvalue (*) ~/r (O) = a $
_ ~0)=~
~s2 -~5 -2097/2~2~2 2 0 ~ ~ 9 2 0 In particular, the following function can be used as ~yr(s): the ~th order spline function. ~Itr, satisfying conditions (*), being once continuously differentiable, and having zero derivatives at leftw~dth, rightwidth and 0.

The information, concerning each ridge (left - right values and widths, ~) is redundant, and usually one can use much simpler models, utilizing only a part of this information. In particular, one can take ~r(s) = a + ~s2. Thus we have a~(au~,Bv+c)2,for ~)Iridge (u, v) = cl-o(c2-cl) < ~u - av < c2 + ~(c2-cl) O, for ,Bu-ov < C1-~(C2-C1), or ,Bu-âv > c2 + ~(C2-Cl) Thus, the model is given by a parabolic cylinder with the axis I, the coefficient ~ and the height a on the axis, see Fig. 21.

Various modifications of this model can be used with no changes in the rest of the process. For e~ample, the quadratic function a +
(ocu + ,Bv + c)2 can be replaced by any function ~r(s) of s = au ~ ~v + c (which is the distance of the point (u,v) to the line I), having approximately the same shape, height and curvature - see Fig. 22.

Instead of the abrupt vanishing out of the o(c2-c1) neighborhood of I at its ends, the profile function a + 31 (ocu + pv + c)2 (or ~(au + ,Bv + c) ) can be multiplied by a weight ~unction, which smoothly -6~- 209~.~20 20g7/24~92 change~ from 1 to 0, as the points (u, v) move away from the strip, bounded by l1 and l2, etc.

The following model is used to represent an edge over one interval of the corresponding polygonal line.

Let ~> ~ and ~h be the numbers, associated to the interval J, as described above. We use the same notations as hereinbefore, with reference to Figs. 19-22, except that the interval therein indicated by I i8 now indicated by J.

The model ql~Jedge (u, v) iB defined by ~e (au ~ ~v ~ c), for q~Jedge (u, v) = -o(c2-c1? ~ cl ,Bu - ~v < c2 + ~(C2-Cl)~
0, for ,Bu - av < c1 - o(c2-c1) or ~Bu - av > c2 +
~(C2- Cl) Here for ~ = au ~ c = the distance of the point (u, v) from the line J, as ~e(s) any function can be taken, satisfying the following conditions:

a) ~e (-1/2 ~) = min (or max, according to the orientation of the component. This in~ormation is associated to it.).

b) ~e (l/2 width) = max (min);

c) The graph of ~e has the shape appro~mately shown in Fig. 24.

,~ 2~920 In particular, let f (~ 3 - 3~. Then~e(s) - l/2 (max ~ min) ~ l/4 (max - m~ 2s/width) (or ~(g) = l/2 (max + min) - l/4 (ma~ - ~ig) f(~s/width), respectively can be used.

Another possible form of the edge model is a combination of two ridge models. Namely, for an interval J given, two intervals J' and J", shifted to l/2 ~h are constructed - see ~ig. 24.

On e~ch of J' and J", a ridge model is construc$ed as described hereinbefore, with a valuè equal to ~nin (max, respectively) and ~
for both ridges equal approximately :t2(~ - ~a) / (~a)2. The values of this combined model at each point is equal to an appropriate weighted B11m of the two lqdge models constructed -see Fig. 25.

The following model is used to represent hills and hollows; it is the polynomial p(x,y) of degree 2, obtained on the lxl - cell, in which the corresponding element has been detected. More accurately, if the hill (hollow) is represented by the grey level a, the center (uO, vO), ~ 2 and 0, then, in a new coordinate system u', v' with the origin at (uO, vO) and the axis ov~ forming the angle ~ with 0v, ~h(u~y)-a~lul2~2vl2.
Thus the model, representing hills and hollows, is an elliptic paraboloid (which coincides with the original approx;mating polynomial p(x,y)).

~il .. j, .. ....... ,.. - . - ..

-6~-20Y7124~2 2 0 91 9 2 0 Other functions having approximately the same shape can be ujed, a9 grephically illustrated in Fi~ . 26.

The submodels hereinbefore described represent hills, hollows and intervals of the ridge and edge-polygonal lines.

Now, for each polygonal line, the model corresponding to it is defined as follows: for a line L(L') consisting of the intervals lilJj).

q)LIidje(U-'~ indge(U.V).
q~L'edge(U.V) ~ ~ Jjedge(U9V)-Here the ~unctions ~ ridge(u,v), ~ ~;edge(u,v) are defined as above, with u,v being the global region coordinates. Such models are illustrated in Fig. 27.

Instead of "rectangular" models hereinbefore defined, one can use other forms; for example, elliptical ones or rectangl~lar ones completed by two semi-dijkj, aj graphically illustrjted in Fig. 28.

To each model representing a ridge or edge, a boundary line is associated in one of the following ways:

a) for each interval I, two lines 11,12 parallel to I and passing at the distances (1 ~ K) lÇ~l~h and (l~ bi~ respectively, .,, .'Pl , . ..
: , 209't~2~ 6~ - 2 0 ~ 1 9 2 0 form the piece of the boundary corresponding to I; for edges on the distance 1/2 (1 + K) width).

b) for each interval I the lines, passing on both sides of I at the distance l/2 ~leftwidth + ri~2~idth) (1 +1C) form~ a boundary; for edges - the same as above.

c) for each interval I the lines, passing on both ~ides of I at the distance computed through ~ by the following formula:
(Hridge/~)1/2 pixels. Here Hridge takes grey level values between 20-60, preferably 30.

d) The lines at the constan~ distance d.

e) The boundary is not computed.

Here, K and d are the parameters. Usually 1C - 0.05-0.2 and preferably 0.1, and d ~ 2-4 pixels, pre~erably 3.

This procedure is illustrated in Fig. 29. For each of the possibilities a- d, the boundar~ is formed by the curves 11 and 12 for each interval I and the orthogonal lines (or semidisks) for the end intervals of the polygonal line, as ~hown ;n Fig. 29.

The model function q~Lridge (q?L~edge) iB defined around the polygonal line L, in particular on its boundary. By construction, it . . . ~ . . .- . .

7~ 2~9192~
~7~2 takes a constant value on the parts of the lines 11 and 12, bounded by the ends of each segment.

For each of the hills and the hollows, the boundary line is associated in one of the following ways: for a hill (hollow), given by Xo1 Yo, ~ 2, ~, a:

i. It is an ellipse centered at (~O~ yO), with the main a~is in the direction ~, and the semiaxes rl, r2, computed through ~1 and ~2 and a by the following ~rmula: rl = (Hh/~ /2, r2 = (Hh/~2)l'2. The parameter Hh takes values between 20-60 grey levels, preferably 30.

ii. An ellipse as above vrith rl, r2 given by the following formula through ~ 2 and a: rl = r, r2 = (~ 2)l'2 r. r usually takes values between 1-6 pixels, preferably 3 pi~els.

iii. An ellipse as above with r1 = r, r2 = r;

iiii. rl, r2 given as the parameters;

iiiii. The boundary line is not computed.

The model ~unction q~h, hereinbefore de~med, takes a constant value on the boundary line.

~ 2~1920 ~;teps 10 and 11-From now on, x,y denote the global coordinates on the region.
By now for each ridge polygonal line L, edge polygonal line L' and each hill or hollow defined in stages 6 to 8, the model ql)Lndge(x~
Y)~ ~)L~edge(X, y) or q)h(x~ y), representing the corresponding picture element, has been constructed. Al80, the boundary line has been constructed ~or each model, as hereinbefore described.
The part of the picture contained inside the boundary line will be called "the support" of the corresponding model. The construction of the global model representing the picture on the entire region is performed in several stages.

A certain part ~ of the ~et of all the models i6 specified; usually, the models with the maximal visual contribution are chosen.
Hereinaf~er a detailed description of some possible cho;ces will be given.

a. The region is subdivided into square cells of the size m x m.
Here, m is approximately equal to the basic scale L. The parameter L is the same as defined above. Below, m is usually between ~-48 pixels.

b. For each m ~ m - cell C, all the models in ~P whose supports intersect C, are considered. This situation is graphically illustrated in Fig. 30.

-72- 2~91920 20~7/2~2g/H~2 c. The polynomial Pc(x,y) of a low degree (usually c 2) is constructed, which minimizes the square deviation from the following data:

- the values "aoo" at all the points in the initial grid in C, which belong to the set Al and do not belong to the model supports intersecting C;

- the values of the models on the boundary lines inside C.

Various rOUtineB to cornpute Pc(g~ y) can be utilized by a skilled person.

In particular, one pos6ible realization i8 as follows: a sufficient number of points is chosen on each boundary line. Now Pc is constructed as the polynomial, providing the minimal square deviation of the model values at the6e points and the values aoo at the Al - points.

Now, a partial global model ~ ~ ,y) i8 constructed, corresponding to the part !Pof all the models chosen be~ore.

~0~7/2~2 Pc (x, y), for (x, y)belonging to the mxm C and not belonging to the supports of the cell models in ~.
, Y) = q~(x, y), for (x, y), belonging to the support of the model ~ in !P and not supports of the other models in ~.
The average of the values of all the models q~(x, y), to which Gupports the point (x, y) belongs, if (x9y) belongs to the support of than one model.

Other methods of "glueing toget~er" the local models can be used by a skilled person. In particular, the one usually known as a "partition of unity" can be applied. It is performed as follows: let q~i(x~ y) donote all the models in ~P, and let ~ be the support of q~i, i = 1, .. , N.

The functions Wi(x,y) are constructed with the follow;ng properties:
a. W;(x, y) ~ 0 b. ~ iWi(x, y) _ 1 c. W;~c, y) is equal to 0 out of a certain neighborhood of S; - i=1,2,.. N.

-74- 2~9~2~

In particular, Wi can be constructed as follows: one can easily construct continuous function~ W'i(x, y), W' > O, which are equal to 1 inside V2 S;, and are equal to O outside of 2S;. Then, Wi =W'i /~iW'i satisfies conditions a, b, c.

(Here ~ S denotes the result of a homothetic transformation of the subset S.) Then, the value of the partial global model q~(x, y) is defined as the weighted sum of the values of the models q~?i with the weights wi ~(x, y) - ~; Wi(x, y) q~i(g. Y) (i=0,1,...N) (The functionWo(x, y) above corresponds to the complement of all Si and ~o(x, y) is equal to Pc(~, y), for (x, y) belonging to each of the m x m cells C).

Construction of a global model by the above partition of unity formula provide6 in particular smoothing out the discrepancies between the local models.

By now the partial global model q~x, y) is constructed. Then the part ~ of the remaining models is chosen (~ee below), and the filtering i~ performed.

~091920 20g7t~4~J~02 a. Namely, for each ridge model (~)r in ~P, given by the polygonal line L, consisting of the intervals Ij:

q~r i8 omitted if either q~r represents the white ridge, and the grey levels a; associated to each interval Ii satisfy:
ai ~ x;, y;) < max (Fl, Ql Hl) where (x;, y;) is the central point of the interval I;. Here, Hl i8 the average value of ai - ~ p(x;, Yi) over the intervals of all the white ridges in ~, Fl and Q1 are the external parameters, or:
q~r represents the black ridge, and a; - (~P(Xi, Yi) >min t-F2, Q2-H2), where H2 i8 the average value of ai - ~P(x;, y; ) over all the intervale of the black ridges in ~' Usually, F1 and F2 are between 5-20 grey levels, preferably 10, Q1 and Q2 are between 0.3 -1.2, preferably 0.6.

b. For an edge model q~e in ~P, ~e is omitted if:
max; - q~p(xi~yi) < max (F3, Q3 . H3) and min; - q~!P(xi, Yi) 2 - ma~ (F3, Q3 ~I3).

Here, (x'i, Y'i) and (x";. y";) are $he points on the sides of the edgeinterval, as shown in Fig. 31, and H3 is the corresponding average. F3, Q3 are parameters.

20Y7/~24J~2 -7 6 - 2 0 9 ~ 9 2 ~

c. The hill q~h iS dropped, if a - ~ , y~ < max (F4, Q4 H4~ and the hollow 5~)h i3 dropped if a - ~(x, y) ? - ma~ (F4, Q4 Hs).
Here (x,y) is the center of the hill (hollow), a = aoo i8 its associated grey level value, and H4 (H5) is the average of a - q~!P(x,y) over all the hills (hollows) in ~'.

F4 and Q4 are the parameters, and usually F4 i~ between ~-20 grey levels, preferably 1 0,Q4 i8 between 0.3-1.2, preferably 0.8.
Denote ~1, the 6et of models in ~' which have not been filtered out.

Now the next partial global model q~!P,!P'(~, y) is constructed. It can be done in various ways. One possible construction is the following:

~x,y), if (x, y) does not belong to a support of any model in !P, q~j(x, y), if (~, y) belong to the support of the '(x, y) = model q~j in ~?', An average of the value~ of all the models ~j(x,y) in !Pto whose supports (x, y) belong, if there i~ more ~an one such model.

Alternatively, q)~ '(x,y) can be constructed, using the partition ofu~ityasdescribed above.

-77- 2~9~
20Y7~4EV92 The part ~ "of the remaining models is chosen, and the filtering and the construction of the ne~t partial model, ~ P', are performed as deæcribed hereinbefore.

This process is now repeated until all the models detected are used or filtered out.

Now will be described some speci~c choice~ of ~ P; etc.

i. ~P is empty. Thus, as the first partial global model we takeq~x,y) = Pc(x,y), for (~,y) belonging to each m~n - cell C.

This partial global model is called a "rough background", and it is given by a piecewise polynomial function which on eacb mxm -cell C is equal to the corresponding polynomial Pc.

~' here consists of all the models detected. Thus, on the second step we filter out insignificant models and complete the construction of a global model.

ii. Here we f~lrst construct a partial global model, oalled "background". It includes the smooth domains as well as the "best smooth approximation" of the domains, containing ridges, hill~ and hollows.

Usually, smooth areas OI the background are separated by the edge~ and ridges with strongly diEerent grey level values on two 2~91920 -7g-~os7/24~vi2 sides. Thus, the ridge polygonal line i8 called "separating" if for at least one of its intervals, the associated sidevalueP" differ for more than Sr Here, Sr iB a threshold, usually between 10-30 grey levels, preferably ~0. A profile of a separating ridge is shown in Fig. 32.

Now ~Pis defined as consisting of all the edges and separating ridges and ~' contains all the remaining ridges, hills and hollows. The partial model q~!P is called a background.

iii. Here the background is constructed as in ii. Then ~' is taken to contain all the remaining ridge~" for which I ai ~ (xi~ Yi) I ~ Q6 H6 for at least one interval Ii of the corresponding polygonal line.
Here H6 is the a~erage of lai ~ i,Yi) over the interval~, of all the remaining ridges, Q6 is a parameter. Usually Q6 is the same as above.
Then, the second partial model i5 ~p'and ~'is taken to contain all the remaining (after a filtering according to (~, ~) ridges, hills and hollows.

iiii. Xere ~ contains all the detected ridges, edges, hills and hollows.

79 ~09~2~
2~/2424aD~2 Among the variants i, ii, iii and iiii above, i is usually computationally simple, while iiii usually provided the best -visual qual;ty.

Version iii is relatively simple computationally and moreover provides high visual quality and high compression. It is preferable.

Various options for the choice of boundary lines have been described above. The option a (c for edges), i for hills and hollows -provides the best quality but usually the lowest compression, since additional information must be memorized.

The option e (iiiii) is compatible only with $he version (i) of the "rough" background construction. It provides higher compression, but usually lower quality.

The option c usually provides both a good quality and a high compression ratio.

Usually the profiles of the ridges and edges look as in Fig. 33.

Thus, if the bounding lines will be chosen as shown in Fig. 33, and the profile functions ~ used in the model'~ construction as described above, this will provide the required shape and usually the inclusion of these edges and ridges to t~e parl;ial model allows one to filter out the "side ridges", as shown in Fig. 33.

-8û- 2091 92~
2~12~2 One of the possible way~ to add the local models (ridges, hills and hollows) to the background i8 to define the resulting models as shown in Fig. 34.

max (q~backglr.(x~ y), white models), if (x, y) belongs to the support of the white modela min (~backg,.(x, y), black models, q~(x, y) = if (~, y) belongs to the support of the black models.
Average value OI all the models to whose support ~3y) belongs, if it belongs both to the support~ of black and white models.

The local approximation in the mxm - cella C can be obtained in various ways, not only using the polynomials Pc(x, y). For example, if two or more models support~ intersect this cell, the gray levels on it can be obtained just by extrapolating the model values on the boundary line~. Thus, no information must be stored for the cell shown in Fig. 35.

The "smooth cella" can be represented by only one value (at the center), and then linearly interpolated, taking into account the models values on the boundary lines.

Other principles of choice of ~, ~; etc., than those described above, can be used. In particular, in the regions, where a large . -- , . . .

2~ 8 1- ~O
number of hills and hollows are densely positioned, ~P can be cho~en to contain only hills. The partial model q~ is constructed in such a way that the hollows of a typical height (average height) are produced between the hills. In the subsequent filtering, usually many of the detected hollows are filtered out (being "well predicted" by q)!P): sea Fig. 36.

Since various smooth domains and the domains with different types of texture are usually separated by edges, it may be desirable to form a partition of the entire region into subpiece~, separated by edge~ and separating ridges. These subpieces can be defined in valious ways. In particular, one can form a polygonal subdivision of the region.

As such a subdiv~sion i~ constructed, it can be u~ed as follows:

a) various average values, as above, can be computed with respect to those naturally defined subpieces, and not to the square region, which is not related to the picture struGture; and b) The size m OI a ~ubdivision of the region into the m~m - cells, can be chosen separately for each subpiece, thus allowing for an economic representation of the smooth area~.

~og7/242~ 2 2 0 ~ ~ 9 2 0 The way of constructing global models, hereinaflcer described, can lead to a particularly simple realization of the process. The structure of the comprefised data, obtained in this way, is convenient for operations on compressed pictures and for videosequence compression.

Hereinafter a more detailed description of this realiza$ion iB
given.

1. Let the set !P of models be chosen. All the models in this set are assumed to be provided wit~ l~heir boundary lines and their values on the boundary lines, as described above.

2. As usual, the region is sub-~iv;ded into m~m cells. The corners of these cells form a grid, which will be denoted by G.

3. To each point w in G, a grey value v(w) is associated, which is the (averaged over a small neighborhood) grey level value of the original picture at the point w. In particular, we can always assume that G is a subgrid of the basic grid, used in the models identif;cation process. Then u(w) can ke defined as the constant term aO0 of the polynomial p~x, y) constructed at w.

-~3- 1 20~7/24:WlV~ 2 ~ ~ 1 9 2 ~ I

4. The value of the global model ~(x,y) at each point x, y of a certain m x m-cell C is defined as follows:

a. Consider all the models in ~P whose supporta intersect ~. The part of the boundary of C not covered by the supports of the~e models, is subdivided into segments with the end point either at the corners of C or on the boundary lines of the models involved.

b. Thus the values of q~ are defined at each endpoint either as v(w) for a corner w, or a~ the value of the corresponding model at its boundary point. Define the values of ~ at the interior points of the boundary segme~ts by linearly interpolating the values of the endpoints: see Eig. 43.

c. Finally, for any (~,y) in C, not belonging to the supports of the models intersecting C, the horizontal line through (~, y) ia drawn until the first intersection on the left and on the right either with the boundary segments of C or with the boundary lines of some of the modela. The values of q~ at $hese intersection points are already defined (either by b. above, for the boundary segments of C~ or by the value of the corresponding model on its boundary line). Then the value of q~ at (x,y) is defined by linear interpolating the values at the intersection points.

d. The values of q )(~c,y) for (~,y) belonging to ~he support ~or one of the models is def;ned as the value of $his mode at (x, y). If (~,y) 8 4 - 2 0 9 1~ 0 belongs to the support of several models, the value q~ (x,y) i~
determined by averaging the values of the model~ involved, as described above.

