FIELD OF THE INVENTION

[0001]
The current invention is generally related to a method and a system for correcting distorted aberration in images, and more particularly related to a method and a system for determining coefficients for correcting distorted aberration based upon corresponding portions of partially overlapping images.
BACKGROUND OF THE INVENTION

[0002]
An image generated by an optical lens is generally susceptible to distortion or aberration. The distorted aberration of a lens is inherent and roughly defined by a positional or geometric discrepancy between a generated image and a true object. In other words, a certain portion of the image generated by the lens does not reflect an original object. For example, a straight line in the original object is often curved in an image generated by a lens. To further define the distorted aberration, referring to FIG. 1, Y is an object placed before a lens, and θ is an incoming angle of the object Y. y′ is a distorted image focused by the lens with an outgoing angle of θ′. Y′ is an assumed true image without distortion with an angle θ_{0}. The distorted aberration of the lens is defined either in terms of angle β or distance D as follows:

tan(θ′)/tan(θ)=β (1)

(y′−Y′)/Y′·100=D (2)

[0003]
Now referring to FIG. 2, the effects of the distorted aberration are illustrated in terms of the above defined equations (1) and (2). FIG. 2A shows an original object, and FIGS. 2B and 2C respectively show an image formed by a lens which caused distorted aberration in the image. As θ in equation (1) increases and β decreases, the original lattice image is distorted like a barrel as illustrated in FIG. 2C. Similarly, as D in the equation (2) is negative, the barrel image is also formed. On the other hand, as θ in the equation (1) increases and β also increases, the original lattice image is distorted like a spool as illustrated in FIG. 2B.

[0004]
Similarly, as D in the equation (2) is positive, the spool image is also formed. In other words, the distorted aberration of a lens causes portions of an image to be variably affected depending upon a distance from the optical center of the lens. To correct the image, certain lenses have been designed to compensate for the distortion. However, such lens design efforts are generally limited in solving the problem due to advanced features such as zooming and autofocusing of the optical systems.

[0005]
Digital processing of the images is more efficient in correcting distorted aberration than lens design efforts. One digital processing technique in correcting a distorted image is to initially determine an amount of distortion in a known image such as a lattice or a test pattern which is taken by an optical system in question. A comparison between the known image and the distorted image yields a distortion amount, and an appropriate distortion correction factor is subsequently determined. In a similar approach, a known straight line in a distorted image is selected, and the distorted line is used to determine the above described distortion amount. Another digital processing technique in correcting a distorted image involves the use of reference markers or control points on a threedimensional object. Instead of using the above described twodimensional lattice, the known reference markers are imaged, and the geometric relation of the markers is digitally corrected according to the known marker positions.

[0006]
In the above described approaches, a series of the images must be generated with reference markers such as a lattice. In other words, the images without the reference markers are not generally suitable for digital correction of distortion. The placement of the reference markers is tedious since the image with the markers must be generated for each particular use of the lens such as a particular zoom distance. Furthermore, the marker placement is not always feasible or practical. For example, when the object is a distant scenery. Another example is old existing images which are recorded by a device that is no longer available. For these reasons, the use of reference markers is not always practical or feasible.

[0007]
The distorted aberration is especially problematic for highresolution wide angle images. Such images include panoramic images and aerial photographs. Since a series of partially overlapping images is taken and later composed into a single image, the distorted portions of the images especially near edges must be corrected prior to joining the edges of the images.
SUMMARY OF THE INVENTION

[0008]
In order to solve the above and other problems, according to one aspect of the current invention, a method of correcting distorted aberration in images taken by a common optical system, includes: a) taking at least partially overlapping images of an object, the images containing a common element; b) determining correction coefficients based upon a relationship between corresponding pairs of information representing the common element in the images; and c) correcting distorted aberration in the images according to the correction coefficients.

[0009]
According to a second aspect of the current invention, a method of correcting distorted aberration in images taken by a common optical system, includes: j) determining a correction coefficient based upon a relationship between corresponding pairs of information on a common element in the images, the images being at least partially overlapping to contain the common element; k) correcting distorted aberration in the images according to a predetermined correction function using the correction coefficient; and l) joining the corrected images so as to generate a composed single image.

