US RE42187 E1 Abstract Image processing utilizing numerical calculation of fractional exponential powers of a diagonalizable numerical transform operator for use in an iterative or other larger computational environments. In one implementation, a computation involving a similarity transformation is partitioned so that one part remains fixed and may be reused in subsequent iterations. The numerical transform operator may be a discrete Fourier transform operator, discrete fractional Fourier transform operator, centered discrete fractional Fourier transform operator, and other operators, modeling propagation through physical media. Such iterative environments for these types of numerical calculations are useful in correcting the focus of misfocused images which may originate from optical processes involving light (for example, with a lens or lens system) or from particle beams (for example, in electron microscopy or ion lithography).
Claims(58) 1. A method performed by a numerical processor for approximating the evolution of images propagating through a physical medium by calculating a fractional power of a numerical operator, said numerical operator being defined by said physical medium and comprising a diagonalizable numerical linear operator raised to a power (α), said method comprising:
(a) representing a plurality of images using an individual data array for each of said plurality of images;
(b) representing said numerical operator with a linear operator formed by multiplying an ordered similarity transformation operator (P) by a correspondingly-ordered diagonal operator (Λ), the result of which is multiplied by an approximate inverse (P
^{−1}) of said ordered similarity transformation operator (P); (c) raising diagonal elements of said correspondingly-ordered diagonal operator (Λ) to said power (α) to produce a fractional power diagonal operator;
(d) multiplying said fractional power diagonal operator by an approximate inverse of said ordered similarity transformation operator (P
^{−1}) to produce a first partial result; (e) multiplying athe data array of one of said plurality of images by said ordered similarity transformation operator (P) to produce a modified data array;
(f) multiplying said modified data array by said first partial result to produce said fractional power of said numerical operator; and
(g) repeating operations (e) and (f) for each of said plurality of images; and
(h) producing a transformed image from the modified data array for each of said plurality of images.
2. The method according to
3. The method according to
4. The method according to
5. The method according to
6. The method according to
7. The method according to
8. The method according to
9. The method according to
10. The method according to
11. The method according to
12. The method according to
13. The method according to
14. The method according to
15. The method according to
16. The method according to
17. The method according to
18. The method according to
19. The method according to
20. The method according to
21. The method according to
shifting an index of said numerical operator so that said zero origin of said numerical operator matches said zero origin of each of said plurality of images.
22. The method according to
23. A method performed by a numerical processor for approximating the evolution of images propagating through a physical medium by calculating a fractional power of a numerical operator, said numerical operator being defined by said physical medium and comprising a diagonalizable numerical linear operator raised to a power (α) having any one of a plurality of values, said method comprising:
(a) representing an image using a data array;
(b) representing said numerical operator with a linear operator formed by multiplying an ordered similarity transformation operator (P) by a correspondingly-ordered diagonal operator (Λ), the result of which is multiplied by an approximate inverse (P
^{−1}) of said ordered similarity transformation operator (P); (c) raising diagonal elements of said correspondingly-ordered diagonal operator (Λ) to one of said plurality of values of said power (α) to produce a fractional power diagonal operator;
(d) multiplying said fractional power diagonal operator by an approximate inverse of said ordered similarity transformation operator (P
^{−1}) to produce a first partial result; (e) multiplying said data array by said ordered similarity transformation operator (P) to produce a modified data array;
(f) multiplying said modified data array by said first partial result to produce said fractional power of said numerical operator; and
(g) repeating operations (c) and (d) for each of said plurality of values of said power (α); and
(h) producing a transformed image from the modified data array for each of said plurality of images.
24. The method according to
25. The method according to
26. The method according to
27. The method according to
28. The method according to
29. The method according to
30. The method according to
31. The method according to
32. The method according to
33. The method according to
34. The method according to
35. The method according to
36. The method according to
37. The method according to
38. The method according to
39. The method according to
40. The method according to
41. The method according to
42. The method according to
43. The method according to
shifting an index of said numerical operator so that said zero origin of said numerical operator matches said zero origin of said image.
44. The method according to
45. A computer-readable medium containing instructions for controlling a computer system to approximate the evolution of images propagating through a physical medium by calculating a fractional power of a numerical operator, said numerical operator being defined by said physical medium and comprising a diagonalizable numerical linear operator raised to a power (α), said controlling provided by said computer system being accomplished according to operations comprising:
(a) representing a plurality of images using an individual data array for each of said plurality of images;
(b) representing said numerical operator with a linear operator formed by multiplying an ordered similarity transformation operator (P) by a correspondingly-ordered diagonal operator (Λ), the result of which is multiplied by an approximate inverse (P
^{−1}) of said ordered similarity transformation operator (P); (c) raising diagonal elements of said correspondingly-ordered diagonal operator (Λ) to said power (α) to produce a fractional power diagonal operator;
(d) multiplying said fractional power diagonal operator by an approximate inverse of said ordered similarity transformation operator (P
^{−1}) to produce a first partial result; (e) multiplying a data array of one of said plurality of images by said ordered similarity transformation operator (P) to produce a modified data array;
(f) multiplying said modified data array by said first partial result to produce said fractional power of said numerical operator; and
(g) repeating operations (e) and (f) for each of said plurality of images; and
(h) producing a transformed image from the modified data array for each of said plurality of images.