For example, in Fig. 43 the values of q~ on AB, CD, DE, EF and GA are defined by linear interpolation of the values at the endpoints. The values at (xl,yl), (x2,y2), (x3,y3) are defined by linear interpolation of the values at Al and Bl (A2 and B2, A3 and B3, respectively).

5. This global model has some important advantages:
a. Its values are continuous, by construction. This prevents certain visual distortions. This continuity is maintained also on the boundaries of the regions, if the same values v(w) are used ~or the bolmdary gridpoints w in both adjacent regions. This eliminates the necessity of smoothening operation~.
c. Computations of the values of ~ are local: for each point in a mxm cell C, only infor~ation concerning C is used in computations. This property i8 important for a hardware implementation of the process.
d. Very simple structure of com~res~ed data: the parameters OI
the local models and the values v(w) at the points of ~, not covered by the supports of the models.

6. Filtering operations can be performed on this model as herein described. In particular, in smooth regions the grid G may be too 2~920 2~g7~429~2 dense. Then one can compare the values v(w) at G with the values interpolated from the, say, twice-spaced grid, and to omit the gridpoint in G, whose value~ are approximated closely enough by this interpolation.

In the filtering stage (Step 11) the following considerations and examples should be kept in mind. In this stage we eliminate some exces6ive information, according to its psychovisual ~ignificance. This involves simplification of the model~, caried out by eliminating some excess*e parameter~, without dropping the model completely. Since the model~ are constructed on the base of a local analysis, a posteriori they may contain "excessive"
parameters. The following e~amples will clarify this step of the process.

Assume that a central line of a certain curvilinear structure is represented by a broken line, consisting of a number OI straight segments. However, if the deviation of the intermedîate points from the straight segment connecting the endpoints is le~s than a certain thre~hold Tg, the broken line can be replaced by said segment with no visual degradation (see Fig. 37), and all the parameters, repre~enting the intermediate point~ of the broken line, can be omitted. T8 is usually between 0.3-1 pixel, preferably 0.5.

:,, ' - :' ~ ' -86- 2~92~
~097~4~2 ~n,ple 2 Hills and hollows may be represented by bell-shaped models with an elliptical base (see Fig. 38).

Let rl, r2 be the semiaxes and 0 the angle of the biggest semia~is with the ox.

For I r2!rl -1 1 s Tg, the ellip~e is visually indistinguishable from the circle of radius r = l/2(rl + r2). Thus we can replace the ellipse by t~is circle and to store only r, in~tead of rl, r2, ~. T~ is usually between 0.1-0.3, preferably 0.2.

l~nlple ~
Smooth regions can be represented by polynomials of degree 2, Z = aO ~ alx' ~ a2Y' + all~'2 ~ 2al2 X'y' f a22y'2 (in local coordinates x', y' at each cell).

If all the coeffilcients of the second order are less than Tlo, all the three second order terms can be omitted with no visual degradation.

For any specific model realization the corresponding thresholds Tl~ can be found in advance in a straightforward psychovisllal experiment. Tlo is usually between 5-20 grey levels, preferably 10.

' `

, :

!
-87 - 2 ~ ~ 1 92 ~

In a further filtering stage, entire models may be l'iltered out, YiZ.
screened out entirely, because of their small psychovisual contribution.

A general scheme is as follows: We choose a part of models and construct on their base a partial global model. Then for each of the remaining models we measure its contribution to this partial picture. If it is less than Tl, the model is omitted. Next we add a part of the ~urviving models to the partial model, and construct a second partial global model. Once more, we measure the contribution of the remaining models, then repeat the procedure.
In detail this procedure is described above.

We con6truct a background u6ing the edges and smooth regions detected (see detailed description below). This background is our partial model. Then we add the ridges, the hills and hollows, dropping those whose height over t~e background is leas than Tl2-~m~
We construct the background and per~orm the first filtering, as inExample 4. Then we add part of the "surviving" ridges to the background (those of greater than average height), using the profile illustrated in Fig. 39.

Here the height of the side channels 1 and 2 i8 proportional (within a certain fixed coefficient) to the height of the ridge.

-8g- 2~91920 2097/2429~2 Experiments show that this profile is typical for ridges on the usual pictures, and in this way we form a second partial model.

Usually the ridges, corresponding to the side channels are detected and stored. However, now they will be screened out, since their contribution has been already provided by the chosen profile.

The thresholds value T in the filtering process depend strongly on the structure of the picture, the distribution of sharpness, brightness, on the geometry of the models, etc. An important point i8 that the primarily compressed data provide an adequate and easily extractable information of this sort. So highly adaptive methods of filtering ean be constructed, providing a high degree of utilization of the visual perception properties.

On the other hand, a very simple form of thresholds T can be given:
T = max (Tabs- KIaver)-where Tab~ i8 the (usually low) absolute threshold, and IaVer i6 the average value of the threshold quantity. Above, a detailed description i8 given.

S~ep ~
In this, the quantization step, we approximate the parameters values of the primarily compressed data by the values firom a certain smaller array, constructed according to the psychovisual significance of each parameter and their combinations.

89 209~a 2~Y7~241~

Since the parameters have very ~imple geometri~ and vi8ual interpretation (position, slope, curva$ure and width of curve position and form of hilla, their brightneas, etc.), the levels OI
admi~sible quantization in their various combinations can be established rather accurately. Thus, the quantization i5 performed in several steps:

~ o~aran~ ~d model~
At this stage we aggregate together those parameters whose psychovisual si~nificance and/or their dynamic range depend on their mutual values.

Part of this aggregation can be built into the model's structure, but another part depends OIl the mutual position of the models on the picture, and hence can be performed only when the models have been identified.

Aggregation can be followed by the ~ubstitution of other parameters andJ'or a coordinate change, to express in a better way a mutual dependence of parameters.

33~m~
This i~ similar to the preceding filtering Example 1 - aee Fig. 40.

Experimenta show that the replacements of the intermediate vertices El, E2, E3 in the direction perpendicular to the segment ~90- 2 ~ 9 1 9 2 1~

AB, are visually much more important than the replacement~
in the AB direction.

Thus, we aggregate the parameters in the following way:
a) The coordinates of the endpoints A and B are quantified with respect to the global region coordinate system.
b) The new coordinate system i8 constructed, its first axis being parallel to AB and the second axis perpendicular to AB.
c) Coordinates of the intermediate points Ei are expressed with respect to this new system.
d) The coordinate values are quantized, the quantization step for the first coordinate being bigger than the step for the ~econd coordinate.
e) The dynamic r ange of the new coordinates i8 also determined naturally. The f;rst coordinate for the point Ei is bounded by its values for A and B. For the second coordinate the experiments show that its range usually ia bounded by approximately 20% of I AB 1.

E~a~
The same broken line as in Example 6 can be described by the following parameters: lengths p of the segments and the angles ~ between the consequent segments: see Eig. 41:

The psychovisual considerations allow one to define the appropriate quantization~ step~ and dynamic ranges for p's and ~'s.

-91- 209~9~
2~Y7~C2 For models that are geometrically close to one another, the experiments show, that of the greatest visual importance are the perturbations, which change the "topological" structure of the picture, viz. create new visual contacts between the models or separate visually unified models, as illusrated in Fig. 42.

On the other hand, parameter perturbations, which alter the positions and forms but preserve the topological structure, are much less detectable.

Thus the parameteræ of nearby models, responsible for their mutual positions, can be aggregated, and their dynamic range and quantized values can be chosen in such a way that the topology of the picture to be preserved by the quantization.

E~m~2~
This is similar to Example 2 under filtering. E~periments show that for the description of the ellipses, ~e c~oser the ratio rl/r2 to 1, the smaller is the visual significance of the angle û.

Thus, we can use new parameters: rl, e = r2/rl and ~. The dynamic range of rl depends on the scales chosen; e is bounded by 1 (r2 < rl), and ~ belongs to ~0,11].

However, the quantization steps for ~ can be chosen in greater size as the values of e are closer to 1.

-92- 2~19~
2ûY~124Z~JHf92 ~B~m~
At this etage the common d~ynamic range of the aggregated variables is subdivided into subparts. In each subpart, one specific value of the parameters is ~xed, which represents this subpart.

The quantization consists of replacing parameter values belonging to a certain 3ubpart by the value representing this subpart chosen above. The f~rst two steps are performed according to the psychovisual signilScance of the parameters values. These steps have partially been described in Examples 5-9 above.

A general procedure, described above, i8 well known under the name "vector quantization" (see [A. Gersho and R.M. Gray, Vector Quanti~ation and Signal Compreasion, Kluwer Academic Publishers, Boston/Dordrecht/London, 1992]). Thus, for any other realization of ba~ic models and their parameters, the corresponding quantization procedllres can be constructed by skilled persons.

S~ep 13 ~
This i8 a procedure which ultimately represents the quantized data, obtained in step 12, in the form of a binary file (or a bit-~tream). This procedure is v,~ithout loss in the sense that after decoding we obtain exactly the quantized values of all the parameters. In principle, any lossless encoding process can be applied at this stage. HoweYer, a correct organization of data 209~92~
20~7/2~24J~2 reflecting the specific nature of the parameters to be encoded, can greatly reduce the volume of the final encoded file.

The order of the models representing the basic elements detected is not important in the construction of the global model. Thi~ fact can be utilized in constructing a more economic encoding.

The geometric parameters must be encoded with respect to a correct acale. For e~zample, in order to encode the positions of the points entering our models, we can subdivide the region into the smaller cells. Since the order of the objects i8 not important, we can encode the coordinates of the points in each cell separately. Since each cell is smaller than the entire domain, for tlle same level of quantization we need fewer bits to encode the coordinates with respect to a cell.

Additiorlal data which must be memolized ~r each cell i8 the number of the points in it. Even if ~he distribution of the points is ordinary, an easy computation show6 that in such a subdivision we usually gain a significant amount of bits.
However, if the cells have approximately the size of the basic scale L, we know that the possible number of objests in each cell iB small, and therefore the number of bits we need to memorize the amount of objects in each cell drops.

Thu8 we encode the geometric parameters with respect to the cells approximately of the size of the basic 6cale L. In this way ~4 2 0 ~ 2 0 2097/2424~92 we utilize the experimentally known uniformity of distribution of the basic elements on the picture.

The parameters, which are (or can be) correlated, must be aggregated in a correct way. For example, the slopes or widths of the curvilinear structures, or the brightness of the models, etc., within one region are usually concentrated around a small number of typical values.

Thus, values can be memorized once per region, and for each model only the difference must be encoded. In particular, the average value can be used as a typical one.

Huffiman-like coding can be used to utilize non-uniformity in distribution of the values of certain parameters. In order that this coding be effective, the parameters must be properly ~ubdivided into groups with similar di~tribution. For e~ample, the values of the ~lope~ by them~elves usually are distlibuted uniformly around the picture and hence Huf~man encoding will no$ reduce the amount of data. However, if in each region we subtract the average value, for the differences dis,tribution, we can expect a strong concentration around zero and application of Huffman encoding will reduce significantly the volume of data.

~3ome transformations of parameters in each region can be used to make their dist~butions over different regions similar.

2091~20 209712424/~2 The size of the regions into which the picture is subdivided i~
important in the e~ectiveness of this Huffman encoding, as earlier detailed.

Finally, the binary data encoding the quantized values of the parameters, as well as the types of the model~, is organized in a binary file. In principle, this file can be further compre~sed by any lossless method.

On the other hand, this file can be organized in ~uch a way that provides easier access to variou~ parts of the encoded picture and in order to make it error-resi3tant. Usually the~e operation~ increase the volume of data insignificantly.

Note that the model~ are constructed in an invariant way. And subdivisions into regions is done only for reasons of convenience. Thus some models from one region can be encoded in a neighbouring one, and the pointer~ can be used to identify the correct regions ~or each model in the process of decompression.

' ~

~qe4ww~2 ~9~ ~ 0 ~ 1 9 2 0 step 14 ~
At this stage - decompression - we produce from the binary file obtained in Step 13 the picture represented by its RGB values at each pixel. The binary file i8 transformed to the quantized values of parameters. This is done by the same encoding procedures which are used in Step 13, since by nature these procedures ara invertible.

The quantized values are interpreted a~ the value~ of "primarily compressed data". ThiB means that the quantized values obtained are represented by the figurea with the nllmber of digital or binary digits which are required by the computer being used and by computations accuracy considerations.

The obtained values of the parameters are substituted into the global model z = q~ , y) (for each region separately). This global model represents an explicit instruction for a computer, how to compute the value of the grey level z (or RGB) ~or each given values x, y of the coordinates of the pixel. The computer find~ the values of z for all the regions. For each region the coordinates (x, y) of each pixel are substituted to the model ~, and the corresponding value z = ~ (2, y) iS computed.

1~ this way, we obtain the value6 z (or RGB) at each pixel of the picture.

At the decompression stage some smootheIling operations can be performed, to eliminate the discrepancies between the adherent models in dif~erent regions.

20Y7/~42~J~g~ ~97~ 2 ~ 9 ~ ~ 2 Q

It can happ2n that the detection (or filtering, or quantization) procedures introduce certain systematic distortions into the parameter values. These distortions can be corrected in the decompression process by introducing appropriate correction~
to the values of the parameters stored.

For example, the detection process with a large "1" scale introduces a certain low-pa6s ef~ect. In particular, the widths of the ridges and edges and the sizes of hills may be obtained at a size which is 20% larger than their correct values on the picture. Then, in the process of decompression? each value of the width or size can be multiplied by 0.~, which will partially correct the undesirable distortion.

An important property of our models is that they are scale-invariant. They are mathematical e~pressions which can be interpreted according to any given scale.

Thus, the size of the pickure to be obtained in the process of decompression i8 a free parameter of this process. In particular, we can obtain the picture of the same size as the initial one, but any desired zoom or contraction can be produced. The same obtains with respect to non-uniform rescalings in x and y directions. In particular, the picture, compressed in one of the TY standards (PAL, NTS) can be decompressed into another one.

-9~ 203l~2a 20~Y7/~92 1/1~2 The same can be said, in fact, of more complicated geometric transformations. See "Operations" below.

S;z~ o~le window~
It is important to 6tress that although in all the examples the linear dimension "1" of the window was 4, it can (and sometimes should) be dif~erent, for example, 3 or 5 pixels. In particular, 1 _ 3 can be used to capture properly the finest scale details on the usual video or pre-press picture~.

The process, in partioular the identification of the basic structures, can also be arranged in the following way: The basic identification is performed with 1 = 4 until all the segments and hills and hollows are identified. Then at each point where ~egment or hill (hollow) has been found, the polynomial of degree 2 i~ constructed which provides the best mean square approximation of the picture on the 3x3-cell around the center of ~e segment or the hill. Then, the initial polynomial (constructed on a 4x4-cell) is replaced by this new one. All the rest of the processing remains unchanged. This is done in order to improve the accuracy of the detection of the parameters of the obiects (their he;ght, position and curvatures).

In the same way, in the process of edge elements construction, the degree 3 polynomials can be computed on 4x4-cells, instead of 6x6 or 5x5. Alternatively7 both the polynomials can be computed and combined to provide better estimating OI the edge para~e.ers.

gg 2091 920 2~97/242*H~2 In the above processes, the weighting functions on the pixels of 4x4 (6x6)-cells can be used (instead of smaller cells), in order to compute approximating polynomials.

,~_ Color pictures are represented by several color separations~
according to various approximations and ~tandard~ (RGB, YIQ, CMYB, etc.; ~e0, e.g. [R.J. Clarke, Transform Coding of Images, Academic Press, 1985, from page 248]. Each color separation usually represents a grey-level picture of a certain quantization. For example, for RGB each of these basic colors i8 represented by 256 grey values (8 bits) on each pixel.

1. The simplest way to compress colo~ pictures consists in compressing each color separately.

2. The compression process described above provides a tool for a much better utilization of a redundancy, existing in color information.

Different color separations of the same picture represent essentially the same object~ in various part~ of the light spectrum. The models described above capture the geometry of these objects. Therefore, each color ~eparation can be repre~ented by exactly the same model6 with exactly the same geometric parameters. Ollly the brightness parameters of these models dif~er ~rom one separation to another.

2~7e4~D92 -100- 2 0 9 1 ~ ~ ~
3. Thus the color compression is performed as ~ollows:
a. One of the 6eparations (for example R~ i~ compressed as a grey level picture. In particular, all the models are constructed.
b. The same models, with the same geometric parameters, are used to represent G and B.
c. The brightnes3 parameters of the~e models are adjusted for each separation of G and B separately, to provide a faithful representation.
d. In the compressed data, a complete information is etored only for R, and only the values of the brightness parameters are stored for G and B.

4. The same method can be applied to other combinat;ons of the basic colors, for example to YIQ (see ~R.J. Clarke, Transform Coding of Imageæ, Academic Press, 1985, from page 248] ).
Then the grey level picture corresponding to the luminance Y
iB compressed, as described above, and only the values of the brightness parameters are stored ~or the chrominance3 I and Q. Since the visual sensitivity to the values of I and Q is much lower than to Y, the required accuracy of a quantization of the brig~tne~s parameters for I and Q is much less than for Y.

5. In the preferred embodiment described above, the geometric parameters of the modei~ are:
- ~he coordinates of the centers and the directions of the eigenYectoræ for hills and hollows;
- the coordinates of the vertice~ of the polygonal lines, representing ridges and edges;

091~20 2Q~7/2424/W~2 - the width (or right(left) width) for each interval of the polygonal line;
- the position of the boundary line.

The brightness parameters are:
- the height aoo and the curvatures ~1 and ~2 for hills and hollows;
- The height a, the curvature ~ and the values left (right) value for each interval of a ridge.
- the min(max) values for each interval of an edge.

All these brightness parameters are the average grey level (or curvature) values of certain points or on certain curves. Thus, to find the values of these parameters ~r a given color separation (say, G), we compute (for this separation) the required averages at the same points and along the same curvea.

In particular, the following procedure can be applied: in the original compression (say, for R) all the b~ightnes~ parameters are obtained by certain calculations described above from the polynomials p(x, y) and q(x, y) of degrees 2 and 3 respect*ely, appro2~imating the original pictures on certain cells.

Then to find the brightness parameters for another separation (say, G), we compute the approximating polynomials p(x, y~
and q(x, y) for G exactly at the same cells and then repeat the same calculations a~ for R.

, -lo~ 2~9192~

In many cases, the valuee of brightnes~ parameters for different color separations are strongly correlated. Thi~ is due to the fact that different color separations are the intensities of the light reflected by the same objects in dif~erent part~ of the spectrum. Then (on small regions of a picture) these intensities are usually related by very simple transformations.

To remove this redundancy we can try to represent the brightness parameters of the separations G and B by means of simple transformations of the brightness parameters of R.