[0010]
According to a third aspect of the current invention, a system for correcting distorted aberration in images, includes: an optical unit having an optical axis for focusing upon an object and generating information representing images which at least partially overlap and contain a common element of the object; a memory storage unit for storing the information; a coefficient calculation unit connected to the memory storage unit for determining correction coefficients based upon a relationship between corresponding portions of information representing the common element in the images; and an image correction unit for correcting distorted aberration in the images according to the correction coefficients.

[0011]
According to a fourth aspect of the current invention, a recording apparatus for correcting distorted aberration, includes: an optical unit having an optical axis for focusing upon an object and generating information including images which at least partially overlap and contain a common element of the object, the information also including image correction data; an image storage unit connected to the optical unit for storing the images; a correction information table connected to the optical unit for containing image correction data in correcting distorted aberration in the images; and an image correction unit connected to the correction information table and the image storage unit for correcting distorted aberration in the images according to the correction data.

[0012]
According to a fifth aspect of the current invention, a system for correcting distorted aberration includes: a memory storage unit for storing information representing images which at least partially overlap and contain a common element of an object; a pattern recognition unit connected to the memory storage unit for extracting a portion of the information corresponding to the common element from each of the images; and a coefficient calculation unit connected to the memory storage unit for determining correction coefficients based upon a relationship between the corresponding portions of the information representing, the correction coefficients including a first coefficient A and a econd coefficient B used in the equation, f (Φ)=1−AΦ^{2}+BΦ^{4 }where Φ is an angle representing the relationship of the common element in the images.

[0013]
These and various other advantages and features of novelty which characterize the invention are pointed out with particularity in the claims annexed hereto and forming a part hereof. However, for a better understanding of the invention, its advantages, and the objects obtained by its use, reference should be made to the drawings which form a further part hereof, and to the accompanying descriptive matter, in which there is illustrated and described a preferred embodiment of the invention.
BRIEF DESCRIPTION OF THE DRAWINGS

[0014]
[0014]FIG. 1 diagrammatically illustrates distorted aberration of a lens.

[0015]
[0015]FIGS. 2A, 2B and 2C respectively illustrate an image rendered by a lens without distorted aberration, with distorted aberration D>0, and with distorted aberration D<0.

[0016]
[0016]FIG. 3 diagrammatically illustrates one preferred embodiment of the current invention.

[0017]
[0017]FIG. 4 is a flow chart for illustrating the steps of operation in the preferred embodiment according to the current invention.

[0018]
[0018]FIG. 5 diagrammatically illustrates a second preferred embodiment according to the current invention.

[0019]
[0019]FIG. 6 is a table illustrating exemplary parameters to be used in determining correction coefficients.

[0020]
[0020]FIG. 7 is a graph illustrating input and output gamma conversion functions.

[0021]
[0021]FIG. 8 diagrammatically illustrate how images are formed when a lens is repositioned.

[0022]
[0022]FIG. 9 is a diagram illustrating the effect of the lens movement on the images.

[0023]
[0023]FIG. 10 is a flow chart illustrating the steps for determining correction coefficients according to the current invention using image data collected during one dimensional movement.

[0024]
[0024]FIG. 11 is a flow chart illustrating the steps for determining correction coefficients according to the current invention using image data collected during two dimensional movement.

[0025]
[0025]FIG. 12 diagrammatically illustrates vectors indicative of parallel shifts of the images.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT(S)

[0026]
Referring now to the drawings, wherein like reference numerals designate corresponding structure throughout the views, and referring in particular to FIG. 3, one preferred embodiment of the image capturing input device or the image recording device according to the current invention includes an optical unit 101 and a correction unit 100 for correcting distorted aberration in the optical unit 101. One example of the image capturing device is a digital camera. The optical unit 101 further includes at least a lens unit 1 and a lens positioning unit 90 for positioning the lens 1 at a desired distance with respect to an object to be imaged.