46. A computer-readable medium containing instructions for controlling a computer system to approximate the evolution of images propagating through a physical medium by calculating a fractional power of a numerical operator, said numerical operator being defined by said physical medium and comprising a diagonalizable numerical linear operator raised to a power (α) having any one of a plurality of values, said controlling provided by said computer system being accomplished according to operations comprising:
(a) representing an image using a data array;
^{−1}) of said ordered similarity transformation operator (P); (c) raising diagonal elements of said correspondingly-ordered diagonal operator (Λ) to one of said plurality of values of said power (α) to produce a fractional power diagonal operator;
^{−1}) to produce a first partial result; (e) multiplying said data array by said ordered similarity transformation operator (P) to produce a modified data array;
(g) repeating operations (c) and (d) for each of said plurality of values of said power (α); and
(h) producing a transformed image from the modified data array for each of said plurality of images.
47. A method performed by a numerical processor for numerically modeling an image propagating through a medium, the method comprising:
representing the image using image data comprising a plurality of spatially-indexed amplitude values, the image data comprising a center located relative to the plurality of spatially indexed amplitude values; providing a propagation medium model comprising quadratic phase properties which are defined relative to a propagation centerline of the propagation medium model; aligning the propagation centerline of the propagation medium model relative to the center of the image data; approximating the propagation medium model with a numerical operator for applying an index-shifted numerical fractional Fourier transform operation on the image data, the numerical operator having original-domain indices and transform-domain indices, wherein the original domain indices comprise a zero original-domain origin that is centered within the original domain indices, and the transform-domain indices comprise a zero transform-domain origin that is centered within the transform-domain indices; aligning the zero original-domain origin relative to the center of the image data to produce transformed image data comprising-a zero frequency-domain origin that is centered within the transform-domain indices; and producing a transformed image from the transformed image data of the image. 48. The method of
49. The method of
50. The method of
51. The method of
52. The method of
53. The method of
54. The method of
55. The method of
56. The method of
57. The method of
58. The method of
Description This application is a reissue application of U.S. Pat. No. 1. Field of the Invention This invention relates to optical signal processing, and more particularly to the use of fractional Fourier transform properties of lenses to correct the effects of lens misfocus in photographs, video, and other types of captured images. 2. Discussion of the Related Art A number of references are cited herein; these are provided in a numbered list at the end of the detailed description of the preferred embodiments. These references are cited at various locations throughout the specification using a reference number enclosed in square brackets. The Fourier transforming properties of simple lenses and related optical elements is well known and heavily used in a branch of engineering known as Fourier optics [1, 2]. Classical Fourier optics [1, 2, 3, 4] utilize lenses or other means to obtain a two-dimensional Fourier transform of an optical wavefront, thus creating a Fourier plane at a particular spatial location relative to an associated lens. This Fourier plane includes an amplitude distribution of an original two-dimensional optical image, which becomes the two-dimensional Fourier transform of itself. In the far simpler area of classical geometric optics [1, 3], lenses and related objects are used to change the magnification of a two-dimensional image according to the geometric relationship of the classical lens-law. It has been shown that between the geometries required for classical Fourier optics and classical geometric optics, the action of a lens or related object acts on the amplitude distribution of images as the fractional power of the two-dimensional Fourier transform. The fractional power of the fractional. Fourier transform is determined by the focal length characteristics of the lens, and the relative spatial separation between a lens, source image, and an observed image. The fractional Fourier transform has been independently discovered on various occasions over the years [5, 7, 8, 9, 10], and is related to several types of mathematical objects such as the Bargmann transform [8] and the Hermite semi-group [13]. As shown in [5], for example, the most general form of optical properties of lenses and other related elements [1, 2, 3] can be transformed into a fractional Fourier transform representation. This property has apparently been rediscovered some years later and worked on steadily ever since (see for example [6]), expanding the number of optical elements and situations covered. It is important to remark, however, that the lens modeling approach in the latter ongoing series of papers view the multiplicative phase term in the true form of the fractional Fourier transform as a problem or annoyance and usually omit it from consideration. Correction of the effects of misfocusing in recorded or real-time image data may be accomplished using fractional Fourier transform operations realized optically, computationally, or electronically. In some embodiments, the invention extends the capabilities of using a power of the fractional Fourier transform for correcting misfocused images, to situations where phase information associated with the original image misfocus is unavailable. For example, conventional photographic and electronic image capture, storage, and production technologies can only capture and process image amplitude information—the relative phase information created within the original optical path is lost. As will be described herein, the missing phase information can be reconstructed and used when correcting image misfocus. In accordance with embodiments of the invention, a method for approximating the evolution of images propagating through a physical medium is provided by calculating a fractional power of a numerical operator. The numerical operator may be defined by the physical medium and includes a diagonalizable numerical linear operator raised to a power (α). The method further includes representing