For example, assume that P~ -, Pn are the brightness parameters of the model, representing the separation R on a certain region, and P'1, ---, P'n and P"2, --, P"n are the corresponding parameters for G and B. We find the numbers a', b' and a", b", which provide the minimum of the mean square n deviations ~ (P'i- a~pi -b')2 and ~ (Pi"- a~pi - b")2. Then we represent the parameters Pi', Pi"
in i=l :
the form (*) P'i = a'p; + b' + r';
p~i=al~pi~b~+ri~.
Usually the correction terms ri', ri" are visually negligible ( if the regions are small enough - say, 24x24). Then, instead of memorizing P'i and P"i~ we store for each region only four numbers: a', b', a", b".

Various modifications of this method can be used. For example, we can try to find representations of the form (*) -lo~ 20~92~
20~712424/~
separately for various kinds of brightnes~ parameters, ~ay for edges, ridges, hills and hollows; or for the white and black models separately, etc.

7. A possibility of a simple expression of one color ~eparation through another on small region~ can be used to provide a color compression scheme, which can be combined with any grey level compression process (for example, DCT). Thi~ color compression procedure is less accurate than the one described above. However, it usually leads to much better re~ults than the color compression scheme used, say, in JPEG standard. It is performed as ~ollows:

a. The picture is subdivided into cells of the ~ize 8~ (usually is between 8 and 24).

b. For the color separations cho~en (for example, YIQ)> on each block the numbers a', b' and a", b" are deterrnined, which minimize the mean quadrate deviations ~I (x,y) - a'Y(x,y) - b')2 and ~; (Q(~{,y) - a"Y(~,y) - b")2. Here (x, y) (x, y) the summation is per~ormed over all the pixels in the cell, and Y(x,y), I~x,y) and Q(x,y) denote the grey level values of the corresponding separations at the pi~el with the coordinates (x,y).

(The numbers a', b', a", b" can be found by standard procedures well known to the skilled person.) , 2097/242~ -10~- 2 0 9 1 9 2 0 c. The basic separation Y is compressed by the chosen compression method. Let Y' denote the grey values after decompression.

d. Then the grey values of I and Q after decompression are given by I(x,y~=a'Y'(x,y)+b' Q(x,y)=a"~(~,y)+b".
Thus, the compressed data for a color picture consists of a compressed data ~or Y, and of four numbers, a', b', a", b" for each cell, to represent I and Q. The example of a color compression according to this scheme is given below.

Picture processing consists in performing on pictures various visually meaningful operations: e~traction of various ~eatures (defined in visual terms), comparison of pictures or their parts, sl;ressing certain visual features and suppressing others, picture enhancement, creation of visual effects (conturing, quantization, etc.), various color operations, geometric transformations (rotation, zoom, rescaling, non-linear transIormations as a "~lsh-eye"), 3D-geometric transformations (perspective projections, etc.), transformations related with a "textllre" creation (i.e.
matching pictures as a texture with 2D or 3D computer graphics primitives~, etc.

-105- 2091~2 ~nY7~H~2 1. In the compression process de~cribed aboYe, picture~ are represented by models, who~e parameters present the (primary) compressed data. These parameters> as described above, have very simple visual meanings. As a result, any picture processing operation defined in terms of a desired visual effect, can be interpreted as a simple operation on the parameters of our models, i.e. as an operation on compressed data. As a consequence, in our compre~sed data structuIe, picture processing is performed much faster than the same operations on the original pictures; the ~ame is true for the memory required.

2. This important property iB not shared by conventional compression methods, like DCT. The rea~on is that the parameters representing the picture in these methods, like the digital cosine transform (DCT) coeff;cients, are related to the visual structure& of the picture in a complicated way.
Therefore, the effect of processing operations on these parameters is difflcult to evaluate.

One of the very important operations in picture processing iB
edge detection, which con6ists in determining the location of abrupt changes of brightness.

In the compression process of the invention, the edges are detected and repre~ented by corresponding models in the course of the compression proce~. Therefore, to show a ~'' , .
. . , .

-lo~ ~0~2~
picture consisting of only the edge r~gions, it is enough to drop all the model~ in our compressed data but the edges.

Our compressed data allows for a much more detailed shape analy~is. The edges or ridges can be classified according to their profile; textured areas can be analy~ed according to the type and the density of the texture elements (usually represented by hills and hollows in our compressed data).

Our compre~sed data comprise~ a very convenient input for a higher level picture analys;s ~picture compari~on, complicated features extraction, computer vi~ion, etc.).

~3harpening a picture (high-pass ~lter). This vi8ual effect can be achieved by increasing the "slope" of all the models involved.
Increasing the brightness parameters and decreasing the width (size) of the models provides the required increasing of the slopes. A low-pass filter ef~ect is achieved by an opposite variation of the parameters.

Much more specific ef~ects can be achieved. For example, all the edges can be sharpened, while the ~mall-scale texture~
(hills and hollows) can be smoothed.

Various visual effect~ can be prodllced. For example, replacement of the u~ual profiles by new ones, like those illustrated in Fig. 44 (a) and (b) leads to a "granulation" ef~ect.

2Q~ 107- 2 ~ g 1 9 2 Once more, if required, this operation can be performed only on the models of a certain type Various color operations require only corresponding transformations of the brightness parameter~ for each color separation, as described above with reference to color pictures compression.

Artificial te~tures of variou~ type~ can be created in our compressed data structure.

Zoo~ and ~ling.
One of the important properties of our compressed data is its scale invariance. I~is means that the picture i~ represented by a mathematical model, which allowa for computation of the brightness value ~(x, y) at any point. This computation is not related to the specif;c po~ition of piYels. Therefore, the scaling and zoom, i.e. the size and proportion of the picture after decompression, are only the parameters of the decompression.
In this sense the rescaling and zoom operations on the compres~ed data do not take time at all.

One of the important consequence6 of this scale invariance is that pictures can be compressed in one standard (PAL, NTSC, .... ) and decompressed into the same or any other standard with no additional processing.

G~Q~i~

2091~0 20~7~11192 Also more complicated transformations than zoom and rescaling can be interpreted as the parameters of decompression. However, in many applications it i8 important to represent a picture after a transformation in exactly the same compressed format as the initial picture.

In order to define this operation precisely, let us assume that a picture A and a screen B are given (not necessarily of the same size). I,et ~: B~A be a transformation which as~ociates to each point p on the screen B the point q = ~ (p) on the picture A.

Now a new picture ~ (A) is defined on the screen B as follows:
at any point p on B, the grey value (or the color) of the picture ~
(A) is equal to the grey value of the picture A at the point q =
~(P).

Intuitively one can imagine that the picture A is plinted on a rubber f~llm. Then the rubber ~llm is stretched by a transformation ~-1 (inverse of ~ ) to match the screen B. The resulting picture on B i5 ~(A).

If we want to represent '~(A) in our standard compression format, using the compressed representation of the picture A
and the transfo~nation ~, we operate in the following way.
1. For each model in the compre~sed representation of A, the geometric parameters are transformed by a linearization of ~.
More precisely, each polygonal line on A with the vertices vi is -log- 2~91~2~

transformed into a polygonal line on B with the ver~ices ~-1~vi) This i~ illustrated in Fig. 45.
2. In the same manner, the central point p of a hill (hollow) on A is transformed into the point ~-l(p) on B.
3. The eigenvectors of hill8 and hollows are transformed by the differential d~-l. Similarly, the curvatures ~ are transformed.
4. Finally, the heights of the hills, hollow~, edge~ and ridges do not change in thi~ tranaformation.
~. The polynomials representing the smooth regions are transformed by the differential of ~ on corresponding cells.

Especially simple form geometric transformations take in the global model structure described above.

1. The models are transformed from A to B as described.
2. For those of the points w of the grid G on B, which are not covered by the support~s of the models, the grey level value~ v(w) are defined as the grey level values of the picture A at the points ~(w) (more precisely, as the values of the global model, representing A).

In fact the procedure des ribed above replaces ~ by its differential, i.e., it i6 based on local linearization of ~. Thi~
inaccuracy is justified, since usually the scale of our model~ is much finer than the scale of non-linearity of ~.

Rotation i8 an especially simple example of a geometric transformation. Here, all the geometry of the models is rotated :llO- 2 0 9 1 3 2 ~

to a corresponding angle, while the ~'g and the re~t of the brightness parameters do not change.

Similarly, zoom and rescaling can be interpreted.

3D-geometric tran~folmation~
Here the picture A is associated with a certain 3D object, and the transformation ~-1 is the projection of the object onto the viewer's screen. This i8 illustrated in Fig. 46.

To produce a picture on the screen in a conventional way, the intersection of the viewing rays through each pixel on the screen with the ob~ect must be computed (as well a~ the grey values (colors) of the picture A at these intersection points -Ray tracing).

The method described above produces from a compressed representation of A and the 3D data the coInpressed representation of the picture on the viewer's screen. It requires computatioll of the ray's intersections with the body only for a very small number of points (vertice~ of polygonal lines and centers of the hill~ and hollows). Thus, the required amount of computations is drastically reduced.

In this way, high quality pictures can be attached to 2D and 3D
graphic primit*es as a texture.

As a result, 3D scenes with a high ~uality realistic texture can be created. It will be possible to produce (with a small amount 2~9~920 of computations) the view of this scene from any given position.
Thus the u~er will interactiqely be able to choose (in real time) his viewpoint, and to "travel" in~ide the te~tured scene created.

In particular, a sombination of a 3D terrain information (digital terrain model) with a high quality picture of a corresponding landscape will allow one to interactively create a view OI this landscape from any desired po~ition. Thiæ
possibility can be u~ed in entertainment and adverti~ing applications.

Variou~ additional effects in computer graphics, requiring long computation in conventional data structure (texture reflection, shading, etc.) can be performed very fast, if our compre~sed picture representation i8 used.

Our compres6ion process for video sequences (~ee below) preserves the abovementioned properties. Therefore, all the operations described above can be applied to compressed video data. Since the~e operations are vely fast (being performed on compresæed data), the real time interactive video manipulation and processing becomes feasible.

1. Generally, moving ssenes are represented by sequences of still pictureæ (called frames), reproducing the ~cene in fixed time interval~.

20~1920 Z~97~2 Various standards are used for moving scene~ representation in TV, video, cinema, computer animation, etc AB an example, we shall discuss below video ~equences consisting of 480x720 pixels RGB still pictures, representing the scene in intervals of 1/30 sec. (30 frames per second). However, the compression method described below can be equally applied to other standards (see, for example, discussion of interlaced frames compression below).

In principle, video sequences can be compressed by compressing each frame separately. However, this approach presents two problem~:
a. Compression of still pictures usually involve~ various discontinuities filtering of models and parameters according to certain ~resholds, quantization, etc. For a single still picture these discontinuities causé no visual problem. However, in a sequence of frames, the jumps in grey values caused by these discontinuities lead to a serious quality degradation ~"flickers", since they usually change from frame to frame in a completely unpredictable way.).
To avoid this effect, the compression must provide a ce~tain continuity in time.
b. Usually, subsequent fraInes in a video sequence are strongly correlated, being the images of the same scene in a very short time interval. This correlation~ which promises a much higher compression ratio for video sequences than for still images, is not utilized by a ~rame by frame compression~

113- 2091~0 Therefore, one expec~s a good video compression proces~ to provide a desired continuity of the compressed pictures in time and to utilize the correlation between the neighboring frames.

Great ef~orts have been made in developing such processe~ (see MPEG documents, in publications of the International Standard Organization - ISO, ISO/IEC, JTC 1/Sl, 2/WG8). In particular, various "motion compensation" methods have been developed to utilize the fact that some part~ of the adjacent frames are obtained ~rom one another by a certain replacement. However, these rnethods reveal only a small part of a similarity between the neighbo~ng video ~rames. Indeed, only nearby part~ of the scene are moving in a coherent way.
Usually various objects are moving in dif~erent directions, including the objects in the finest scale (leaves of a tree under the v~ind, waves on the water surface, texture details on two overlapping objects moving in different directions, etc.).
Typically, brightness of the picture can be changed from frame to frame gradually, or sharpness of the frame can change as the result of refocusing of a camera, etc. The~e types of frame evolution, which certainly present a strong correlation between the neighboring frame6, ca~ot be captured by conventional "motion compensation" procedure~. (We shall call below these type~ of evolution "generalized motions".) As a re6ult, the e~isting video compres~ion methods utilize only a small part of interframe correlation, give a low compression ratio and encounter serious quality problems.

20gl920 2. We propose a method for video sequences compres~ion, based on a still compression scheme, described above. Its baaic advantage is that representation of the adjacent frames in our compressed data structure reveals much deeper similarity between them than in any conventional method. In fact, experiment~ show that in video sequences, representing natural moving scenes or resulting from animation, scientific visualization, etc., the neighboring frames can be represented by essentially the same models, with slightly different parameters.

Thus in our method the neighboring frames are always obtained from one another by a "generalized motion" - i.e., by a variation in time of the parameter~ of the same models.

Therefore, to represent several neighboring frames in a video sequence, it iB enough to compress one of them (i.e., to represent it by our models) and to f;nd the variation of the parameters of these model~ in time.

Of course, this representation will be visually faithful only for a small number of frames, BO we have to repeat this procedure several frames later.

3. Let us give some examples of a representation of a generalized motion by our models.

a. A usual motion of a part of the picture. In this case, the "position parameters" ~coordinates of the ver$ices of the 20sl~a polygonal lines for ridges and edge~ and coordinate~ of the centers for hills and hollows) of our models are described in a fir6t order approximation by ~(t) = x(O) ~ vl-t (*) y(t) - y(O) + v2 t where v = (vl, v2) i8 the motion vector of this part of the picture.

b. In a case where different model~ are moving in dif~erent directions, we have the ~ame representation (*), but the motion vector changes fi~om model to model.

c. In some ~ituations, dif~erent vertices of the ~ame model are moving in dif~erent directions. For example, (see Fig. ~0).
for a curvilinear structure in Fig. 50, vertices A and B are moving in opposite direction~. In such æituation~, for each vertex its own motion vector i~ def;med.

d. Gradual changes in sharpneas. Here, the "widths" of the models involved decrease and their slopes increase iD time. In a first order approximation, this evolution can be described by the same e:epression (*).

e. Gradual chaDges in brightness or color - the aame as above, but for brightnesæ and color parameters of the models invclved.

Uaually a Plr~t order approximation (*) of the param~ters evolution in time provides a visually faithfill representat;on of video sequences for 3 to 6 frames or more.

209~ 920 20~7/2424~2 4. The compression method as above provides a 6erious additional compression factor (in comparison with a still compression). Indeed, for each 3 to 6 frames we need to memorize only the parameters of one frame and the corresponding generalized motion vectors, which have exactly the same structure as the corresponding parameters. Thus, the data to be stored iB leBS than twice that of a still compressed picture. Since this information i8 enough to reconstruct 3 to 6 frames, we get an additional factor of 1.5 to 3.

5. However, mo~tly one does not need to memorize the generalized motion vector~ for each model separately, since the parameters of different models usually change in time in a coherent way. Utilization of this coherency we call "generalized motion compensation". It is performed in the following way: We analyze the individual generalized motion vectors for each model and try to ~nd correlation~ between these vectors.

For example, on certain blocks OI the picture we repre~ent these vectors in a ~orm v;= V+ri~
wh~re v is an average general;zed motion vector over all the models in the block, and vi are the correc~ion vectors for each model.

Usually on relatively amall blocks the corrections vi are negligible, and then only the global generalized motion vector "v" must be memorized per block.

.

.

2091~2~

2~97~2 6. In 60me situations models can appear, disappear or change their type in the process of their evolution in time. Such change6 cannot be described by variations of the parameters.
However, most of these situations can be covered by simple mathematical model6 (time-depending), which we call "bifurcations".

For example, a ridge component can be split into two parts (see Fig. 48).

Topology of a curvilinear structure can be changed ~see Fig.
49).

To represent such situations, we memorize the type of a bifurcation model involved and itæ parameter~.

7. The general compression scheme above can be implemented in various ways: as the prediction of the following f~ame on the basis of the previous ones, as an interpolation (not of grey values, but parameters of the models), or as a combination of these and other methods. Below we describe in more detail one specific realization of the 1 nethod.

8. The specific realization iB ba~ed on an e~plicit compression of some subsequence of the control frames and an intrerpolation of the models parameter6 from the control frames to intermediate ones. Thus, for a seguence Fo, Fl, ....
F; of frames, we define the frarnes Fo, Fs, F2s~ F3S, --- a~ the llg 2~92~
2~97~9~
control frame~ and the regt as the intermediate ones. Here B iB
usually between 3 and 6.

a. Each control frame is compressed as a still picture, as described above.

b. Consider a certain control frame Fj8. For each model q~, representing this frame, we estimate the motion of it~
geometric "skeleton" (which consiets, by definition in the underlying component for edges and ridges, and in the central point for hills and hollows). Below, this operation is described in more detail.

c. On the basis of a motion estimation obtained in stage b, we predict a position of a model q~ considered (or of its geometrio skeleton) on the next control ~rame Ftj+1)s (see Fig. 47).
d. We try to "match" the predicted model position with one of the models actually detected on a frame F(j+1)s. This i~ done by ~alculating the "Hausdor~' distance of the predicted component (center to the actually detected ones and choosing the neare6$ component ~ . If its distance to the predicted component (center) is less than a certain threshold, we "match" the model ql~ on the frame Fjs with the model ~' on the frame F(j+1)s. Hausdorf distance of two sets A and B is defined as max(x in A) min(y in B) distance (~,y).

Now we construct a representation of intermediate frames by the models: for each pair of matched models q~ and q~3'on the control frame~ Fjs and F~j~1)s, we define a model q~"of the 2~Y7n~æ -1~9- ~ 0 919 2~
same type on each frame between Fjs and F(j+l)s, with all the parameters obtained by a linear interpolation OI the parameters OI ~ and ~'.

f. For each unmatched model ~ on Fjs we define a model ~ of the same type on each frame between Fjs and F(j+l)s vwith parameters, obtained by an extrapolation from the previous control frames, and with th~ brightness parameters multiE)lied by a factor, linearly decreasing from 1 to O from Fjs to F(j+l)s.

For unmatched models on F(j+1)B, exactly the same procedure i8 performed, but in an opposite time direction.

g. The background or ~mooth regions data (approximating polynomials or values at the grid G-point~ - see above) are linearly interpolated from the control frames to intermediate ones.

h. Finally, the color parametera (see above) are al~o linearly interpoiated from the control ~rames.

At stage b above, we estimate the motion of each component (for ridges or edges) and each central point for hills and hollows. Thi~ is done as follows:

a. For ridge components, at the center of each segment in this component we compute (on a 3-dimentional cell of the size nxnxn) a polynomial P of degree 2 of three variables, providing the minimal square deviation from the 3D - grey level distribution, formed by the subsequent frames7 where the time t is considered as the third coordinate. Here n is measured in pixels in the frame directions and in frame numbers in time direction. Its typical values are between 4 to 6. Then the quadrated part of P is transformed to the main axis (diagonalized). We define the plane of a motion of the segment as the plane, spanned by the two eigenvectors of P, corresponding to the smallest eigenvalues (Note that a segment was spanned by the eigenvector, corresponding to the smallest eigenvalue in a two-dimen6ional case.).