[0027]
Still referring to FIG. 3, the image correction unit 100 further includes an image capturing unit 103 such as a chargecoupled device (CCD), a driving circuit 105 for driving the optical unit 101 and an image signal circuit 104 for initially processing image data received from the CCD 103. The image signal circuit 104 performs any combination of an analogtodigital (A/D) conversion, a gamma conversion and amplification of the image data. An onboard central processing unit (CPU) 110 controls the driving circuit 105 and receives the image data from the image signal circuit 104. The image data may be stored in an image memory such as a detachable memory card 102 via the CPU 110. The correction unit 100 is interfaced with an external central processing unit or a peripheral device via an interface unit 109. A data memory unit 107 such as a ROM stores predetermined sets of data including a gamma conversion table and other predetermined parameter values as discussed below. These parameters are used in determining coefficient values used in a predetermined function for correcting the distorted aberration in the image data. An onboard key board unit or an input unit 108 is used to input or select a set of parameter data which is later used in correcting the distorted aberration in the image data.

[0028]
Referring to FIG. 4, a flow chart illustrates a series of actions taken for determining the coefficients so as to correct the distorted aberration in images. In a step 50, a first image of an object is taken using an optical unit. In a step 51, a second image is taken after the optical unit is rotated by changing the angle of the optical axis with respect to the object but without changing a distance from the object. In the alternative, in the step 51, the second image is taken by moving the optical unit in parallel to the object without changing the angle and without changing the distance. In either case, the second image is partially overlapping the first image. In other words, the second image contains at least a common element of the same object in the first image. After the above described first and second images are taken, in a step 52, a user decision is made as to whether an onboard computer which is located in the same system for correcting distorted aberration as an optical unit is used for taking the images. If the decision is affirmative, in a step 53, the above described correction coefficients denoted as A and B used in a predetermined function for correcting distorted aberration in the images are determined using the onboard computer, and in a step 54, the coefficients A and B are stored in a memory unit.

[0029]
Still referring to FIG. 4, when an external computer is to be used, the image data as well as other information such as the above described distance is sent to the external computer via an interface unit in a step 55. In a step 56, the above described coefficients A and B are determined by the external computer, and in a step 57, the coefficient values are sent back to the system for correcting distorted aberration. The coefficients are stored in a data memory in a step 58.

[0030]
Referring to FIG. 5, according to another preferred embodiment of the current invention, a system for correcting the distorted aberration includes an image file 51, an image buffer 52, a calculation control unit 54 and a parameter memory 55. The image data file stores image data representing a series of partially overlapping images. The image buffer 52 holds the image data for processing. The pattern extraction unit 53 extracts a common element or pattern from the partially overlapping images before the calculation control unit 54 determines coefficients A and B used in a function which corrects distorted aberration based upon the relationship between a common element in the images. The coefficients A and B are respectively related to a squared and a fourth powered terms in a function which will be described below in reference to equation (4). In addition, the calculation control unit 54 employs certain parameter values stored in the parameter memory 55 during the determination of the coefficients A and B.

[0031]
Referring to FIG. 6, one exemplary set of the above described parameter data is shown in a table format. The parameter values are used in determining the correction coefficients and include fixed value parameters as well as image captured condition parameters. The fix value parameters are generally predetermined at the time when an optical unit is assembled. Examples of the fixed value parameters include a pixel pitch size Px and Py respectively indicating a distance between pixels such as in a CCD in the horizontal and vertical directions as well as a lens focal length f. On the other hand, the image captured condition parameters are determined when an image is captured. Examples of the condition parameters include a lensimage distance R indicating a distance between the center of the lens and an imageforming surface, a F value indicating luminance, a gamma conversion table value and an amplification value for processing the image data and an average color value or a dynamic range of the image data.

[0032]
The pixel pitches Px and Py are generally constant. In order to correct the images, the pixel pitches Px and Py correlate imageforming surface coordinates x′ and y′ to digital coordinates k′ and l′ as indicated in the equation: (x′, y′)=(k′·Px, l′·Py). The digital coordinates k′ and l′ are obtained from positions on an image recording device such as a CD while imageforming surface coordinates x′ and y′ are calculated. In the following description of the current invention, the above described digital coordinates are generally used to determine the correction factors or coefficients for correcting the distorted aberration in the images. Since the pixel pitches Px and Py are generally fixed on an image recording device such as a CCD, the pitch information is stored in a memory at the time of the recording device assembly. In the alternative, the pixel pitch information is stored as a part of the image data for each image when a plurality of imagerecording devices are used.