a plurality of images using an individual data array for each of the images. The numerical operator may be represented with a linear operator formed by multiplying an ordered similarity transformation operator (P) by a correspondingly-ordered diagonal operator (Λ), the result of which is multiplied by an approximate inverse (P The above and other aspects, features and advantages of the present invention will become more apparent upon consideration of the following description of preferred embodiments taken in conjunction with the accompanying drawing figures, wherein: FIG. 1 is a block diagram showing a general lens arrangement and associated image observation entity capable of classical geometric optics, classical Fourier optics, and fractional Fourier transform optics; FIG. 2 is a block diagram showing an exemplary approach for automated adjustment of fractional Fourier transform parameters for maximizing the sharp edge content of a corrected image, in accordance with one embodiment of the present invention; FIG. 3 is a block diagram showing a typical approach for adjusting the fractional Fourier transform parameters to maximize misfocus correction of an image, in accordance with one embodiment of the present invention; FIG. 4 is a diagram showing a generalized optical environment for implementing image correction in accordance with the present invention; FIG. 5 is a diagram showing focused and unfocused image planes in relationship to the optical environment depicted in FIG. 4; FIG. 6 is a block diagram showing an exemplary image misfocus correction process that also provides phase corrections; FIG. 7 is a diagram showing a more detailed view of the focused and unfocused image planes shown in FIG. 5; FIG. 8 is a diagram showing typical phase shifts involved in the focused and unfocused image planes depicted in FIG. 5; FIG. 9 shows techniques for computing phase correction determined by the fractional Fourier transform applied to a misfocused image; FIG. 10 is a block diagram showing an exemplary image misfocus correction process that also provides for phase correction, in accordance with an alternative embodiment of the invention; FIG. 11 shows a diagonalizable matrix, tensor, or linear operator acting on an underlying vector, matrix, tensor, or function; FIG. 12 is a flowchart showing exemplary operations for approximating the evolution of images propagating through a physical medium, in accordance with embodiments of the invention; FIG. 13 is a flowchart showing exemplary operations for approximating the evolution of images propagating through a physical medium, in accordance with alternative embodiments of the invention. FIG. 14 shows a simplified version of the equation depicted in FIG. 11; FIG. 15 shows an exemplary matrix; FIGS. 16A through 16C are side views of a propagating light or particle beam; FIG. 16D is a top view of contours of constant radial displacement with respect to the image center of paths associated with the image propagation of an image; FIG. 17 shows an exemplary matrix for a centered normalized classical discrete Fourier transform; FIG. 18A is a graph showing a frequency-domain output of a non-centered classical discrete Fourier transform for an exemplary signal; FIG. 18B is a graph showing a frequency-domain output of a centered classical discrete Fourier transform for the same exemplary signal; FIG. 19 shows the coordinate indexing of an image having a particular height and width; and FIG. 20 is a block diagram showing the isolation and later reassembly of quadrant portions of an original image. In the following description, reference is made to the accompanying drawing figures which form a part hereof, and which show by way of illustration specific embodiments of the invention. It is to be understood by those of ordinary skill in this technological field that other embodiments may be utilized, and structural, electrical, optical, as well as procedural changes may be made without departing from the scope of the present invention. As used herein, the term “image” refers to both still-images (such as photographs, video frames, video stills, movie frames, and the like) and moving images (such as motion video and movies). Many embodiments of the present invention are directed to processing recorded or real-time image data provided by an exogenous system, means, or method. Presented image data may be obtained from a suitable electronic display such as an LCD panel, CRT, LED array, films, slides, illuminated photographs, and the like. Alternatively or additionally, the presented image data may be the output of some exogenous system such as an optical computer or integrated optics device, to name a few. The presented image data will also be referred to herein as the image source. If desired, the system may output generated image data having some amount of misfocus correction. Generated image data may be presented to a person, sensor (such as a CCD image sensor, photo-transistor array, for example), or some exogenous system such as an optical computer, integrated optics device, and the like. The entity receiving generated image data will be referred to as an observer, image observation entity, or observation entity. Reference will first be made to FIG. 3 which shows a general approach for adjusting the fractional Fourier transform parameters to maximize the correction of misfocus in an image. Details regarding the use of a fractional Fourier transform (with adjusted parameters of exponential power and scale) to correct image misfocus will be later described with regard to FIGS. 1 and 2. Original visual scene Original image data For a human operator, this typically would be a matter of adjusting a control and comparing images side by side (facilitated by non-human memory) or, as in the case of a microscope or telescope, by comparison facilitated purely with human memory. For a machine, a systematic iterative or other feedback control scheme would typically be used. In FIG. 3, each of these image adjustments is generalized by the steps and elements suggested by interconnected elements High level actions With this high level description having been established, attention is now directed to details of the properties and use of a fractional Fourier transform (with adjusted parameters of exponential power and scale) to correct misfocus in an image and maximize correction of misfocus. This aspect of the present invention will be described with regard to FIG. 1. FIG. 1 is a block diagram showing image source -
- separation distances
**111**and**112**; - the “focal length” parameter “f” of lens
**102**; - the type of image source (lit object, projection screen, etc.) in as far as whether a plane or spherical wave is emitted.