Then the motion plane of the component i8 defined by an averaging of the motion planes of its segments.

Finally, if between the motion planes of the segments are pairs, forming sufficiently big angle between them, we define the motion vector of the component as t~e average of the vectors corresponding to the intersections of the motion planes of these pairs of segments.

b. For edge components, at the center of each edge element, forming this component the same polynomial P of three variables as in "a", is constructed. The motion plane of the edge element is defined as the tangen~ plane to the level surface of P. The motion plane and the motion vector ~or the component are defined from the motion planes of the edge elements in the same way as in a for ridges.

~ , 20!~9~
~s7n~
c. For the central points of hills and hollows, the same polynomial P as above is computed and diagonalized. The motion vector of the center is defined as the eigenvector of P, corresponding to the smallest eigenvalue.
For interlaced sequences the corresponding 3-dimentional array iB con~tructed by subsequent half-frame~ (each on its corresponding pixels). The approximating polynomials is then constructed on a cell of the si~e n pixels x n pixels x 2n half frames. The rest of the compreBsion proceBB iB performed as above.
The compressed data according to the procedure described above, consists of:
- compressed data for each control frame;
pointers of watching, showing ~or each model on the control frame the watched model on the next one (or indicating that the model is unmatched~.
Since the pointers require a relatively small percent of a compressed data volume, the additional compression factor here is approximately s - the distance between con~rol frames.

Usually, almost all the pointers can be eliminated by the following procedure:
a. Using the compressed data on the two previous control frames, one can predict a motion of each model, and its position on the next control frame. Usually, this prediction is accurate enough to find the neare~t actual model, which is a matched one, thus eliminating the necessity of a pointer.
b. Moreover, the same prediction can be used $o simplify encoding of the ne~t control frame - for each its model only -12~ 209~2~
2~97~42~92 corrections of the parameters with respect to the predicted ones must be memorized.
c. A "generalized motion compression" as described above can be applied.

l~oom~s~on For each control frame its compressed representation is stored in a compressed data. For each intermediate frame, its (compressed) representation is constructed, a~ described above. Finally, each frame iB decompressed as a still image.

9. Operatio~s on co~pres~ed data All the properties, which allow for image processing operations to be performed on a compressed data ~or still images, are preserved in a video compression scheme described above. Therefore, all these operations can be performed on a video compressed data. Moreover, they become relatively even more eff!ective, since they must be performed only on the control frames: the interpolation procedure described above automatically extends them to the intermediate frames.

10. The video compression procedure, described above, is computationally effective, for the following reasons:
a. Only the control frames are entirely compressed as still images (Note that all frame~ can be analyzed to provide a better detection.).

123- 2091~2~

b. The most computationally intensive part - motion detection -iB per~ormed only for components and centers - i.e., for a significantly reduced data.
c. Finally, the "motion compensation" part i8 performed on a compressed data.

The results obtained by the process according to the invention and its stages are further illustrated by FigB. 51 to 67.
The following pictures repre~ent the re~ults and ~ome intermediate steps of a compression of still image~ and video sequences, performed acsording to our method. All the operation has been performed on a SUN sparc 1 workstation.

1. Figure 51 repre~ent6 the original RGB still picture of the size approximately 480x720 pixels.

Figure 52 repre~ent~ the same picture after compre6sion by our method to 1/50 of the initial volume and decompre~sion.

Fi~re 53 represents the same picture after compression to 1/40 by the standard JPEG method and decompression.

2. Figures 54, 55, 56 show the stages of analysis of the 48x48 pixel~ region, indicated by a black square in Figure 51. They represent thi3 region at an enlarged scaleO

Each white point represents a pi~el. White star6 on Figures 54 and 55 repre~ent the points of the basic grid in the region Al.

2091~2~
20~7~24~332 The blac~ and white segmen~s detected in the region A3 are shown by 6mall intervals of the co~Tesponding color. Hills and hollow~ are shown by ellipse, and the gridpoints in the region A'2 are shown by the triangle~ (Figure 54).

The green intervals on Figure 55 represent the edge elements detected.

Figure 56 represents the r~sult of the edge-line procedure, as described above.

3. Figure ~7 represents the original of the 48x48 pixels region, its represexltation of the global model before quantization and after qsantization (small pictures from left to right). Big pictures represent the same picture~ in a 1:5 zoom form.

4. Figure 58 represents the re6ult of a picture processing operation, performed on a compre~sed form of the above picture:

The width of the models has been enlarged and their brightness has beeIl decreased.

The zoom, repre~ented on Figures 57 and 58, give~ another example of an operation on compressed data. It has been performed by changing the decompression parameters, as described above.

2097/2424~92 5. Figures 59, 60, 61 represent three color separations (R,G,B) of Figur0 52 compressed and decompre~sed by our method.
The R separation is compressed according to the complete process described above. Separations G and B are represented through R, as described hereinbefore.

6. Figures 62-67 represent the video sequences compression.

Figure 62 represents 10 Irames of a videosequence (on a 48x48 region). The upper line i8 the original sequence, and the bottom line is the sequence af'cer compression (1:150).

Fi~res 63 and 64 show the results of analysis of the frames 4 and 7 (control frame~) as still pictures (the arrows represent the endpoint~ of the ridge components detected).

Figure 65 6hows (by yellow and blue components) results of the motion predicted for the components ~om the frame 4 on frame 7.

Figures 66 and 67 show the interpolated positions of the components (blue lines) agains~ the real position (yellow lines) on the intermediate frames 5 and 6.

In another embodiment of the invention, in which the ba~ic 6tructures are nct identified as in the previous embodiment, the following operations are performed:

-12~ 2 ~ 2 0 2~7/2424~D2 A p;cture i8 always considered as an array of pixels, Pjj, of various ~izes, for example 480x700, viz. 1 s i ~ 480 and 1 ~ j ~
70Q. This array is assumed to be contained in the plane with coordinates x,y. Thus each pixel has discrete coordinates x,y, though the coordinates themselves are considered as continuous. The grey level brightness distribution z = ffx,y) assumes at each pixel Pjj the value zjj= ftxj,yj), where the values of z vary betweeIl 0 and 2~5. A colour picture i~
generally defined by three intensity functions R(x,y), G(x,y) and B(x,y), each assuming values between 0 and 2~5, or by equivalent expressions obtained by the transform coding methods hereinbefore mentioned.

Considering now a single cell, one of the possible implementation processes according to the invention iB carried out as follows. The simplest basic element type is chosen and the corresponding model's parameters are determined by minimizing, by known minimization routine~, the devia$ion thereof from the actual object contained in the cell. As a measure of said deviation it is convenient to assume what will be called the "square deviation", viz. the sum of the ~quares of the differences between ghe values which the function z=f~x,y),defining the object, has at the various pixels and the corresponding value of the function q> defining the model, viz.:
~(f,~) = ~; [f~x,y) ~(x~y)]2 iB minimized for each cell with respect to the parameter~ of ~, e.g. by a standard minimization routine such as those from the IM~;L library. If c~ is not greater than a predetermined threshold value T, the model is assumed to represent the object -127- 2~9~92 20Y7/2424~'92 and the processing of the cell is stopped. The threshold value T
may vary ~or various applications, but in general it iB
comprised between 5 and 1~, and preferably is about 10, in the scale of the z or RGB values. If ~ > T, the procedure is repeated with another basic element model, and if none OI them gives a small enough square deviation, the procedure is repeated with a model which is the sum of all the previously tried models. If, even thus, the square deviation is greater than the threshold value, the scale L is decreased. Experience has shown that, if the characteristic scale L is small enough, every object can be represented by a few basic element models.

Each model or combination of models is identified by a code number. Said number and the parameters of the model or combination of models assumed to represent the object of the cell, constitute the "primary compression" data relative to said cell.

At this stageJ part of the primary compression data may be omitted and another part simplifiled, depending on their psychovisual contribution. Thus some small structures may be neglected, some others may be approximated (e.g. ellipsoidal ones by spherical ones), etc.

After the said secondary compression, quantization may be carried out. "Quantization" means herein using only a number of the possible parameter values, e.g. appro~imating each value by the nearest among an array of values differing from one another by a predetermined amount, e.g. 0, 32, 64 etc, 9 1 9 2 ~
20~71~24/EI/~2 thus considerably reducing down from 2~6 the number of possible values.

At this stage, the correlations between parameters relative to di~erent cells may be taken into account, whereby larger models, which extend throughout regions that are bigger - e.g.
2-3 times bigger - than a single cell, may be defined. This further simplifies the compression data by extending the validity of certain parameters to larger areas. E.g., the ~ame polynomial may represent a smooth curve extending through several neighbouring cells, curvilinear structures extending through several neighbouring cells may form regular nets, and the like. Small corrections on a single cell level may be required and stored.

All the aforesaid appro2~imation method~ may lead to discrepancies between neighbouring cell6. These discrepancie~ may be smoothed out during the decompres~ion stage, during wkich the basic elements of the same type (e.g.
smooth regions, curvilinea~ regions etc) are smoothed out separately.

A particular application of the invention is the compression of TV pictures. It has been found that in any such picture, one can find certain smooth regions, curvilinear 6tructures and local simple objects, such that:
a) the number of ~uch basic elements in any standard cell is small, usually 5 or 6, the value of L being~ as stated hereinbefore, preferably about 12 pixels;

-1~9- 2~192~
20~7/~12 b) any array of models, which represents ~aithfully (in a visually undistinguishable way) each of the above elements, faithfully r0presents the whole picture.

Rather high compression of IrV-pictures is thus possible: the models contain a small number of parameters; each of them must be defined by at most 256 values.

An example of the implementation of the process according to the invention vwill now be given.

The picture to be represented by compression is a colour picture, as shown in Fig. 68 or Fig. 70. The following basic elements and models are uæed:

Model: smooth regian - z = q~ 1 (x,y) = P1 (x,y) aoo+alox+aoly+a2ox2+allxy~ao2y2+a3o~3+a2l~2y+al2xy2+ao3y3.
This model has 10 parameters.

2 - Model: curvilinear structure - z = ~P2(x,y) To define this model we use an orthonormal system of coordinates u,v that iB rotated by an angle ~ counterclockwise with respect to the system x,y. The central curve of the "line" is given by the equation v = r+ku2. The model is defined by:
= ~S2 (x,y) = Zl = poo+plox+poly for Y 2 r+ku2~h Z2 = Poo +Po1 x+pO1 y ~or v S r~ku2-h Z3 = tz~ t)z2, where t - (v-ku2+h)/2h, for r~ku2~h > v > r+ku2-h, 209~92~

2W7/24~2 and z = ~2(x,Y) = ~2 (x,y) + cA, where A i~ the "pro~lle funcgion", equal to [(2t)2-1]2, with t as above, for 0 ~ t s 1, and A
- equal to 0 othervise~
Thus the model ~2 is completely determined by the parameters H~ r~ k~ h~ poo, P1o, Po1, Poo, Plo', Po1', c - altogether 11 parameters.

3 - Model: simple local object~ - z = <~3(x,y).
Firstly we define a "supporting ellypse" by the angle ~ which the short semiaxis makes with the x-axis, by the coordinates x~), yO of the center, and by the values rl~ r2 of the semia~is.
The (P3 is defined as ~ -~3(x,y) = c(u2/rl2 + v2/r22), where u and v are the coordinates of khe point (x,y) in the coordinate sy6tem (u,v) hereinbefore defined. Here ~(s) has the form (s2-1).
Finally, the model generated by the objects contains "l" objects, with 1<1 ~ 6. The value of z that is finally obtained is the sum of the values of q)3 for all objects.
UBUa11Y, one stores the coefficients of the orthonormal system (u,v) rotated by the angle 9. Thus the model i5 characterized by:
1 - the number of objects;
2 - for each object, the coordinate~ xO, yO, the angle ~, rl < r2, and the coefflcient c of each object in ~he linear combination.
3 - thus the model on a cell of the third type is given by Z = qO ~ qlg ~q2Y + C1~31(X"y) + .. ~C1~3l(x,y) It is more convenient to store the coeff~lcients q and c in a transformed form. We consider the functions 1, x, y, (p31, .....
~31 as vectors in the inner product space of all the functions z =
g(x,y) on the pixels of the cell under consideration. Then we .. .

-131- 2~9~92~
20~Y7/~24/HU~2 apply the Gramm-Smidt orthogonalization procedure to the vectors and store the coefficients of the orthogonalized system.
Those are the coefficients given in the Tables that will follow.
In the following examples L = 10, the cells are squares 10x10.

Fig. 68 shows an original picture - a landscape - to which the invention is applied. The procedure hereinbefore described has been followed. Table 1 shows the grey level values of a 40x40 region of the original,which is marked by a black dot in Fig.69, which shows the decompre~sed reproduction. It is seen that this latter i8 quite undistinguishable from the original. Table 2 lists the grey levels of the models of the same region. Table 3 lists the same data of Table ~, but after quantization of the models. Table 4 lists the pararneters of the models, which are of the type 3 (simple local obJects) described above, indicated by code H, which consitute the compressed data. Each group of success*e 5 rows contains the coefficients of one cell. Table ~
lists the same data as Table 4, but afcer quantisation. Fig. 70 showæ aIlother original picture - a girl sitting at a desk - and Fig. 71 the decompressed reproduction. Once again, the two images are undistinguishable. Tables 6 to 10 respectively correspond to Table 1 to 5 of the preceding example. However it is seen in Tables 9 and 10 that models indicated by E and T
have been used: these respectively indicate models of the type 2 (curvilinear structures) and 1 (smooth regions) described hereinbefore.
While particular embodiments have been described by way of illustration, it will be understood that they are not limitative and that the invention can be carried out in different ways by -13~ 2Q91~20 2~gql24~
persons skilled in the art, without departing ~rom it~ spirit or exceeding the ~cope of the claims.

..
' ' .

2091~0 Table 1 125 57 58 89 74 5~ 64 57 50 64 73 66 72 78 74 75 76 95 195 45 145 177 65 95 153 ~ 18q 240 171 183 198 157 75 S9 113 84 97 137 l22 166 175 186 193 171 112 112 18?3 174 147 188 190 191 199 182 146 73 38 8q 155 126 100 207 161 44 119 208 202 197 221 195 124 132 156 116 81 86 185 176 173 179 192 1g3 127 90 165 198 159 155 138 109 92 69 96 151 139 67 34 85 139 158 118 50 ~2 218 227 16q 162 159 110 103 124 109 114 87 : 91 158 155 141 124 120 155 172 161 168 188 197 201 168 85 63 135 161 82 185 182 154 113 114 148 180 190 151 79 70 141 184 192 171 84 67.117 99 149 212 120 85 1q4 162 182 201 181 186 193 166 158 135 93 74 80 87 56 182 179 179 166 164 190 20~ 166 116 83 116 184 1.85 189 201 127 52 63 148 .205 179 109 102 176 161 150 207 158 128 122 75 77 62 60 69 44 49 35 39 1~3 189 178 185 190 206 209 145 148 196 136 137 204 174 186 214 147 66 56 1~3 211 118 81 147 187 135 14'i 166 77 65 110 83 45 65 '94 59 46 42 47 189 191 193 208 200 152 ~ 188 153 11?3 195 192 184 ~24 156 71 54 121 154 77 96 183 145 85 111 131 ~1 79 142 145 143 133 62 87 108 51 113 ' ~ . ' ' .

-~3~- 2091~2~
~r~lble 1 ! cnt d ) 166 196 1~36 160 128 97 6~ 6q 160 150 104 195 200 182 204 117 69 63 57 -~9 90 150 159 71 69 93 66 60 95 ~.66 162 116 121 105 140 188 90 50 /0 128 117 90 122 99 59 1i7 183 120 100 172 201 130 86 99 138 115 53 109 146 lOf3 99 59 103 14fi 50 33 88 136 146 81 93 165 199 199 128 57 3.~ 69 74 43 49 82 94 112 160 1~4 190 168 119 158 183 79 38 123 185 137 115 16~ 105 49 97 111 122 137 74 Jl 71 100 104 146 182 192 188 146 108 96 71 99 146 178 165 154 154 11~ 56 72 127 83 95 188 165 108 91 105 153 165 104 130 175 190 210 194' 154 159 186 110 48 105 137 167 186 107 119 190 113 1~2 196 172 lg3 186 170 168 156 143 120 133 115 48 115 211 212 199 116 74 137 lq7 141 191 181 182 194 184 190 188 18~ 189 172 125 47 87 207 229 178 121 117 169 190 186 17'7 135 104 71 75 123 151 120 72 81 101 143 204 188 178 187 176 183 180 181 1~36 182 187 1~5 1~0 203 204 158 66 69 189 225 142 121 166 182 1~9 189 20~3 203 161 69 66 162 120 43 106 157 103 108 191 192 175 204 190 179 '84 17'~ 1~5 183 1~3 186 1~7 199 214 195 109 50 111 160 102 113 185 180 -135- 20~2~
Table 1 (cnt'd) 1~3 190 206 210 158 92 137 164 65 85 190 193 147 100 144 210 188 209 202 171 198 186 186 196 182 l9S 196 198 191 173 143 61 33 62 49 53 lol 12g 12~ 164 19~ 193 193 172 125 131 17:3 lo.~ 85 183 187 172 171 99 llo 194 192 180 188 ~4 89 187 169 144 101 137 190 108 60 151 200 164 154 155 98 58 119 169 147 15g 95 107 66 64 113 89 66 156 l9S 171 190 187 187 197 187 175 133 98 91 89 86 55 55 123 172 142 ~6 110 180 187 191 215 179 103 116 185 118 68 144 139 121 101 115 127 lol 110 174 194 185 201 216 197 114 88 147 152 172 105 127 125 ?1 119 170 162 190 185 175 190 177 185 189 189 187 175 191 196 188 148 96 139 201 181 184 192 175 187 184 180 187 1~33 194 172 137 168 201 ~8 209 187 215 175 125 159 1~2 ~1 175 222 133 129 221 218 204 235 224 182 15~ 192 -136- 2~9~2~

Table 1 ( cnt ' d ) 19S 17~ 183 177 186 181 184 183 179 184 187 188 171 lS0 153 155 110 gl 165 ~07 lY8 213 ~ 0 78 125 178 218 201 133 166 220 162 176 228 189 142 68 149 1~6 145 106 q7 40 118 211 227 162 129 198 228 146 109 199 159 102 185 179 18~3 183 186 185 180 184 180 182 185 184 188 194 191 169 lSS 156 141 182 173 182 186 188 191 181 189 184 180 188 182 187 lg2 192 l9S 192 198 202 147 61 36 45 52 122 164 lOS 74 69 42 72 126 119 139 204 192 132 81 9g 176 -137- 2 0 9 1~2 Table_ 94 8~3 94 ~5 86 65 J5 25 ~0 7~ 66 66 67 67 68 69 69 70 71 71 120 124 g9 75 81 157 249 205 164 174 llS 52 36 68 121 142 126 110 111 122 138 146 140 116 7~ 67 127 130 118 141 149 139 116 88 73 73 74 75 120 126 131 109 86 102 181 216 178 175 188 128 86 81 lOS 126 112 96 131 151 173 186 180 155 113 100 165 17~1 130 173 209 222 202 163 120 85 77 78 121 127 133 139 133 74 119 210 207 175 193 177 lSl 129 121 114 98 82 145 166 190 204 199 173 130 111 18~ 199 91 143 188 200 177 146 121 96 82 150 167 187 198 196 181 128 lOo 169 202 145 171 164 129 91 82 101 152 121 88 122 153 189 142 90 85 136 172 192 177 lSl 136 119 103 106 129 150 96 91 ].22 178 183 130 107 127 159 165 172 177 171 174 135 so 101 137 160 82 98 123 129 135 141 147 153 159 165 172 178 142 163 136 75 58 92 100 3g 57 1'5 200 186 173 159 151 186 2~2 166 103 78 103 157 205 211 169 113 99 lOo 101 lol 124 130 136 142 148 154 160 166 172 178 95 86 69 47 3~ 48 55 17 164 189 202 204 l9S 179 155 1:38 138 153 161 176 191 206 200 139 85 78 129 220 108 109 145 171 115 111 lil 112 112 113 102 101 100 9'3 98 97 96 95 94 9:~