[0033]
The correction coefficients A and B are also generally constant for a lens at a predetermined distance with respect to an object. At a fixed distance, the correction coefficients A and B are substantially the same for the lenses. For this reason, the correction coefficient values are stored in a memory device such as a ROM at the system assembly time according to one preferred embodiment. However, because of slight variations in optical characteristics of each lens, the correction coefficients A and B are initially determined prior to use according to another preferred embodiment. As described above, the correction coefficients A and B are determined using either an onboard computer or an external computer via an interface unit.

[0034]
Other information is generally determined as the images are taken. For example, the distance R between the imageforming surface and the center of a lens is obtained from a driving unit for positioning a lens with respect to an imageforming surface. Another example includes a gamma conversion table value and an amplification value. Because these values are determined based upon the input image data, the values must be determined for each input image. Since an imagerecording device such as a CCD generally does not have a large dynamic range, the range should encompass the range of human perception. For example, shades of one color are coded by eight bits, and the maximal and minimal input values of an image are expressed by the eight bit range while the overall average is color balanced to make the sum of color signals appear monochromatic. In other words, a median value of the luminance distribution lies on a middle of the dynamic range by adjusting the amplification gain value together with the gamma conversion table.

[0035]
Referring to FIG. 7, a graph illustrates how a dynamic range affects input reproduction signals. In general, the input signal I is defined by the equation, I=[pixel signal]·[an amplification gain value]. Given three output signals O_{1}, O_{2}, and O_{3}, depending upon a gamma table such as γ1 and γ2 and its associated dynamic range such as Dlm_DlM and D2m_D2M, the input reproduction signals are different. The input reproduction signals are used to generate a corrected image and respectively include I_{11}, I_{12 }and I_{13 }for the output signals O_{1}, O_{2}, and O_{3 }when the γ1 table is used and I_{21}, I_{22 }and I_{23 }for the output signals O_{1}, O_{2}, and O_{3 }when the γ^{2 }table is used. Both the gamma conversion table and the amplification value are stored in the above described parameter table. In the alternative, this information is stored in the image data itself. In processing the image data representing a series of partially overlapping images, the above described dynamic range conversion should be applied after the distorted aberration of the images is corrected and the corrected images are joined together to form a single image. In other words, since input values in each of the series of images such as panoramic pictures are different, if each image is treated before joining them, the joined image may not have a uniform appearance. By the same token, the color balancing process should also be applied after the corrected images are joined together into a single image. Color balancing information is stored in a memory device or in each image data.

[0036]
Referring to FIG. 8, in order to determine correction coefficients used in a predetermined function for correcting distorted aberration in images, at least two images are formed through a common lens 1 at two angles with respect to its optical axis 2. According to one preferred process of the current invention, an image input capturing device having the lens 1 such as a camera is employed, and the image input capturing device is moved from an original position in such a manner that an optical axis of the lens at a new position forms a known angle with respect to the optical axis at the original position. To illustrate how the two images are formed in the above described manner, two incoming beams of light A and B reflecting respective objects are shown in solid lines. When the lens 1 is at a first angular position α, the incoming beams A and B exit the lens 1 and project beams A′ and B′ towards an image forming surface 3. When the lens 1 is moved to a second angular position β as shown in a dotted line, the incoming beams A and B now exit the lens 1, and beams A″ and B″ as shown in dotted lines are projected towards an image forming surface 3′. An arrow R indicates a distance between a center of the lens 1 and the image forming surface.

[0037]
Still referring to FIG. 8, the above described light beams form certain angles with respect to an optical axis. The beams A and B respectively form angles φ_{j }φ_{i }with respect to the optical axis 2. The exited beams A′ and B′ respectively form angles φ_{j}′ and φ_{i}′ with respect to the optical axis 2 when the lens 1 is at the first position α. Similarly, the exited beams A″ and B″ respectively form angles φ_{j}″ and φ_{i}″ with respect to the optical axis 2′ when the lens 1 is at the first position β. The above described angles φ_{i}′, φ_{j}′, φ_{i}″ and φ_{j}″ are measured values. The above measured values are not true values and generally include errors due to distorted aberration of the lens 1.