172 111 llg 189 lss 92 9~3 1~)4 104 115 116 99 98 97 96 ss 92 88 92 -138- 20~192~

Table 2 ( cnt ' d ) 160 17S 1~0 158 123 45 90 11() 145 167 144 159 174 189 187 lZ8 78 59 6g 135 114 196 lS~ 76 73 10-~ 74 59 103 118 96 g5 94 99 132 149 94 50 81 ~37 115 126 115 as 68 68 95 1 ~3 161 173 135 150 165 152 10~ 101 110 82 70 78 fls 73 65 75 112 155 178 1.77 180 196 169 161 73 38 111 168 119 91 108 101 ~5 79 110 121 121 79 29 62 123 90 105 166 210 204 lS0 96 99 81 87 87 94 131 178 204 205 l9q 188 185 207 126 45 60 155 199 158 88 80 113 133 163 169 159 lq2 116 71 84 126 87 115 151 153 119 84 110 140 140 10~

88 100 10~3 120 148 175 187 185 177 182 183 161 76 54 120 184 190 129 72 99 1~5 164 193 209 207 187 157 125 120 129 84 87 98 108 105 89 125 170 lOg 121 133 145 156 163 162 159 157 160 80 91 113 128 123 90 102 16~ 202 ~17 132 132 133 133 134 134 135 135 136 136 80 132 191 212 182 119 70 87 14~ 174 181 182 173 161 152 148 138 125 118 65 85 168 216 187 118 108 120 166 177 1~35 189 185 184 182 182 176 159 137 54 104 203 247 208 133 138 166 196 19~3 178 130 94 8~3 107 102 97 110 107 74 104 158 164 171 183 190 190 1.80 179 185 191 190 192 lg7 202 204 193 169 53 98 181 211 169 122 155 187 192 l9S 197 194 161 109 81 g5 94 116 128 125 153 166 171 184 196 202 198 184 184 18~ 189 1.89 1~5 190 199 2~9 206 189 63 65 110 123 98 102 145 173 18r3 1~3 -1~39- 209~2~

Table 2 ~ cnt ' d ) 1~3 1~1 19q 185 153 114 142 llS 93 lZ7 172 173 132 125 169 19() 200 202 194 176 1~JO 1~5 185 186 182 174 179 190 194 180 83 49 44 46 61 89 118 136 142 lJq 180 168 14~ llS 129 201 125 84 125 172 180 179 168 125 77 96 159 169 156 138 160 128 121 1~2 171 177 172 ~12 61 86 146 144 157 103 70 88 108 116 121 1~9 9S 97 132 144 118 135 182 181 176 214 198 174 152 138 133 133 131 162 150 100 1~2 189 195 192 187 183 174 184 186 187 189 189 173 146 178 186 199 184 176 162 156 150 175 206 194 196 122 99 64 100 178 213 21.3 196 163 199 l~S 196 176 149 143 170 176 144 118 130 113 126 178 208 222 226 211 183 187 197 ~lS 204 170 1~8 142 129 1~1 171 193 149 161 189 210 221 222 208 189 1~3~ 168 ~091920 Table 2 (cnt'd) 1/6 177 177 178 179 1~ 181 182 183 184 185 186 188 179 152 lS0 164 126 84 149 1~ 215 220 179 118 77 107 179 230 223 149 166 189 200 198 191 179 175 178 179 180 181 182 1~33 184 185 186 187 185 187 188 188 175 154 139 127 70 92 1`18 157 143 104 53 34 103 187 219 180 149 163 179 180 164 145 137 143 181 182 1~3 184 185 186 187 188 189 190 185 187 188 190 191 182 165 152 122 184 185 186 187 188 189 190 191 192 193 185 187 189 190 192 193 l9S 195 191 -141- ~09~920 'l~dble _.3 89 76 67 69 67 54 ~1 9~ 95 78 65 67 68 69 70 71 73 74 75 76 124 l29 118 ~33 77 1~.1 2()9 ~04 161 168 84 40 39 7~ 121 139 123 108 92 102 118 123 110 80 1~2 145 120 80 108 127 126 105 80 75 76 78 79 126 130 135 12~ 85 7~ 157 219 172 169 156 97 63 65 92 118 110 94 79 h3 11.5 llS 139 164 173 160 126 159 193 153 85 143 205 235 220 171 117 81 80 81 1~'7 132 137 141 145 118 86 203 216 170 190 165 129 106 103 108 97 80 o3 50 128 135 164 193 204 191 157 196 225 172 104 130 lSl 164 152 123 99 82 83 q6 45 153 148 166 186 193 181 151 191 237 201 169 165 13~ 104 83 84 95 185 153 166 154 154 162 .164 151 125 207 255 193 142 123 97 87 86 87 141 220 137 92 132 1 j7 142 147 137 147 161 166 171 176 202 184 135 91 102 128 85 52 94 134 1~8 143 148 153 158 163 168 172 177 208 199 1S2 90 84 101 55 35 192 179 167 154 141 158 193 184 lS0 111 86 92 159 212 192 122 94 95 96 100 136 141 146 151 156 161 165 170 175 180 ~.07 104 82 51 34 20 0 0 148 178 199 209 209 200 18~ 163 150 148 171 181 192 202 178 119 74 70 113 ~33 79 174 200 216 2~0 210 184 158 143 143 1.54 165 175 186 196 190 1~6 82 61 88 j 74 75 -14~
~able 3 (cnt'd) 209192~
.

1.5~ 177 1~8 178 ~S0 124 11~ llg 1~1 162 158 169 l7g 190 184 130 106 7Z 8 135 106 138 17~ 132 10~ 110 ~8 57 110 113 102 99 95 92 g6 138 136 7~ 72 llZ 126 122 ~g 79 7q 91 127 iS8 1?1 152 163 173 160 lOG 139 140 100 ga 131 113 156 152 112 113 114 82 1~ 83 117 98 95 92 117 197 212 laO 75 73 ~ 6~ 58 6~ ~6 lqO ~68 171 180 146 157 15~ ~9 7~ 184 176 13~ lZ2 lq3 140 171 151 141 131 121 10~ 40 80 121 g5 91 9~ 159 1~ 131 7a 71 67 6~

78 79 74 78 108 162 199 201 188 189 lqO 1~0 91 a2 129 193 198 177 137 130 157 17~ 190 193 183 16~ lal 107 113 1~4 91 87 86 93 79 7~ 71 53 10~ 79 87 g2 98 llZ 152 191 206 20~ 186 19~ 134 132 70 103 176 1~6 lg7 1~6 100 1~8 144 17Z 195 208 207 lga 170 145 128 128 87 83 80 8~ 87 73 76 133 ~5 106 lOg 10~ lZ2 136 136 123 118 133 128 121 118 lS9 170 180 127 83 133 212 12~ 139 159 176 l~a 180 167 1~9 13~ 131 83 80 1}~ 160 1~1 113 73 139 1~ 116 119 114 106 100 89 ~0 81 8g 121 132 1~2 15~ 1~3 13~ 90 124 2 216 127 128 129 13~ 1~0 1~2 1~0 136 13~ 135 79 g~ 17~ 224 207 133 6~ 10 ~54 153 1~3 125 13~ 1~6 lg6 139 130 122 117 ~ 17 llS 126 13~ 1~7 lSS 130 147 189 199 210 131 132 132 133 13~ 135 136 137 137 13~ 75 106 176 20$ 17~ 9~ 55 55 93 ?7 112 175 208 201 178 151 131 1~3 163 98 Sl Sl 95 144 1~7 1~4 139 131 lB~ lBl 177 174 170 167 163 160 156 153 S~ Sg 152 2~5 254 170 116 128 180 120 7~ 7~ lZ7 168 15Z 124 107 lOS 12~ 65 ~6 76 131 l~g 162 163 1~8 146 187 1~4 180 177 173 170 156 163 159 156 50 61 163 2~ 240 177 1~8 ~ 82 lgO 175 ~

19~ 196 177 127 90 81 100 106 95 1~1 109 73 85 lZ9 16~ 173 182 18~ 176 160 190 187 18~ 180 176 173 169 166 162 1~8 57 5Z 123 175 16~ 148 182 209 ~16 1~7 191 193 19~ l9G ~ 69 113 78 ~0 9B 110 lZ7 111 118 162 175 186 195 19~ lB5 165 193 190 186 183 17g 176 172 lG8 165 161 75 ~ 61 82 85 128 176 197 -14~ ~9~920 T~ble 3 ( cn' ' d ) 188 l90 792 190 170 135 113 95 102 129 168 1?3 141 109 151 190 156 19~ 781 160 1~6 19~ 189 1~5 182 178 175 171 168 162 102 67 51 ~5 7~ 105 143 156 185 187 189 174 13a 138 14~ 92 113 151 lg~ 188 177 108 85 lq3 186 181 167 149 199 1~5 190 188 la5 181 178 17~ 1~2 10~ 136 96 71 66 78 98 }16 li7 182 180 160 130 134 l9Z ldl 97 137 178 196 190 18a 167 ~4 76 132 16a lSZ
142 193 165 150 160 181 1~4 181 118 67 101 172 13~ 1~0 g7 89 103 117 116 172 1~6 111 gS 186 190 101 124 1~8 1g7 1~8 192 186 180 16~ 99 85 134 150 1~3 1~3 1~ 9q 111 153 1~ 113 55 91 166 204 166 13~ 114 108 114 123 ~2}

1~1 10~ 71 119 181 124 11~ lS7 189 196 2~1 19~ 188 182 176 167 124 117 1 146 142 85 62 8Z l3a 12~ 77 108 175 176 22~ 198 168 1~5 132 130 132 127 . .
109 75 59 118 115 105 la6 180 190 193 Z03 197 190 184 17~ 172 166 150 1~7 14$ 151 96 73 92 114 120 147 185 laZ 179 23} ~16 195 173 157 1~8 14Z 132 11~ 104.

16g 171 165 107 114 17~ 17~ 177 178 179 176 178 18~ 181 1~3 180 186 188 189 191 204 1~9 193 188 183 178 17~ 182 19~ 178 153 149 1~5 127 130 l~g.181 166 1~0 116 170 167 91 73 1~7 176 177 178 179 180 177 179 ~1 182 184 171 16a 189 191 lg2 193 1~ 182 177 772 167 17~ Zl l 207 163 1~3 1~ 81 6~ 137 200 195 172 139 11~

171 103 78 121 176 177 178 17~ 130 181 179 1~0 18Z 184 1~5 174 131 l7a 192 ~93 18~ 177 171 166 1~1 161 202 2~9 185 137 1~ 5~ 3~ 113 20~ 200 1~8 132 90 168 193 177 178 179 180 181 18Z 180 182 1~3 ~85 1~6 186 120 ~ 2g 193 lg~ 195 183 163 155 15~ 172 Z~4 190 1~2 124 9~ 5Z 122 201 2Z0 219`199 1~5 132 12g 110 13~ 218 182 178 179 1~0 181 182 183 181 18~ 18~ 186 188 189 1~0 98 171 1 ~6 2~g Z06 181 150 1~ 7 171 139 118 12~ 125 }g9 185 21~ 226 224 203 172 1~8 175 178 177 179 ~ 80 181 l~Z 183 18~ 182 1~Z 177 172 16g 170 15g 92 133 197 211 ~25 209 1~0 135 138 125 128 1~3 182 152 16~ 192 212 223 217 198 177 ,:

-144~ 2~9 1 9~0 Table 3 ( cnt ' d ) 175 176 177 178 179 181 182 183 1~4 185 184 181 170 156 144 137 137 87 82 167 194 224 214 169 lOD 51 95 167 219 216 153 169 190 201 203 194 179 175 177 1 ~1 176 177 178 179 180 18~ 183 1~34 185 186 185 186 181 167 150 134 125 111 71 163 I S~

173 42 28 So 78 107 136 137 114 78 58 150 150 149 141 133 128 124 120 -l4.~-2o~l92~
T~b:le ~

~ 20.070~ 1 1 18.4366 -6.0736 1 O.O~S0 ~ --0.3473 -0.1166 ) ~ 0.9746 0.8863 ~ 15.61g8 --0.1108 ~ -0.2950 0.8697 ! 1 0.4769 1.2215 ~ 26 5451 2.3714 ~ 0.6283 0.5195 ) ~ 0.9928 0.3900 ~ 17.4865 o . s044 ~ o 2422 -o .0007 1 ~ o .3550 o .4786 ) lo .5647 I -7.0400 ) ~ 10.2532 -lq.4972 1 0.1672 ~ -0.4982 -0.3300 ) ~ 1.0439 0.3757 ) 23.5290 o .5170 ~ 0.176~ 0.4417 ) ~ 0.3524 0.5732 ) 18.8828 2.6203 ~ o .8412 -o .3778 ~ ~ o .6141 o .5696 ) 15.1243 2.6043 ~ -0.0078 -0.7210 ) ~ 0.9375 0.4282 ~ 18.1785 ~ 18.2800 ) ~ 3.2~13 19.5803 ) 1.2052 ~ -o .5718 o .4028 1 ~ 1.0955 o .3574 ~ 16.2537 2.0697 ~ -0.2350 0.1817 ) ~ 1.0141 0.3891 ) -20.6531 o .5814 1 -o .829fl -o .1252 ) I o .9681 o .3517 ) -12.9288 o .52s8 ~ o .2235 -o .57~4 ) ~ o .2771 o.5063 ) 9.1028 ~ -23.1000 ) ~ -4.8080 -25.0567 ) 2.1579 ~ -1.0736 -o .7244 ) ~ o .i4985 o .9907 ) -18.7614 o .7606 ~ o .6251 o .7408 1 ~ 1.2097 0.8611 ) 15.3114 2.0238 -~ 0.s290 -0.6723 ) ~ 0.4226 0.5152 ) 13.5875 1.7335 ~ 0.5390 0.6098 ) ~ o.gogo 0.4148 ) -19.6480 ~ 7.1100 ) ~ -13.3326 8.7161 ) -0.1533 ~ -0.7812 -0.6231 ) ~ 1.4083 0.7261 ) 22.7463 -o .4063 ~ o .2195 o .2612 ) ~ o .6989 0.5217 ) 14.3231 0.1167 ~ 0.6426 0.7914 ) ~ 1.5990 0.4263 ) -22.0065 2.5504 ~ -0.3412 0.0391 ) ~ 1.0334 0.4872 ) -13.6814 H

~ 1.4300 )~ -8.4689-~ . ~ 372 ) 1.1469 ~ -0.6161 0.5776 ) ~ 1.6198 0.6623 ) -27.6468 2.2347 ~ 0.1120 -0.260~ ) I 1.0931 0.3968 ) -26.5535 2.373s~ o . s480 o .4953 ) ~ 1.0602 o .3667 ~ -21.1788 --0.0810 ~ 0.2136 -0.6939 ) ~ 0.8490 0.4722 ) 15.7904 H
~ -3.4100 )~ 13.9906-7.3165 ) 1.2618 ~ -0.2077 o .5029 ) ~ 0.6444 o .3881 ~ -19.4196 0.2360 ~ 0.3595 -0.1964 ) ~ o.9ol9 0.4604 ) 19.6177 0.so43 ~ -0.684s -0.4466 ) ~ 0.2806 0.7137 ) 12.4950 2.4883 1 -0.3120 -0.4211 ) ~ 0.8983 0.3157 ) -11.7791 ~ - 2~9192~

Table ~ (cnt'd) ~ -9 1600 1 1 15.0612 -2.1272 ~
1.030-3 ~ -0 1806 -().1529 ~ ~ 0.3927 0.8443 ) 21.3891 ~.5530 ~ ~.7719 --0.2gl5 ~ ~ 0.-~043 0.5868 ~ 20.1406 2.6333 ~ 0.4417 0.6637 1 ~ 0.5780 0.7297 ) 21.4013 2.1815 ~ --0.3739 0.5440 1 ~ 0.2365 0.6158 ~ -8.5414 1 6.3900 ~ ~ -7.8074 --3.0968 1 -0.6211 1 0.8587 -0.3780 ~ ~ 1.8641 0.7945 ~ -24.7811 -0.9027 ~ -0.4325 0.6556 1 ~ 1.5081 0.8819 1 -11.7734 2.0542 1 -O. ~i373 -0.5476 1 ~ 0.3064 1.3196 1 -23.8431 0.6488 ~ 0.2881 0.1207 1 ~ 0.2702 0.8305 ~ 22.1091 H

~ 25.8600 1 ~ 6.5314 5.5600 ~
2.2851 ~ -0.7313 -0.5791 1 ~ 0.9312 0.4245 ~ -22.5825 0.7607 ~ 0.3073 -0.0321 ~ ~ 1.0390 0.3156 ~ -21.3682 2.4944 ~ -0.2122 0.6079 ~ 1 1.1660 0.7908 ~ 12.5409 0.0589 ~ 0.8791 0.0193 1 ~ 0.9180 0.4529 1 10.2552 H
1 27.8800 1 ~ -20.4576 -1.1141 ) 1.0858 ~ 0.6873 -0.6222 1 ~ 0.6976 0.7652 ~ -18.5806 2.4152 ~ 0.5361 0.4932 ~ ~ 1.2377 0.3132 ~ -24.2860 0.4854 ~ -0.3912 0.5750 1 ~ 1.0226 0.5740 1 lI.4463 1.0505 ~ -0.7026 -0.4049 ~ ~ 1.1138 0.6097 ~ 6.0134 H

1.3800 ~ I -2.6355 2.7156 ~
1.0233 ~ -0.3111 -0.6606 ~ ~ 1.8593 0.9083 ~ -29.0218 2.0049 ~ -0.6356 -0.3608 j ~ 0.6317 0.5278 ~ 24.0265 -0.0782 ~ -0.5204 0.8698 ~ I 0.8911 0.6808 ~ 23.9088 0.5359 ~ ().4443 -0.5323 ~ ~ 0.3578 0.4830 ) 14.7053 H
g 4S.6700 ~ ~ 9.9799 9 3010 ~
2.2610 ~ -0.5g87 -0.529L ) ~ 1.1727 0.2857 ~ -15.5455 0.0466 ~ -0.4726 -0.4gg7 1 ~ 0.3919 0.5131 ~ 8.1386 1.5708 ~ -0.7000 0.0000 ~ ~ 0.3000 0.5000 ~ 3.2578 2.0944 ~ -0.7000 0.5000 ~ I 0.9000 0.5000 ~ 3.3765 ~ 42.5900 J ~ -4.6566 -11.6684 ~
1.1429 t 0554 0.5966 ~ ~ 1.3173 0.3226 ~ -20.5178 0.1971 ~ 0.5506 0.6132 ~ ~ 1.0509 0.4018 ~ -12.3190 0.6258 1 0.0528 -0.2921 ~ ~ 0.7666 0.2815 ~ -8.2027 2.6180 ~ -0.7000 0.5000 ~ ~ 0.9000 0.5000 ~ -2.5159 -l~7- 2 Ogl ~2~