[0038]
For the purposes of the current invention, the true values are defined as values that are free from the effect of distorted aberration of a lens and are denoted by φ. In contrast, approximated true values are denoted by φ_{i0}′, φ_{j0}′, φ_{i0}″ and φ_{j0}″ and calculated based upon the measured values φ_{i}′, φ_{j}′, φ_{i}″ and φ_{j}″. In other words, the true value Φ′ is related to an approximated value Φ′ as described by the following equation:

Φ′=Φf(Φ) (3)

[0039]
The function f in the equation (3) for converting a true value to an approximated distorted value is expressed as follows:

f(Φ)=1−AΦ ^{2} +BΦ ^{4} (4)

[0040]
where A and B are predetermined values or distorted aberration coefficients.

[0041]
Furthermore, the following squared difference between the true values should be equal without distorted aberration when the lens position is rotated in the above described manner.

(Φ_{io}′−Φ_{jo}′)^{2}=(Φ_{io}″−Φ_{jo}″)^{2} (5)

[0042]
However, the sum of the difference between the right side and the left side of the equation (5) as expressed below is used as an evaluation function:

E=Σi,j{Φ _{i0}′−Φ_{j0}′)^{2} (6)

[0043]
where E is a minimum error value because the above described equations (3) through (6) are expressed in terms of angle, the relationship between the angle values and the coordinate values is described below:

{square root}{square root over (x ^{2} +y ^{2})}=R tan(Φ′) (7)

[0044]
where R indicates a distance between a center of the lens and the image forming surface while x and y respectively indicate coordinates. Based upon measuring multiple pairs of the x and y coordinate values and the R value, φ_{i0}′, φ_{j0}′, φ_{i0}″ and φ_{j0}″ values are determined, and these approximated true values, in turn, determine A and B.

[0045]
As an alternative process according to the current invention, the image input capturing device or an optical system is moved in parallel to an object without rotation. For the alternative process, the equations (5) and (6) are respectively expressed in the equations (8) and (9) as follows:

(X _{io} ′−X _{jo}′)^{2}=(X _{io} ″−X _{jo}″)^{2} (8)

[0046]
where X_{io}′, X_{jo}′, X_{io}″ and X_{jo}″ are respectively a corresponding X coordinate of the beam position on an image forming surface. However, the sum of the difference between the right side and the left side of the equation (8) as expressed below is used as an evaluation function:

E=Σi,j{(X _{i0} ′−X _{j0}′)^{2} (9)

[0047]
The current invention is not limited to either of the above described movements of the image capturing device but also includes a combination of the movements.

[0048]
In order to determine correction coefficients for correcting distorted aberration in the images, a separate process is performed based upon how images are generated. Assuming that at least a pair of partially overlapping images have been recorded in the above described manner and a common element or the corresponding portions in the images have been identified, if corresponding pixels in the common element are located along an optical axis of the lens used to form the images, the correction process is performed in one dimension. On the other hand, if the corresponding pixels in the common element are not located along an optical axis of the lens used to form the images, the correction process is performed in two dimensions. Referring to FIG. 9, a flat image forming surface 3 shows the corresponding point locations of the images as a result of the lens movement. Point P_{i}′ corresponds to point P_{i}″ after the lens 1 has been moved by rotating the direction without changing a distance to an object. Similarly, point P_{j}′ corresponds to point P_{j}″ after the lens 1 has been moved. In contrast, points P_{i0}′, P_{i0}″, P_{j0}′ and P_{j0}″ respectively represent correct or true locations (i.e., without distorted aberration) of the corresponding points P_{i}′, P_{i}″, P_{j}′ and P_{j}″ For purposes of the illustration, distortions in the direction are overly exaggerated.