Tabl~ 4 ~ cnt ' d ) __ ~ 2~.4600 )I -32.81360.8704 ) 2.87.~4 ~ 0.6706 --0.2593 ) ~ 1.1037 0.4549 J -23.2053 0.9069 ~ 0.5096 -0.4137 1 ~ 1.1957 0.56ss ~ 14.8349 2.53~.0 1 0.4107 0.7021 ) ~ 1.0094 0.4634 1 22.686z 2.22Ufi ~ --o.4671 o .4722 ) ~ o .9759 o.3955 J 14.2152 ~ 21.~3000 ~ ~ 7.1581 0.1462 1 2.121~ ~ -0.1:~47-0.0398 ~ ~ 1.4448 0.6800 ) 29.3055 2.4651 ~ -0.5664 -o .4291 J ~ I .0687 o .3643 ) -21.2993 0.0539 ~ o .1672o .6776 ) ~ o .9691 o .6396 ! 15.3907 1.5710 ~ o.a884 0.0184 j ~ 0.2705 0.4484 1 10.9159 20~ 1 920 T~b1e 5 ~ 25.0000 1 ~ 22.000~ fi .0000 ~
1.57~)~3 ~ -0.~54S 0.0000 ) ~ 0.9000 0.9000 1 16.5000 O.O'l91 1 -0.2545 0.76.16 ~ ~ 0.5000 1.92~30 1 31.5000 O.U345 ~ 0.5(!91 0.5091 ~ ~ 0.3000 0.8568 1 lfi.5000 0.5~39() ~ 545 ().0000 1 ~ ~.3000 0.3900 1 0.0000 ~ -13.0000 1 ~ 10.0000 -10.0000 j 1.7181 1 -0.5091 -0.2545 ~ I 0.3000 0.8568 ) 22.5000 0.4909 ~ 0.2545 0.5091 ) ~ 0.3000 0.5070 ) 16.5000 1.5708 ~ 0.7636 -0.2545 ) ~ 0.5000 0.5000 1 16.5000 1.0308 ~ 0.0000 -0.7636 ) ~ 0.5000 1.0985 1 19.5000 H

~ 21.0000 i 1 6.000() 18.0000 1 2.7980 ( -0.5091 0.5091 1 ~ 0.3000 0.8568 J 16.5000 0.5400 ~ -0.25~5 0.2545 ) ~ 0.3000 0.8568 1 -16.5000 2.1108 ~ -0.76~6 0.0000 ) ~ 0.3000 0.8568 j -13.5000 0.4909 ~ 0.2545 -0.5091 ~ ~ 0.3000 0.5070 1 0.0000 ~ -29.0000 ~ ~ -6.0000 -26.0000 ~
2.1108 ~ -1.0182 -0.7636 1 ~ 0.5000 1.0985 ) -22.5000 2.1598 ~ 0.5091 0.7636 1 ~ 0.9000 1.1700 1 13.5000 2.1598 ~ 0.5091 -0.7636 1 ~ 0.5000 0.6500 1 16.5000 0.1473 ~ 0.5091 0.5091 ~ ~ 0.5900 1.0985 ~ -22.5000 ~ 9.0000 1 ~ -18.000() 10.0000 ~
1.4235 1 -0.7636 -0.5091 ~ ~ 0.7000 1.5379 1 22.5000 0.9817 1 0.2545 0.2545 ~ ~ 0.5000 0.6500 1 16.5000 1.6935 ~ 0.7636 1.0182 ) ~ 0.5000 1.8565 1 -19.5000 0.9327 1 -0.2545 0.0000 1 ~ 0.5000 1.0985 j --13.5000 H
~ 15.0~()0 ~ ~ 2.0000 -2.0000 ~
2.6998 ~ -0.7636 0.509I I ~ 0.7000 1.5379 ~ -25.5000 0.6381 ~ 0-000() -0.2545 1 ~ 0.3000 0.8568 ~ -22.5000 0.8345 ~ 0.5~91 0.5091 1 ~ 0.3000 0.8568 1 -lg .5000 1.4726 ~ 0.2545 -0.7636 ~ ~ ().5000 0.845V ) 0.0000 ~ 3.0000 ~ ~ 14.000() - lO .0000 1 2.8471 ! -0.2545 0.5091 ~ ~ 0.3000 0.5070 ) -16.5000 1.8162 ~ () .2545 -0.2545 1 ~ 0.5000 1.0985 ) 19.5000 () .5400 ~ 0'~1 -0.5091 ~ I 0.3000 0.8568 1 10.5000 0.9327 ~ -0.2545 -0.5091 j ~ 0.3000 0.8568 ~ 0.0()00 I

-149- 2~9 1~2 0 'I`~ble ~i ( cn~ ' d ) --2~ 000 ~~ lo oooo-2.0000 1 ~.()3~ 1 -0.2s4s o.oooo ) 1 0.3000 0.6591 ) ls.sooo ().4~31`7~ 0.~6.J6-0.2545 1 ~ 0.5000 0.6500 ~ 22.5000 52~ ~ 0.50~l 0.7636 1 ~ 0.5000 0.6500 ~ 22.5000 .20~ -0.,~95 0.7636 1 ~ 0.3000 0.8568 1 o.oooo ~ l3. ~ooo I 1 2. oooo -6.0000 ) 0.93~ ~ o.~ 6 -0.2545 ~ ~ 0.7000 1.5379 ~ -16.5000 0.6872 ~ -0.5091 0.7636 ) ~ o.sooo 1.5210 1 -16.5000 ~ .0~7 l~ -o .7636 -o .5091 1 ~ 0.3000 1.4480 ) -25.5000 O.fi3f31~ 0.5()91 0.0000 ~ ~ 0.3000 0.8s68 1 19.5000 ~ 23. oooo ~ ~ 14.0000 2.0000 ~
0.7363 ~ -0.7636 -0.5091 ~ ~ 0.5000 1.0985 ~ -25.5000 ~.~317 ~ 0.2s45 o.oooo ~ ~ 0.3000 1.1139 ~ -22.5000 o s8l7 ~ -0.2545 0.5091 ~ ~ 0.7000 0.9100 ~ 10.5000 1.6199 ~ 0.76~6 o.oooo ~ ~ o.Sooo 1.0985 ~ O.oO00 H

~ 2~3.0000 ~ . oooo -6.0000 ) 1.57()8 ~ o .7636 -o .5091 1 ~ o .7000 o .7000 ~ -19.5000 0.8540 ~ 0.5091 0.5091 ) ~ 0.3000 1.1139 ~ -22.5000 2.061-~ ~ -0.5091 0.5091 ~ ~ 0.5000 0.8450 ~ o.oooO
2.6507 ~ -o .7636 -o .5091 ~ ~ o .7000 1.1830 ~ 0.0000 H

~ 5. ooOo ~ ~ -2. oooo 6. oooo ~
2.6016 ~ -0.5091 -0.7636 ~ ~ o.sooo 1.9773 ~ -40.5000 o .5890 ~ -o .7636 -o .2545 ~ ~ o .5000 o .6500 ~ 19.5000 1.3744 ~ -0.5091 0.7636 ~ ~ 0.7000 o.sloo ~ Z2.5000 0.5890 ~ 0.5091 -0.5091 ~ ~ 0.3000 0.3900 ~ o.Oooo H

~ 43.0000 J ~ lo.oooo lo.oooo ~
0.7118~ -0.5091 -0.5091 ~ ~ 0.3000 1.1139 ~ -19.5000 o .1963~ -o .2545 -o .5091 ~ ~ o .3000 0.3900 ~ 7.5000 1.4726~ -0.7636 o.oooo ~ ~ 0.3000 0.5070 ~ o.0000 o .4409~ -o .7636 o .5091 ~ ~ o .5000 o .8450 ) 0.0000 H
~ 41.0000 1 ~ -6.0000 -lo.oooo ~
2.7243~ o.oooo 0.5091 ) ~ 0.3000 1.1139 ) -19.5000 1.8162~ 0.5()91 0.5091 I g 0.5000 1.4280 ) --16 5000 2.2084~ o . oooo -o .2545 ) ~ o .3000 o .8568 ) o . oooo 1.0799~ -o .7636 o .5091 ~ ~ o .5000 o .8450 ) o . oooo '' ;' ' ' ~ ' ' -,', ' , -150- 20~192V

Table 5 ( cnt '_ ~ 23.5000 1 1 -31.5000 0.5000 1 1.3()0f~ ~ 0.6533 -0.2800 1 ~ 0.4750 1.1761 1 -25.5000 2.~78~ ~ 0.46~ -0.4667 1 ~ 0.5750 1.2711 1 15.5000 0.9572 ~ ().4667 0.6533 1 ~ 0.4750 1.0501 ~ 23.5000 0. ~j627 1 --() qfi67 0.4667 1 ~ 0.3750 0.9285 1 13.5000 ~ 29.0000 1 ~ 2.0000 2.0000 1 0.5400 ~ -0.2545 0.0000 ~ ~ 0.7000 1.5379 ~ 28.5000 O.~ j~7 ~ -0.5091 -0.5091 1 ~ 0.3000 0.8568 ~ -l9.S000 1. fi690 ~ 0.2S45 () .7636 ; ~ 0.7000 1.1830 1 13.5000 1.66~0 ~ 0.7~j~6 0.0000 1 ~ 0.3000 0 5070 1 0.0000 ~able_6 ~O~)~g20 14 1 ih 137 139 135 1 ~ 9 13~3 134 137 136 135 137 137 139 138 135 13t3 138 138 3.3/ 1.3~ 1 i6 134 139 13t3 134 135 137 136 140 137 131 138 140 136 136 1341 ~ 7 I ~ 5 l ~f 13f) 1 ~ 134 135 134 139 ~ ` lJ ~ 141 137 139 137 134 13g 136 136 141 136 137 138 137 141 137 136 135 l ~ 9 136 137 110 1 i6 136 141 135 131 142 138 133 137 13fi 137 136 136 ~ 3~; 134 136 1)6 135 1.38 136 136 1~9 1:~ ) 141 1.38 1~8 l i9 1.3.3 137 140 137 139 139 137 139 137 138 141 137 138 134 ~3~ 7 133 138 136 131 139 134 131 137 14~ 1:19 136 140 141 135 138 142 135 13S 137 137 139 139 140 136 135 14~
139 l:i ) 13~3 138 137 138 135 136 137 134 140 138 134 141 138 138 136 132 140 1 i8 142 139 135 138 141 136 137 137 136 140 136 135 137 141 139 135 139 137 136 1 i~ 136 136 138 135 137 139 136 136 139 137 134 137 139 134 137 136 137 140 1 :j9 137 138 136 137 133 138 135 136 128 l~t i 136 144 139 135 142 138 138 137 139 137 136 140 136 135 136 136 139 139 3.3~ 134 136 140 137 136 135 135 138 ~5 1~1 141 139 139 140 137 135 136 141 136 135 139 135 139 138 135 138 134 138 13~3 134 139 140 137 136 135 138 139 139 136 139 141 132 136 141 135 135 137 1 iS 136 138 138 135 136 136 133 137 3~3 ~j~ 108 136 140 138 144 137 137 141 137 139 138 133 139 137 135 142 135 136 li~ 135 139 138 135 139 137 138 140 136 136 138 133 138 140 135 138 136 135 14() 13i3 137 136 135 137 135 136 136 72 i j 58 104 135 144 140 139 138 139 138 138 138 136 137 131 136 13g 134 1:39 1 i~ i 36 138 136 139 140 133 132 140 139 134 140 137 135 143 141 135 135 138 ll~i lig 137 134 136 137 138 135 129 11.6 /~ 4~3 48 96 135 138 141 142 137 137 137 140 141 134 135 140 139 137 136 1~0 1 i6 136 141 135 139 136 134 142 135 135 141 137 134 139 139 134 138 139 1.33 I i5 141 137 132 141 13i3 132 13a 145 12~ i34 43 48 91 ~31 142 140 144 139 133 138 142 140 137 136 138 137 137 141 135 136 141 134 139 139 134 139 138 135 136 138 136 13g 138 135 138 136 l~ 136 134 138 140 134 133 139 144 19~ i 92 46 41 ~2 1~4 142 141 141 140 138 139 1:3~i i4~ 1.3~3 1.36 1.3tj l.i9 138 1 :i6 138 136 137 1.1~i 135 140 138 134 139 140 135 1 i2 140 139 132 135 139 138 ! 37 1 i6 136 ].:35 l.if 1.35 135 137 -~. 5~-Ta~le 6 (cnt'd) 2~9~92~
_ l9 3 14q Iqi 132 9f3 t-2 3U 72120 139139 19113913f3138!38139 134 13/
] .1;~1~6 13~3l ~f~I 3613-/13g13913613'~14()136134 137134136141 13 1 ~'i !91 1 :32 13fi 137 135 i33 137 137 131 14.3 llf3 143 146 139 110 60 36 62 111 L33 1.38 142 139 136 141 139 135 137 140 !.lU 136 138 138 136 136 137 136 138 139 136 139 135 134 142 135 135 141 1:37 ! 35 l38 13fi 13~ 1.39 136 136 135 135 IJ`'~ / .1431~12 1461~13112 67 37 5810'~135145 137137138136 139 139 13`,~14() 139 138 140135 136141136 1371313135138 137137138134 138 13~3 I :~t~l 3~J 134 135 139135 140136134 139 1.3f~ 13~ 139 141 140141 146118 75 45 4S 90~33 1401421qO141 143 135 13'3141139 ~38 13314~)138 13213813f3134135 142137 133141139136 1:36 1411.~139 140 137 143148 145126 86 48 47 84125 144144139137 137 14214114() 137 131 139141 134140135 133141 139134 138138133134 140 13c6I~6139 135 138 138137 140136138 13613c'~140 138 138 140139 143148130 96 53 41 76 120139140142 1q2 1371~4138 139 134 133136 140134138 1~9137139 143 137 137137 137139138 144144 138121 8448 44 80 128 13Ci138144 138 137 143140 137139135 137140 1331381.39133137140 134 137140140 137 139 138138 137137138 140142 145143 13296 5q 38 63 1141~6142 142 137 139138 138138136 138139 138134 138139132138 1.33 132142135 1:36 1391.'34139,13713S140 1361~8140 138 140 138138 139133138 141136 141144 146137106 70 40 4C3103136 136 138 142137 138142137 134140 138136 136137137130 13' ~40135135 139 135 137142 137135137 14413913S 139 13~3 136135 13913813S 1401381.37142 143143143119 76 q4 4~ 79 124 140 139140 136139137 136142 135135 14013513'71.15 1 ;';
1421:361:32 138 13C~ 139135 137135133 20~1~20 ~2ble 6 ( cnt ' d ) 143 139 1~5 139 1~0 1~9 137 136 138 ~38 133 138 139 13~ 1~2 143 13g 1g6 1~3 9~ 53 ~5 68 11~ 136 1~2 1az l~o 139 139 137 '35 137 138 138 139 135 136 136 135 135 139 138 133 13~ 138 13~ 133 136 139 las 137 136 l~o 136 137 135 135 137 1~6 1~0 137 134 143 lao lao 149 135 105 72 ~0 51 lOZ 130 139 142 1~0 140 13~ 137 1~0 137 137 138 137 138 138 136 137 13~ 138 137 ~35 135 138 l~o ~ ~7 1~2 l~S 136 73~ S 13~ 1~6 ~ 37 136 133 1~ 138 135 139 l~o 141 la 146 14~ 120 78 5~ a6 76 118 140 1~0 lqo 13~ 139 13q 13; 1g2 135 137 13 13~ 142 133 135 140 131 135 141 13~ 135 137 139 142 1a2 1~0 139 1~2 13~ 13~ 13~ 137 13~ 137 13~ 13g 137 138 137 135 143 14~ 1~3 131 ~9 56 3g 65 109 132 1~1 142 ~0 l~o 138 138 13g 136 133 1~0 140 142 13~ 13~ 139 138 136 1~3 137 135 1~1 ~37 136 138 138 lso 136 138 la3 139 1~2 149 136 112 79 ~3 47 86 123 139 13g 1~0 139 137 13~ 138 138 13~ 137 136 136 139 13~ ~3~ 137 138 139 lq3 143 13g 142 137 136 142 136 138 138 134 141 13? 136 137 137 138 136 1~2 141 738- 139 lq~ 147 1~1 128 sz 53 ~2 sg 107 132 138 14~ 1~7 1~1 14g 133 134 138 145 142 ~39 141 138 140 lgl 138 138 138 13~ }3~ 136 ~38 13~ 137 138 134 13~ 13~ 140 1~3 139 146 136 107 81 48 43 84 118 131 1~1 144 137 139 136 138 143 134 132 138 1~7 135 136 138 96 121 142 147 13~ 1~1 142 139 136 l~Q 1~0 136 13~ 135 13~ 139 136 140 136 12~ 131 139 139 136 141 143 144 1~4 12~ 98 58 37 5~ 97 lZ6 138 13~ 141 139 135 139 13g 132 737 a40 135 136 138 ~7 75 119 1~2 142 143 141 13g 13g 13g ~a~ 190 13B 13as 137 140 138 1~5 137 136 135 1~0 137 136 lq~ g 1~4 137 116 ~3 50 3~ 71 111 128 13 1~1 13g 1~2 1~0 138 ~37 7 38 139 133 135 5~ ~ 63 1~6 136 139 146 141 138 14Z ldO lq2 13~ 136 14~ 1 4 139 }39 13S
137 13fi ~3~ 137 135 138 135 137 1~3 1~ 1 147 133 101 70 ~1 47 86 118 131 135 1~3 l~g 140 136 138 14~ 135 137 116 7g 4~ ~5 8~ 13~ 135 141 146 138 1~1 138 138 139 138 ~39 136 139 137 135 13~3 136 137 13~ 136 13~ 137. 13~ 137 140 144 19.Z 14C) 128 95 58 39 53 90 116 130 1~2 141 ~3~ 142 141 134 137 ~g~ 132 9~ 51 40 71 111 131 1~3 139 ~39 143 137 142 1~1 135 13g 13~ 136 13B 136 ~37 137 135 1~6 138 13~ 135 137 136 1~0 141 144 145 133 122 91 55 43 53 93 lZ0 12~ 1~3 141 139 lq3 140 :

-1 5a~
209~920 T lble 6 ( cnt ' ~ ) I~IS14~13~1114 /1 41 52 ~ 13113f~ 14413913'~ IqSlJ9 lJt313t3136 ~90 l'~fi 137 136 134 13713fi 140 136 134 14()134 135 1441.'~81381'17115 133 117 ~3~ i 37 55 ~ 0 131 13~ 141 140 L ~143147 lq41~'4 138 46 45~3.J 122 136140 141 13~3140 140 139136 136 I~li) 137 138 1361 15 13913fi 138I 38 137136 114 13fi140 139 136142 148 142 ~ 3~ 90 56 1~3 58 ~1 116 132 137 14~)13914314'7141 133 99 57 41 60 1091321.37 14q139 136 142lql 13e 1 ~813G139136 135 142 133 135 1431.35 13S137 133 136137 138 139140 144 13~314913613g 148 lg5 137 113 75 45 48 85 119 133138 143 1411.38 140 142134136 143 137 144 148 143 1311()2 67 42 55 9~ 120 134 141143 139 141140139 139 136 138 138 134 132 138 138133 137 13~;138 139 133137 136 137138135 146 144 139 136 llS 8~3' 54 140138192 13g 140 137 137 136 136 137 133135 137 133139 135 136140 133 136138143 134 139 139 143 143 142 lS0 147139 121 86 56 44 6811.5 131 132137139 136 133 137 1421.41 144 144 1:~4133134144 143 142 139 139 140 137 137140 136 136138 136 136139 135 134139136 134 140 134 133 141 139 14~