[0049]
In the case of one dimension, the distorted aberration correction coefficients A and B are determined as illustrated in a flow chart of FIG. 10. In a step S1, coefficients A and B are respectively initialized to a predetermined number A_{0 }and 0. Similarly, a distance R between a lens and an image forming surface is set to a known value. In a step S2, from each of at least two images, a portion including a common pattern is extracted. Within the common pattern portion, two pairs of corresponding pixels are selected from the two images. The two pairs of the pixels are denoted as (P_{i}′, P_{i}″) and (P_{j}′, P_{j}″). Since the two images are moved in one dimension, the above corresponding pixels are located on an optical axis at (x_{i}′, y_{i}′) and (x_{i}″, y_{i}″). Still in the step 2, the above selected two pairs of the pixel coordinates are converted into angles using the following general equation (10):

{square root}{square root over (x ^{2} +y ^{2})}=R tan(Φ′) (10)

[0050]
Applying the above coordinates to the above equation (10) yields:
$\begin{array}{c}\Phi \ue89e\text{\hspace{1em}}\ue89e{i}^{\prime}=\mathrm{arctan}\ue89e\text{\hspace{1em}}\ue89e\left(\sqrt{{\mathrm{xi}}^{\mathrm{\prime 2}}+{\mathrm{yi}}^{\mathrm{\prime 2}}}/R\right)\\ \Phi \ue89e\text{\hspace{1em}}\ue89e{i}^{\u2033}=\mathrm{arctan}\ue89e\text{\hspace{1em}}\ue89e\left(\sqrt{{{\mathrm{xi}}^{\u2033}}^{2}+{\mathrm{yi}}^{\mathrm{\u20332}}}/R\right)\\ \Phi \ue89e\text{\hspace{1em}}\ue89e{j}^{\prime}=\mathrm{arctan}\ue89e\text{\hspace{1em}}\ue89e\left(\sqrt{{\mathrm{xj}}^{\mathrm{\prime 2}}+{\mathrm{yj}}^{\mathrm{\prime 2}}}/R\right)\\ \Phi \ue89e\text{\hspace{1em}}\ue89e{j}^{\u2033}=\mathrm{arctan}\ue89e\text{\hspace{1em}}\ue89e\left(\sqrt{{\mathrm{xj}}^{\mathrm{\u20332}}+{\mathrm{yj}}^{\mathrm{\u20332}}}/R\right)\end{array}$

[0051]
In a step S3, by substituting the above obtained values Φi′, Φi″, Φj′ and Φj″ into the following equation (11):

Φ′=Φ−AΦ ^{3} +BΦ ^{5} (11)

[0052]
The above equation (11) is obtained by substituting the equation (5) into the equation (3). As described in the step S1, since the coefficient B is 0, the above equation (11) is now AΦ^{3}−Φ+Φ′=0. In order to solve this equation, the following p and q are introduced:

p=−1/(3A), q=Φ′/(2A)

[0053]
Furthermore, using the above defined variable, the following equation set (12) is defined as follows:
$\begin{array}{cc}s={q}^{2}+{p}^{3}\ue89e\text{}\ue89eu={\left\{q+\sqrt{s}\right\}}^{\frac{1}{3}}\ue89e\text{}\ue89ev={\left\{q\sqrt{s}\right\}}^{\frac{1}{3}}& \left(12\right)\end{array}$

[0054]
The solution for the above equation set (12) are obtained as follows:

Φ_{1} =u+v Φ _{2}=−(u+v)/2+i{square root}{square root over (3)}( u−v)/2 Φ_{3}=−(u+v)/2−i{square root}{square root over (3)}( u−v)/2 (13)

[0055]
However, since only real numbers are used, it is only necessary to consider a solution Φ1 in the above equation (13). In order to determine the error of the solution Φ1, the equation (6) is used and the following evaluation equation is obtained:
$\begin{array}{cc}\begin{array}{c}E=\text{\hspace{1em}}\ue89e4\ue89e\sum \text{\hspace{1em}}\ue89ei,j\ue89e\{{\left({\left(\theta \ue89e\text{\hspace{1em}}\ue89e{\mathrm{io}}^{\prime}\right)}^{\frac{1}{3}}{\left(\theta \ue89e\text{\hspace{1em}}\ue89e{\mathrm{jo}}^{\prime}\right)}^{\frac{1}{3}}\right)}^{2}\\ {\text{\hspace{1em}}\ue89e{\left({\left(\theta \ue89e\text{\hspace{1em}}\ue89e{\mathrm{io}}^{\u2033}\right)}^{\frac{1}{3}}{\left(\theta \ue89e\text{\hspace{1em}}\ue89e{\mathrm{jo}}^{\u2033}\right)}^{\frac{1}{3}}\right)}^{2}\}}^{2}\end{array}& \left(14\right)\end{array}$

[0056]
When solutions θ′ and θ″ which correspond to the measured values Φ′ and Φ″ in the equation (11) are used, the parameter or correction coefficient A is expressed in terms of the measured coordinates (x′, y′) and (x″,y″). In other words, E in the equation (11) is a function of the coefficient A. Thus, from the equation (11), ∂E/∂A as well as ∂E/∂AB is obtained.

[0057]
Still referring to FIG. 10, in a step S4, it is determined whether the error value E in the equation (14) substantially converges. If the error value E does not substantially converge, in a step S5, the coefficient A value is adjusted by adding ΔA to the current coefficient value, and ΔA is obtained from the relation ΔE=(∂E/∂A) ΔA. Then, the step S3 is repeated. On the other hand, when it is determined that the error value E substantially converges in the step S4, the correction coefficient value A is also determined. In order to determine a value of the correction coefficient B, in a step S6, the correction coefficient B is now initialized to B_{0}.

[0058]
In a step S7, it is determined if there exists another common pattern or portion between the two images as similarly decided in the above described Step 2. If there in no additional common pattern, the value of the correction coefficient B is determined to be B_{0}. On the other hand, if the additional common patter exits, two pairs of corresponding pixels are selected from the two images, and steps comparable to the above described step S3 through step S5 for correction coefficient A are repeated for the correction coefficient B.

[0059]
The above selected two pairs of pixel coordinates are converted into angles in a step S8 using the above equation (10). In a step S9, the equation (11) is solved, and it is determined whether the error evaluation E is converging in a step 10. In a step 11, the correction coefficient B is adjusted by adding AB to the current value of the correction coefficient B, and AB is obtained from the relation ΔE=(∂E/∂B) ΔB. The above adjustment is repeated until the E value substantially converges in the step S10.

[0060]
Now referring to FIG. 11, a flow chart illustrates one preferred process of determining correction coefficients for image data which is generated by two dimensionally adjusting the optical system. In a step S21, the correction coefficients A and B are respectively initialized to A_{0 }and D_{0}. Similarly, a distance R between the center of a lens and an image forming surface is initialized to R_{0}. In a step S22, from each of at least two images, a portion including a common pattern is extracted. Within the common pattern portion, two pairs of corresponding pixels are selected from the two images. The two pairs of pixels are denoted as (Pi′, Pi″) and (Pj′, Pj″) while their coordinates are denoted by (xi′, yi′) (xi″, yi″) and (xj′, yj∝) (xj″, yj″). Still in the step S22, the above selected two pairs of the pixel coordinates are converted into angles using the above equation (10). In a step S23, equation (11) is solved for Φ_{i0}′, Φ_{i0}′, Φ_{j0}″, and Φ_{j0}″. Using these solution, in a step S24, an error evaluation is performed using the following equation:

E=(∠pjo′Cpio′−∠pjo″Cpio″)^{2} (15)

[0061]
In the alternative, the error evaluation is made based upon the measured coordinates:
$\begin{array}{cc}E={\left(\frac{\text{\hspace{1em}}\ue89e{\mathrm{xio}}^{\prime}\ue89e{\mathrm{xjo}}^{\prime}+\text{}\ue89e{\mathrm{yio}}^{\prime}\ue89e{\mathrm{yjo}}^{\prime}+{R}^{2}}{\sqrt{{x}^{\prime}\ue89e{\mathrm{io}}^{2}+{y}^{\prime}\ue89e{\mathrm{io}}^{2}+{R}^{2}}}\frac{{\mathrm{xio}}^{\u2033}\ue89e{\mathrm{xjo}}^{\u2033}+\text{}\ue89e{\mathrm{yio}}^{\u2033}\ue89e{\mathrm{yjo}}^{\u2033}+{R}^{2}}{\sqrt{{x}^{\u2033}\ue89e{\mathrm{io}}^{2}+{y}^{\u2033}\ue89e{\mathrm{io}}^{2}+{R}^{2}}}\right)}^{2}& \left(16\right)\end{array}$