14113614~)141 lJ7 135 141 141 136 137 14314q 144 14.~142 131 96 66 43 49 92 116 131 143 139 142 141 13813~3 139141 134 136140 134 136138 137 1..37 134 137 139 137 135 133 137 137 133 138141138 13613~3 139 138 137 13913~3 138136 140 141139 147 1391.0 ' 93 55 42 53 84 122 135 137 143 14214() 138137 140 141lJ6 134 13h137 1.37 138134137 137 133 139 13413.3 141 136 137138130 143~4() 133 139 1371 39 138 14()li9 135 139138 142 147144 135 11990 58 42 S0 89 120 12913C~ 14~ 14~;141 141 1.3:1135 140 136135 138 1361.36134135 13g1 35 135 13R 1371.16 lJ~13813~ 140 140 1371.35 142I.i~14(` 1421.3~:~13~I'i5140 147 1181~4 11 1401 38121 87 56 40 52 94 122 12g 14014614:1 14()138 138 1411 ~ ~ 134 1371.35135136 135'l3713t3 133 1341 39 -l55- 2~9~9~

I`~ble 7 I 1~.1 1 ! 81 J tl13 tl I 37 1 37 l J 7 1 J 7 137 1.~ 6 1 J 6 136 137 13 f3 138 1 J 8 13 ~ 13 t~ 138 713'~I .i61 i6I .i~ 137 1 ~ 137 136 136 136 136 136 137 135 13fiI !o 1 171 J 713013fi1 J t; I l h l .3 '5 l 3 'i I 3 5 1~9lJ~ 38 I~tl 138 LJ8 137 IJ7 1~7137 136 136 137 138 138 138 13R 138 138 .l ltlI .l)13~ 13713ti lJ7 137 lJ713'7 137 137 136 136 13fi 136 137 lï'7 135 13fi 13f~L.J7L3'7 l:J6 136 1.30 136 1:35 l35 135 L3~ 9 lï91.3~31 ~8 1~8 138 137 1 ~7lï~ 13~ 13~7 137 1.37 138 138 138 138 13~
13t3L~ 7 137 137 13~ lï7 137 13713~ 137 136 136 136 136 i37 137 135 136 13~;13~137 137 136 1~6 136 1 i5 135135 13713'7137 137].36 13G 136 135 135135 140139139 13'~ 139 138 138 138 138138 137 137 137 137 13'7 137 137 137 ~37 1371:37137 137 137 137 137 137 137137 137 137 137 137 137 137 138 135 136 13'7137137 137 136 136 136 135 135135 1~714013~ 1.391:39 139 139 138 133138 138 137 137 1:37 137 137 137 13'7 13'7 13'713713'713& 13813'7 137 137 137137 137 137 137 137 137 137 138 135 13f-1:37137137 137 137 136 136 136 135135 73120 140 140 139 139 139 139 13813f~ 1.38 138 137 137 137 137 137 137 137 137137137 138 138 137 137 13-~ 137137 137 137 137 137 137 138 138 135 136 13'7137137 137 137 136 136 136 135135 ~11 65 112 14() 140 139 139 139 139139 138 138 137 137 137 137 137 137 137 13/ 137 137 138 13813'7 137 137 137137 137 137 137 137 137 138 138 ]35 136 I.i7 137 137 137 137 136 136 1.'~6 136 13S

fi~ 41 58 103 139 190 139 139 139139 139 138 138 137 137 137 137 137 137 13'~ 137 137 138 138 137 1:37 137 137137 137 137 137 137 138 138 139 135 136 13~ ~37 137 13'7 137 137 136 136 1:~6136 2 43 53 9S 136 140 140 1.~9139 139 139 138 138 138 137 1:37 lJ7 137 1:371 37 137 13813f.~137 137 137 13-7117 l37 137 137 137 1-Jt3 138 139 13S i i6 1371.J7 137 1371 j 71,i 7130 136 1.loI 3h 1461;'~ 80 45 4f3 88 1.32 140 lq(~9 139 139 139 138 138 13~3 1~7 1:37 137 1~/ L~7 137 13;313~31.37 lJ7 1.37 1.j'7 1 J7 137 1~7 137 137 137 138 139 135 ~36 I 371 371.3't 137! ~'7I lGI .jG 1 3fi1 .~t; 1 3G

146 I16l'.~S t37 4~ ~5 81 128 14()!4(j 14() 1.39 140 1:J9 139 138 !38 1.37 137 1.~7 13713~i 138 11~l 37 137 137 1.~7137 137 137 137 1.37 137 13t3 13t3 1 I.'i I lfi l.i6 137 137 l:i7 137 136 136 136 1 ~ti ! '35 ,. . .

-la~;- 2~91920 'I`able 7 ( cnt_d Iq.' 1~2 14.3 1~ 9 59 42 '/() 118 138 13~3 13~3 140 140 1.~ 39 138 l-i~1 ~3 131 137 137 I 17 137 1 37 13'7 13'7 137 1 37 137 1.37 137 1 .17 136 13S 138 1 37 13'~ 137 1 3G I 36 135 135 13r> 1 35 135 1 35 141 1~12 1:-12 142 142 107 fiO ql 63 110 138 138 140 1~3(~ 140 139 139 139 13~3 1 ~3 1 ~3 117 137 137 137 L3'7 13'7 137 137 137 137 13'~ 1 37 137 136 136 138 137 1.~7 L ~7 1.3'7 136 136 136 13~j 136 136 136 ~9 ~ ll 142 142 14~ 66 41 56 100 13'7 140 14~ 40 1~0 139 139 139 13~3 13~3 138 138 137 137 137 13-~ 137 1 ~7 137 137 137 1.37 117 137 136 137 13711~ 137 137 13'7 137 137 1.:17 137 137 137 140 140 141 141 142 142 142 123 74 43 50 90 136 14 l 14~ 140 140 139 139 139 138 138 138 138 137 1~7 137 137 137 13'~ 137 137 137 1.37 137 137 137 137 i37 137 137 137 137 137 137 1 ~7 137 137 140 140 140 141 141 141 142 142 130 84 47 45 82 1~9 141 140 140 140 139 l j7 137 137 137 137 137 137 137 137 137 139 139 140 140 140 141 141 141. 142 137 94 52 41 7~ 119 141 140 140 140 13~ 139 139 139 138 138 137 137 137 137 137 137 137 137 137 137 137 136 137 l~j7 137 137 137 137 137 137 137 137 137 I j8 119 139 139 140 140 140 141 141 142 141 105 57 40 61 108 140 140 140 1~() 139 139 139 139 13i3 138 137 137 137 137 137 137 137 137 137 137 136 136 33~ 13'~ 137 137 137 137 137 13'/ 137 137 13~3 138 138 139 139 140 140 140 141 141 141 142 114 67 41 53 94 136 140 14~ 140 139 139 139 139 138 138 137 137 137 137 137 137 i37 137 137 136 136 137 138 138 138 139 1~9 139 140 140 140 141 141 142 12t; 79 45 4fi 81 126 140 140 140 139 139 139 138 138 138 137 137 137 137 13'7 137 1.37 137 135 136 130 1 37 137 137 13'7 13'7 137 137 137 137 137 1 3'7 137 138 138 138 139 139 139 140 140 140 142 143 1 36 ~3 52 42 fi8 113 140 140 ~40 140 139 139 138 138 137 137 137 137 I,17 137 137 137 13S 136 l.io 1 37 137 1.'37 1.37 137 137 137 137 136 l~o 1 36 137 137 137 138 138 138 139 139 140 14() 141 14~ 14~ 143 108 63 41 Ij6 ~3 13fi 140 14() 140 139 138 138 137 137 137 137 137 137 137 137 135 136 1..1t 1.37 1 .17 137 1 37 1 37 137 136 13fi 13fi I :~5 ! ~r~ 136 137 I J7 137 138 138 138 139 1 39 139 14() 141 ' q 3 144 145 123 7fi 45 48 82 126 140 141 140 139 138 138 1 37 137 137 137 ! 37 136 136 135 136 l s/ 137 137 1.37 137 ~3'/ 1.3'7 136 1'3fi 136 -15~-2~9~92a Table 7 (cnt'd) 1 J'~I ~Yl.J~3IJ~31~3J.~llJ~ 1~713t~1.3~ 613S 14014()140140140141 134 "25~ 4 ~ 681()914()14()lqO139 1391 ~9138138 138137 137137136136 33 ~ t~ 1361.361 ~ ;13 ti1 ~ 6 140l.~i134 13913~31381313137 1371:37 136136139 139140 140190190190 14()10863 42 56 99 136140 140 139 139139138 138138 137137136136 1361361.36 136136 136 136136 136 136 1~1014013(~13g139 138 138138 13713'7 137136139 139139 139190140140 lgO140123 7i3 46 49 8112414() 14014()139139 13913R 138138136136 l~i13fi136 136136 137 137137 137~ 3~

1'11140140 14013'J13913i31381.38 137 137137138 139139 1391391391qO
14()140140 135 9S 56 43 64 105 138 140140139 139139 138138137137 1~7137137 13713'7137 137137 1371.37 1 37137137 1371~'7137 137137 137 137 141141141 1401401.40 139139 139 138 138138138 138138 138138139139 1:~91391391.40140 144 132 90 54 44 64103137 140140 139139137137 14()142141 14114~)140 140139 139 139 138138137 137138 138138138138 137137137 137137 137 137137 1371.37 ~0132 142 141141 141 140140 139 139 139138137 137137 137138138138 138138139 139139 190 141142 144 132 92 56' 43 S9 94 131140137137 1 371~7137 137137 137 1371371.37 138 46 75 119 142~41 141 141140 140 140~.39139136 137137 137137137138 1 ~8138138 139139 139 140141 142 143 144117 76 48 46 69107137137 137~37138 1381381 ~313~3138 138 138 138138138 138138 137 138139 140 ~41 142143137 102 64 44 S0 85122 1381.38138 13813~313~31.~13~31381313 1~.8'7346 51 85 127 141141141. 140 14014013613~i136 136137137137 1.37137138 138138 135 136137 138 140 141142~.43144128 90 56 4354 8712.3138 138138 138 138138 138 138 1441.3290 53 45 69 111141 141 141 140140135 1361.3613613613613'~
I ,,i ~I i 7137137138 33313513 ti1371 38139140 141142 14314411987 S S
4.~Li5c36 1221~3 1~813~313~313~.~i38 - 2~9~0 Ta~le 7 (cnt'd) lqJ 191 11.J 1.2() ~ 44 58 100 137 139 139 139 139 139 139 138 138 i38 137 137 1.~7 ~ 3~ I'Jt. 13tj 137 1:17 13-~ 137 137 137 1.37 137 137 137 137 137 140 1~3 ~ 4 11 ~3 13 ~ 13 S ~ 3 '7 139 14~ 14~ 14.3 1~ 32 48 48 81 123 139 139 140 139 139 139 139 13~ 13i3 i7 1i7 137 1..~'7 137 137 137 137 137 137 137 137 137 137 137 137 140 140 ~ 1 J 'i'7 `38 12;~ 137 139 I'll 14.~ 142 143 14:~ 139 99 58 93 62 100 134 140 140 140 139 139 139 139 i.38 13~ 13~ 137 137 137 137 137 137 1:37 137 137 137 137 137 137 137 1:~9 Iq() 1'1() 140 14() 118 ~2 5~ 45 59 90 125 141 141 142 14~ 14~ 143 lq3 119 75 47 47 74 111 140 140 140 140 134 139 1~9 138 138 138 138 137 137 137 137 137 137 137 137 137 137 137 137 139 139 139 13q 140 140 1:39 117 82 55 47 61 1.40 14~ 141 141 142 142 143 143 137 100 62 44 51 85 126 140 140 140 140 134 139 139 138 138 137 137 137 ~.37 137 137 137 137 137 137 137 137 138 138 1:~9 I:i~ 139 139 ~ 3~ 140 139 118 8q 57 139 lqO 14() lql lql 141 lq2 lq2 lq3 143 127 89 60 44 61 99 135 140 140 lq() 140 139 139 139 13'7 137 137 137 138 138 138 138 138 138 138 138 13.~3 13 1~9 139 140 140 140 141 141 142 142 142 143 143 126 83 50 46 71 111 139 140 140 14() 139 139 138 138 138 138 138 138 138 138 138 138 138 138 13'7 137 1 j7 13~ 138 138 1.38 1:~8 139 139 139 139 138 139 139 139 140 140 141 141 141 142 lq2 143 143 143 112 70 46 50 79 137 137 137 137 1'38 138 138 1.38 138 139 137 1 i~3 1~8 13~ 13~ 14~ lqO 140 141 141 142 142 142 143 143 137 100 62 44 I ~f~ 136 1:~7 13'7 137 137 137 138 138 138 1.37 138 138 138 1.39 133 140 lq0 141 lql lql lql 141 142 142 143 1.30 S7 44 56 87 123 138 1,38 138 138 138 1:38 138 138 138 138 138 1.3~ 135 I.-i~
i36 L:36 136 1.36 136 137 137 137 13'7 1 :37 I jf) 137 137 13'7 1~3 138 139 139 139 140 140 141 139 140 141 141 14~ 14~ 14~
1~5 ~37 55 44 S~ 90 1'~5 138 138 138 138 138 1:38 138 1,38 138 138 135 135 ..JS 136 13t~ 6 136 136 137 137 137 ~ 136 1 J6 1 ~7 1 ~7 138 138 138 1:19 139 140 14() 138 139 139 140 lqO 14; 14 14~ 14.~ 123 86 55 J9 55 88 123 138 138 1~8 138 138 138 138 11~'~ I.i4 i '4 1 ~ 5 I 35 135 135 1~5 136 13h 136 136 136 . -15~-209~0 Table 8 J.~9 I:~9 ~ 3 ]38 ]3~ 137 136 136 136 ~37 137 137 137 137 137 13, ~)7 1.37 1 ~7 137 137 137 137 137 137 137 137 137 137 137 137 137 137 13q 13c :130 137 13~ 137 137 ~ 37 137 136 135 134 139 139 1.39 138 1:38 138 13~3 137 1.-J7 137 136 l~i 137 137 137 137 137 137 137137 137 137 137 137 137 lJ7 13-/ 1.37 137 1~7 137 ~37 137 ~37 137 134 135 L 3~i 137 1.37 137 137 1-7 137 136 l:)S 134 I ~0 1:.19 l 19 ~ ] ~ 118 138 138 137 lJ7 137 136 lJ7 137 137 137 137 137 137 ~ ~7 137 137 137 1~7 137 137 137 13/ 137 13/ 1:~7 13~ 137 137 137 137 134 135 l~ti 137 137 137 13~ 137 137 136 135 134 140 140 13~ 139 139 138 138 138 138 137 137 137 137 137 137 137 137 137 137 1:~7 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 134 135 0 140 140 139 139 13g 138 138 138 138 137 137 137 137 137 137 137 137 137 134 lqo 140 140 139 139 139 138 138 138 138 137 1:37 137 137 137 137 137 137 137 137 137 137 137 137 137 137 l3? 137 137 137 137 137 137 137 137 134 135 136 137 137 1:37 137 137 137 136 135 134 82 127 lqO 140 140 139 139 139 138 138 138 138 13'7 137 137 137 137 137 137 136 137 137 ~37 ~37 137 137 136 lJS 134 137 137 137 137 137 137 137 137 137 137 137 137 ~37 137 137 137 137 134 135 ll(J 64 42 56 98 137 140 140 139 139 139 138 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 ~37 137 137 137 137 137 137 137 134 135 136 137 137 137 137 137 137 1-~6 135 134 146 120 73 43 50 88 131 140 140 1.39 139 139 1.37 137 1.37 137 1:37 137 137 137 137 137 13'7 137 137 137 1:37 137 i37 137 137 137 1:37 137 137 137 134 135 146 146 129 132 47 45 73 123 14(J 140 139 139 L37 137 137 137 137 137 137 137 137 137 137 137 1.37 137 137 137 137 137 ~37 1~7 ~37 137 137 137 134 135 1~ 137 137 137 13'7 1:37 137 1~6 1 ~s 1:3~

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142 Iq2 143 14313999 56 41h~l 109 138 138 140140 140 139 139 139 139 I i~3 l38 138 13713-;117137 137137 137 ~.37 1371-.171-7 137 137 137 1.37 137 ! 37 137 137 1371.37137137 1371.i7137 Iql 14;~ 142 14i~14 3 142 10762 41 57 101 136141 140 140 140 135 139 139 ~39 1.38 138 13813713'7137 13713~ 137 1371.37 13~137 137 137 137 137 137 141 141 141 14214~142lq3 1166Y 42 52 92 127141 140 140 140 li9 139 139 139 138 138138137137 137137 137 137 137 1371::~7137137 137 137 137 117 137 137 1371371371i7 1371:~7137 14~) 141 141 141141142142 lq2123 76 qq q7 71118 141 140 140 140 139 139 139 139 138138137137 1371:37137 137 137 137137 137 137 137 137 137 140 140 140 141lql141lql lq2142 130 . 8447 39 62 108 140 140 14() 140 137 137 137 137 137 137 1:37 137 137 137 1.39 139 140 140140141141 141141 142 135 93 61 39 55 98 137 140 14() 140 139 139 139139].37137 137137 137 137 137 137137 137 137 137 137 137 1:37 137 137 1371:37137137 137137 137 1 ~3 138 139 13g13913C.j140 140140 141 141 141 142127 80 45 44 77 123 140 140 140 139139137137 1:37137 137 1.37 137 137137 1371.37 137 13'7 137 .i J/ 137 137 13'71.37 137 1:37137 137 1.3'7 13-/ 138 138 138139139139 13'~140 140 lqO lql 142lq3 135 91 51 ql 68 114 140 140 140139137137 13'/13'7137 137 13713'713? 137 137 137 137 l.37 13/ 137 137 137137137137 li71'~'713'7 1 37 137 137 1381 38138139 1391:19139 1qO14() 1ql14~ lq3 141 103 58 41 lOq 139 140140137137 137l'i7137 1371.37 1371371:37 137 137 137 137 1.37 1 :17137 137137137137 1 J71 i713'7 136 137 137 1371371.38l'i81 i~ 9 139 139 139 14014214:3 144 lq5 114 h7 q~ 'j3 9q 135140137137 '1371i713'7 1371.~71'371 ~ 137 137 137 137 1371.37 137 137 1.371 ~7 137 137I:i7li7 137 ..'', "