[0062]
Until the above E value substantially converges in a step S24, the correction coefficient A is adjusted by repeating steps S23 through S25. In step 25, the first coefficient A value is adjusted by adding ΔA to the current coefficient value, and ΔA is obtained from the relation ΔE=(∂E/∂A) ΔA. After the E value converges, in a step 26, it is determined whether another common pattern exists. If it does not, the process ends. On the other hand, if an additional common pattern exists in the images, in a step 27, a second correction coefficient A′ is now initialized to a predetermined value A_{0}′. Steps S28 through S31 are substantially corresponding to the above described steps S22 through S25. In the step S31, a second correction coefficient A′ is adjusted by adding ΔA′ to the current coefficient value, and ΔA′ is obtained from the relation ΔE=(∂E/∂A′) ΔA′. The steps S29 through S31 are repeated until the error evaluation value substantially converges in the step S30.

[0063]
In a step 32, after the first coefficient A and the second coefficient A′ are determined, it is determined whether the absolute difference between these coefficients substantially converges. If the difference does not substantially converge, in a step 33, the first coefficient A is again initialized to A_{0 }while the distance R is incremented by ΔR. Following the step S33, the preferred process now starts in the step S22. On the other hand, if the absolute difference converges, the process now moves to a sep 34 where the third coefficient B is initialized to a predetermined value B_{0}. Steps S35 through S37 are comparable to the steps S23 through S25 as well as the steps S29 through S31 and use the values Φ_{i0}′, Φ_{i0}′, Φ_{j0}″, and Φ_{j0}″ determined in the step S28. The third coefficient value B is adjusted until the error evaluation value E substantially converges. Upon the convergence, the coefficients A and B as well as the distance R are all determined.

[0064]
Referring to FIG. 12, an alternative method of determining the correction coefficients A and B is based upon the images which are generated without rotating the lens about the optical axis of a lens. Since the distorted aberration of the lens is generally small, under the above condition, when the two partially overlapping images are superimposed, the position of the common elements show a group of vectors S indicating the direction of the positional change of the lens. These vectors S are substantially parallel with each other. Using these vectors in relation to the following relation, p_{j0}′ p_{j0}″//P_{i0}′ p_{i }p_{j0}′ p_{j0}″//S, the following equations are derived:

(xi′−xi″)/(yi′−yj″)≈(xj′−xj″)/(yj′−yj″)≈direction(S)

(xio′−xio″)/(yio′−yjo″)=(xjo′−xjo″)/(yjo′−yjo″)=direction(S)

[0065]
The above equations are useful in improving the accuracy in determining the correction coefficients when the lens does not rotate about its optical axis during the image generation. By using this or other improvement techniques, the determination of the correction coefficients is improved by shortening the calculation time and increasing the accuracy. It is also possible to combining these techniques to improve the task or to repeat the task on a series of common patterns. For example, using a facial image, a series of common patterns such as an eye area and a nose area is extracted and prioritized based upon contrast.

[0066]
For each of these areas, the process for determining the coefficient values is repeated. The results are then used by modifying a criterion or a threshold for error evaluation. Such variable thresholds include a high threshold value for a center portion of the image and a low threshold value for a peripheral portion of the image.

[0067]
The above described preferred embodiments are directed to images formed on a flat surface. However, the concept of the current invention is applicable to images formed on a curved surface as well as a spherical surface for correcting distorted aberration in the images.

[0068]
It is to be understood, however, that even though numerous characteristics and advantages of the present invention have been set forth in the foregoing description, together with details of the structure and function of the invention, the disclosure is illustrative only, and changes may be made in detail, especially in matters of shape, size and arrangement of parts, as well as implementation in software, hardware, or a combination of both within the principles of the invention to the full extent indicated by the broad general meaning of the terms in which the appended claims are expressed.