Tabl~ ;~ (cnt'd) 2091920 I s~3 l i~.sl.~Ss1~7l.S/ 137 13~ 1361:s6 IJ633'jl~'J1 s's14!)140I40 J40 130 '.)6',o 4 3tit)1()/IJ9IJ'S13~JL.3') 13l~I 3S~138 IJ/1~'7 13/137 13f~!3j J:s5 1 s '~ ',I 351.5 '~ 51 '~ SI :J ')1 :I ',13 ' I:J'3 1.39I3fll.~.s1 ~$3 13S3 137 1'371.3'71.3613fi136 1391:~q 13914()14() !'J() 14U
141112 6'/4 S',',1()413~i 139 lJ9 1:39 IJ9 13813f313fs 137137 I3~lJ6 13S
13ti1 S6 1:3ti 136 136 1 36 136 136 1~6 136 IqOI ~'3 l I~(S1 ~S313f~13S31:3fs1~I7 13713'7 136IJS.3139 139139 19()140 140 14()141 1~6c~,'qs~~io s37 130 1~10 139 139 139 139 lJ8 1381:3S313~1.36 136 I S61.36 1.~61.St! 136 136 1361 :~t- 13fi 136 1q()14~) 1401341:3913`,S13S3 138 1.38 138 137 137 138 138 139139 1:34140 140 140190 1411'3'797 54 4 3 69 113 140 140 13913C/l `s9139138 13813t; 136 1:3t-1 13613613613t> 136 136 1.36 136 136 141140 140140140 139 139 139 138 138 138 138 138 138 138139 139139 i40 13'7137 13713'f1~7137 137 137 137 137 141141 141190140 140 140 139 139 139 138 13813i3 13813~3138 13g139 134 140140 14014()141143 131 c,s7 51 44 69 111 140 140 140139 139~37 137 13'7137 137l~71 S7137 137 137 137 137 137142 141141141 140 140 140 140 1.39 139 139 137 138 138138 13813g 139 139140 140140140 142 143 143 110 67 44 51 8S 125 140140 14()137 137 13'7137 1371371 S'71.171.37 137 1.37 137 85126 142142141 141 14114s') 140 140 1401:391.37 137 138138 1381 s~3139 1 s9139 140140]40 140 141 142 143 132 91 54 43 61 98133 140137 137 137137 1371371.3'7137 137 13'7 1.37 137 457() 111141142 142 141 141 141 140 140 14013-.~1:37 137138 13Ss138 138 I S'~ 139139140 1qO 138 139 141 14214.3 1q4 117 76 48 45 691071.3~3 138 13f313~3 13813~,13~31.'.s81381381 Sf313~`s 61 44 579S13.~ 142 142 14;~ 141 141 141 140I:j~,137 137137 1 j~ 3 138 1 sfs 1 S913913Y14(J 137 138 139 140 141 142 143 134 10i 67 454fs s3-/ 122 1Jc,13S3 13f~13f31~S31381381.3~3 13S3 li'3 11/'/S 4'749 7~s12() 142 142 142 1.42 141 1411 ~6 136 137137 1371 ~S~s13c31381~8 1341341 i9135 1:36 137 13813~:~ 141 14214.'~1'14133 97 62;s 3 54 83118 1~813f~1'3813813~s 138 13~s 1:38 144131 '~1~5 44 64 1()5 139 1~2 142 142 14213fi13~; 136137 137137 1313 1 :38 1 ~81 ~f31 iC31 JC~I 31 1 ~4 1351 :371.3813914(j 14114~ 14314S 129 87 58 44 51 /~311~13~134 1:33 13Y 1`3~ 1:39 2 ~ 2 ~
Table 8 _cnt ' d ) 1 'l i 14 JI 'i ~I 2()~ 144 l~b'~61 i41 i91 J'J1 J91 i91 i91 i~3 1 J4 ~ 38 1.3f~ 1 ifl 13-~ 13'~ 13~13-~13613813fi13~3 1~313f3138138138138138138138 131 10 67 4~ 9~'5f~f~'7119li3li5 137139 i42 1~ 14)1"313185 49 477~j 118l j~)13~140140139139139139138 13 f3 138 11 /1171 J 713813813813 f113813313 f1138138138138 138 139 139 ~32 103 7() 9~ 93 S9 89i2113'~1:39 191 142 14214;~ 143140103t;l 93 S7(34 130191140140140139 139 139 li9 138 13813813713813813813813f3138118138138138138138138 139 i'j 13') 133lOS73 S0 95 61 91 124 11 lql14214214~1431:'3~31S0 45 67 111140141lqO140140 139 li9 139 139138138138138138l i8138138138i38138138138138138 138 13~3 139 13913913S10875 524'7 62 14() 139 139139139138138138138138138138 138138138138138137 138 140 140 140 139 139 138 138 138 138 138 1.38 138 1.38 138 138 138 138 137 137 141 lql 14014014013813813-31:381381381381:38138138138138136136 137 137 137li8lJ8138138139139139 i i9 139 139139140140140141141141141142 144139105 67 46 50 76 112 139 14114114013813813f.~ 1381381.38138338138138138138 136 136 l.i6 136 13713713713813813f31 j81:39 138 138 13913913913914014()140141141141142143143132 96 61 45 S~ 78 113138141138138138138138138138 138138138138138135 135 13fi 136 13613613713713713~138138 1-7 138 13813813913913913~140140140141 141141142142143127 91 S9 44 S1761091381381381 i8138138138 13813813813813l3134135 l35 13S ~.3613~'j13613613713713'7138 l'i7 li7 li'713fi13~3lifi~.39l:i~3li9li4140140139140140141 142 142 143 125 <:J0 6044 49 1001331:38138I3U13f3138138138138138138134 134 ~'14 1 i5 l:iS1351 ~613613~136 1371:~7 13h 1 i ~ 1371 i 71371381381 ~f31 :i91391.-14139l.38138139140140 141 141 342 142 12f~ 94 63 416.i 493 il1381381 ~8138138138138138 133 134 1:14 134 1~43.~S 13513513h13~j136136 -: . . ~ ' , ' ' -~63- 2091~2~

'rc~ble 9 !
-O . -~V~3~ H O .7976 K: 0. 0094 H:0. 3366 Proflle type: 1 a: ~ .9j14 1 17.~itj67 ~
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~ 0.117~3 -0.24~50.324~3 -0.0091 1 'I`
~ 9.0694 1 ~ O .213~O .1408 I -0. ~)93C~ 0.1978 0.1984 ~
-O.iO13 -0.06980.0375 0.1925 ) ;3.090~ ~ ~ 0.1559 -O.2263 ) -O .1065 O.1015 -O .5252 1 ~ --0.0788 0.0514--O . O j46O .2444 ~

Q 2.4427 R. 0. 7269 K: 0. 0125 H:0. 3295 Profile type: 1 a: -99.85fi9 10 . 0000 ~ 12.5987 ) ~ -2.0156 1.2151 ) Q -0.6614 R: 0.5372 K: -0.0131. H:O.3295 Profile type: 1 a: -103.4667 22.3493 1 ~ -2.3316 4.0633 ~
11.9938 ) ~ 1.1234 -1.0171 ) T

9.270~3 ~ ~ O.2726 -0.40~3 --0.0481 -0.2758 0.1060 ~
O.()ltj4 -0.2178 0.2337 -0.1412 1 8.6944 1 ~ -0.0141 0.0181 --0.2915 0.2640 -0 125fi 1 ~ O 15~9 -0.2775 -O.2572O. 071~3 ~

Q --0.6265 R: 0. 9160 K: -0.0100 H:0.:329S
Profile type: 1 a: --99 fi7 7 3 16.0000 ~
11.-~162 ~ ~ 1 502C,~ -1.2395 1 Table 9 (cnt'd) 2091920 ~: 2.5066 R: 1.0541 K: 0.0103 H:0.3117 E~rotile type: 1 a: -92. i3743 O . 00()0 1 ~ 1().5651 1 1 --1.3324 ().6983 1 Q: -0.5703 R: 0.0839 K: ~() .0214 H: 0.3237 E~-of~l~ type: 1 a: -101.589fi ~0. i~ 5.9137 3.7597 13.1990 ~ ~ 1.7771 -1.1217 E

Q: -0.4601 R: 1.0753 K: -0.015Q H:0.2796 Profile type: 1 a: -90. 3296 O.0000 ~ 9.1003 1 ~ 0.6131 0.0838 E
Q: 2.5517 R: 0.7077 K.: 0.0638 H: 0.3138 Profile type: 1 a: -98.5962 11.4V00 1 ~ 13.34r~S ¦ ~ - 2.2201 1.4149 ¦

Q: -0.5143 R: 0.3120 K: -0.0245 H:0.2994 Profile type: 1 a: -99.8304 i 20.6s34 1 ~ -4.5809 2.0873 ~ 12.5008 1 ~ 1.7990 -1.0-01 1 Q: -O.q397 R: 1.2347 K: -0.0270 H:0.2897 Profile type: 1 a: -94.4909 () . O(j()O
~ 9.5~31 ~ ~ 0.4157 0.035.3 E`
Q: 2.~754 R: 0.6984 K: 0.01.03 H:0.2578 Profile type: 1 a: -92.1178 --0.061(j ~ ~ 0.0000 6.8199 10.4()21 ~ ~ -1.9~67 0.6955 'L`cl~r~)l.e~

--() 687$ R: 0 812') K: 0.0000 fi:0.34i8 Ylotile type: 1 a: -102.0000 i.~.0000 11.000() ~ ~ I.()000 -1.0000 1 '3 . ()()O() ~ I O ~)()UO O . 0000 ~).0000 0.0000 ().0000 ().~)000 0.0000 ().0000 0.00(~0 9.0000 1 ~ O.0000 0.0000 O.0000 0.0000 (~.0000 O.0000 0.0000 0.0000 0.0000 T

8.0000 1 ~ 0.0000 0.0000 O.0000 0.0000 -1.0000 O.0000 0.0000 0 0000 0.0000 ~: 2.4375 R: 0.71f38 K: 0.0000 H:0.3438 Profile type: 1 a: -10().0000 10 . 0000 13.0000 ~ ~ -2.0000 1.0000 1 ~: -0.6875 R: 0.5312 K: 0.0000 H:0.3438 Profile type: 1 a: -104.0000 ~2.0000 1 ~ -2 0000 4.0000 ) ~ 1~.0000 1 { l.oooo --1.0000 ) T
9.000() i ~ 0000 i O.0000 0.0000 O.00()0 1 0.()00() 0.0000 0.0000 O.0000 1 9.0()00 ~ ~ O.0000 0.0000 O . 0000 0 . 0000 -O.0000 0.0000 0.0000 0.0000 ~: -0.62S0 R: 0.9062 K: 0.00()0 H: 0.34~8 Profile type: 1 a: --100.0000 }6.0000 i 1~.0000 ) ~ 2.0000 --1.~000 --166- 2~i92~

Table 1 0 (_cn t ' d ) ~: 2~5000 Q: 1.0625 K: 0.0000 H: O . 3125 ~ofile type: 1 a: -92.()000 O . 00()0 11.0000 ) ~ -L.0000 1.0000 1 ~ 0.5625 ~: 0.0938 ~(: -0.0312 1~:0.3125 Plofile type: 1 a: -102.0000 21.0000 1 ~ -6.0000 ~.0000 ~
~ 13.0000 ) ~ 2.0000 -1.0000 ) 1, Q: -0.4375 R: 1.0625 K: 0.0000 H:0.2812 Profile type: 1 a: -90.0000 O.0000 9.0000 ~ ~ 1.0000 0.0000 E

Q: 2.5625 R: 0.7188 K: 0.0625 H:0.3125 Profile type: 1 a: -98.0000 11.0000 J
13.0000 1 ~ -2.0000 1.0000 ) E

~: -0.5000 R: 0.3125 K: -0.()312 H:0.3125 Profile type: 1 a: -100.0000 21.0000 1 ~ -5.0000 2.0000 1 ~ 13.0000 ; ~ 2.0000 -1.0000 E
Q: -0.4375 R: 1.2500 K: -0.0312 ~:0.7812 Profile type: 1 a: -94.0000 O.0000 ~ 10.0000 ) ~ 0.0000 0.0000 1 E
Q: 2.6375 R: ().6875 K: 0.00~0 H:O. 31;'5 Profile type: 1 a: -92.000() 000 1 ~ 0.0000 7.000~ 1 10.0000 ~ ~ -2.00()0 1.0000

Claims (32)

1 A process of picture representation by data compression which comprises the steps of:
1 - subdividing the picture into regions;

2 - registering for each region a set of brightness values;

3 - fixing for each region a characteristic scale in terms of a number of pixels;

4 - dividing each region into cells, each comprising a number of pixels defined by two coordinates, said cells having a linear dimension in the order of L said characteristic scale;

5 - identifying in each cell the basic structures, as defined in the specification;

6 - in each cell, representing the basic structures by models, as defined in the specification; and

7 - storing and/or transmitting the data defining each model, said data representing the primary compression of the picture.

2 - Process according to claim 1, comprising application of image processing operations to compressed data.

3 - Process according to claim 1, comprising further quantizing the primary compression data.

4 - Process according to claim 1, wherein the brightness values are chosen from among those of one of the basic colours and one of those obtained by transformation of said brightness values.

5 - Process according to claim 1, which comprises identifying the basic structures and representing them by models, by:

I - identifying structures which comprise smooth areas, edges, ridges, positive and negative hills, and optionally saddles, as defined in the specification;

II - constructing geometric models representing said structures;

III - associating to each of said geometric models a mathematical model representing it;

IV - condensing said models to define a global model for the cell; and subsequently V - quantizing and encoding the data defining said global model, wherein steps III and IV may be partly concurrent.

6 - Process according to claim 1, further comprising decompressing said compressed representation to reconstruct the picture.

7 - Process according to claim 5, wherein the geometric and/or mathematical models representing picture elements or other geometric and/or mathematical models, approximate these latter to a degree determined by predetermined absolute or relative parameters.

8 - Process according to claim 5, wherein the basic structures include:

"background" areas, wherein the values of the brightness function change slowly;

edges, which are curvilinear structures on one side of which the values of the brightness function undergo a sharp change;

ridges, which are curvilinear structures defined by a center line, the cross-sections of which in planes perpendicular to the center line are bell-like curves;

hills, which are points or small areas at which the brightness function value is a maximum or a minimum and decreases or increases, respectively, in all directions from said point or small area; and optionally saddles, which are curvilinear structures which comprise a central smooth region bounded by two edges, wherein the brightness function values increase at one edge and decrease at the other.

9 - Process according to claim 5, wherein the basic structures are mathematically defined through the derivatives of the brightness function z = f(x,y), x and y being the coordinates of a coordinate system of the region considered.

10 - Process according to claim 5, wherein the brightness function is approximated by an approximating function p(x,y).

11 - Process according to claim 10, wherein the approximating function is a polynomial of the second degree having the form p(x, y) = a00 + a10x + a01y + a20x2 + a11xy + a02Y2.

12 - Process according to claim 5, comprising eliminating structures and/or portions thereof.

13 - Process according to claim 5, comprising omitting part of the models.

14 - Process according to claim 1, comprising defining a number of models and associating to each type of model a code identifying it, the model being defined by said code and by its parameters.

15 - Process according to claim 1, comprising eliminating information having a limited psycho-visual significance by dropping some models and eliminating some excessive parameters of the models that are retained.

16 - Process according to claim 5, comprising assigning each grid point to one of three domains A1, A2 and A3, wherein:
domain A1 contains all the points where all the derivatives of the approximating function p of orders 1 and 2 are do not exceed a threshold T1, domain A2 includes the grid points where the gradient of the approximating function p is large; and domain A3 includes the points in which the second order derivatives are bigger than a threshold T3, while¦gradp¦does not exceed another threshold T2; and identifying in the domains the basic structures.

17 - Process according to claim 16, wherein the domain A1 is the set of the points of the grid for which ¦?p¦2<Gabs,.lambda.12+.lambda.22<Sabs?
wherein Gabs and Sabs are thresholds and .lambda.1 and .lambda.2 are the eigenvalues of the matrix the domain A2 is the set of the points of the grid for which ¦?p¦2?max[Gabs,Grel]
where Gabs is a threshold and Grel is a relative threshold, and the domain A3 is the sum total of five sub-domains defined by the following conditions:

(1 )¦.lambda.2/.lambda.1¦<M ratio, .lambda.1 + .lambda.2<0 (2) ¦.lambda.2/.lambda.1¦<M ratio, .lambda.1 + .lambda.2>0 (3) ¦.lambda.2/.lambda.1¦?M ratio, .lambda.1<0,.lambda.2<0 (4) ¦.lambda.2/.lambda.1¦?M ratio, .lambda.1>0,.lambda.2>0 (5) ¦.lambda.2/.lambda.1¦?M ratio, .lambda.1?.lambda.2<0 wherein Mratio is a threshold.

18 - Process according to claim 5, which comprises the steps of:
1) registering the values of the brightness function z = f(x,y);
2) determining a window and a grid related thereto;
3) for each grid point, approximating the brightness function z(x,y) by an approximating function;
4) fixing a number of thresholds;

5) assigning each grid point to one of three domains A1, A2 and A3;
6) identifying in the domains the basic structures;
7) approximating the curvilinear basic structures by lines related to their center lines and parameters related to their profiles;
9) representing the basic structures by mathematical models;
10) interrelating the models thus constructed to construct a global model;
12) quantizing the data thus obtained; and 13) encoding the same.

19 - Process according to claim 1 wherein the characteristic scale is comprised between 6 and 48 pixels.

20 - Process according to claim 18, wherein the window has a linear dimension comprised between 2 and 6 pixels.

21 - Process according to claim 20, wherein the window has a linear dimension chosen from among 3 and 4 pixels.

22 - Process according to claim 3 or 18, wherein the data quantization consists in substituting the values of the compressed data by the closest of a predetermined set of values.

23 - Process according to claim 3 or 5, comprising encoding the compression data by representing the corresponding quantized data in the form of a binary file.

24 - Process according to claim 1, comprising repeating its operations severally for a set of brightness functions chosen from those representing the basic colors or those representing a monochrome signal and color data information carrying signals.

25 - Process according to claim 5, comprising carrying out its operations for one of a set of brightness functions chosen from those representing the basic colors or those representing a monochrome signal and color data information carrying signals, retaining the geometric parameters found for the models of said one brightness function and repeating said operations for the other brightness functions of the same set and using the same models with the said geometric parameters and the appropriate brightness parameters for each of the other brightness functions of the same set.

26 - Process according to claim 18, wherein the window is a square having a side of a few pixels and the grid is constituted by the pixels themselves, if the side is an uneven number of pixels, and is constituted by the central points between the pixels, if the side is an even number of pixels.

27 - Process according to claim 2, wherein the processing comprises one or more operations chosen from among further compression, picture comparison, feature stressing, picture enhancement, creation of visual effects, color operations, geometric transformations, 3D-geometric transformations and texture creation.

28 - Process according to claim 1, wherein the information defining the brightness distribution of the various colors is a function of time.

29 - Process according to claim 1, wherein the characteristic scale is between 6 and 48 pixels.

30 - Process for video sequences compression, wherein only a subsequence of (control) frames is compressed by the process of claim 5, and the intermediate frames are represented by the same models as the control frames, with the parameters obtained by interpolation from the control frames.

31 - Picture compression process, substantially as described and illustrated.

32 - Process according to claim 1 wherein the steps 5 and 6 are performed by choosing among the prefixed list of models, those which (after minimization with respect to the parameters) provide the best approximation of the picture.