- separation distances
As is well known, in situations where the source image is a lit object and where distance As previously noted, the Fourier transforming properties of simple lenses and related optical elements is also well known in the field of Fourier optics [2, 3]. Classical Fourier optics [2, 3, 4, 5] involve the use of a lens, for example, to take a first two-dimensional Fourier transform of an optical wavefront, thus creating a Fourier plane at a particular spatial location such that the amplitude distribution of an original two-dimensional optical image becomes the two-dimensional Fourier transform of itself. In the arrangement depicted in FIG. 1, with a lit object serving as source image As described in [5], for cases where a, b, and f do not satisfy the lens law of the Fourier optics condition above, the amplitude distribution of source image The fractional Fourier transform properties of lenses typically cause complex but predictable phase and scale variations. These variations may be expressed in terms of Hermite functions, as presented shortly, but it is understood that other representations of the effects, such as closed-form integral representations given in [5], are also possible and useful. Various methods can be used to construct the fractional Fourier transform, but to begin it is illustrative to use the orthogonal Hermite functions, which as eigenfunctions diagonalize the Fourier transform [17]. Consider the Hermite function [16] expansion [17, and more recently, 18] of the two dimensional image amplitude distribution function. In one dimension, a bounded (i.e., non-infinite) function k(x) can be represented as an infinite sum of Hermite functions {h Since the function is bounded, the coefficients {a More generally, as the power α varies (via the Arccosine relationship depending on the separation distance), the phase angle of the n Through use of the Mehler kernel [16], the above expansion may be represented in closed form as [5]:
Thus, one aspect of the invention provides image misfocus correction, where the misfocused image had been created by a quality though misfocused lens or lens-system. This misfocus can be corrected by applying a fractional Fourier transform operation; and more specifically, if the lens is misfocused by an amount corresponding to the fractional Fourier transform of power ε, the misfocus may be corrected by applying a fractional Fourier transform operation of power −ε. It is understood that in some types of situations, spatial scale factors of the image may need to be adjusted in conjunction with the fractional Fourier transform power. For small variations of the fractional Fourier transform power associated with a slight misfocus, this is unlikely to be necessary. However, should spatial scaling need to be made, various optical and signal processing methods well known to those skilled in the art can be incorporated. In the case of pixilated images (images generated by digital cameras, for example) or lined-images (generated by video-based systems, for example), numerical signal processing operations may require standard resampling (interpolation and/or decimation) as is well known to those familiar with standard signal processing techniques. It is likely that the value of power ε is unknown a priori. In this particular circumstance, the power of the correcting fractional Fourier transform operation may be varied until the resulting image is optimally sharpened. This variation could be done by human interaction, as with conventional human interaction of lens focus adjustments on a camera or microscope, for example. If desired, this variation could be automated using, for example, some sort of detector in an overall negative feedback situation. In particular, it is noted that a function with sharp edges are obtained only when its contributing, smoothly-shaped basis functions have very particular phase adjustments, and perturbations of these phase relationships rapidly smooth and disperse the sharpness of the edges. Most natural images contain some non-zero content of sharp edges, and further it would be quite unlikely that a naturally occurring, smooth gradient would tighten into a sharp edge under the action of the fractional Fourier transform because of the extraordinary basis phase relationships required. This suggests that a spatial high-pass filter, differentiator, or other edge detector could be used as part of the sensor makeup. In particular, an automatically adjusting system may be configured to adjust the fractional Fourier transform power to maximize the sharp edge content of the resulting correcting image. If desired, such a system may also be configured with human override capabilities to facilitate pathological image situations, for example. FIG. 2 shows an automated approach for adjusting the fractional Fourier transform parameters of exponential power and scale factor to maximize the sharp edge content of the resulting correcting image. In this figure, original image data Typically, this element would initially set the power to the ideal value of zero (making the resulting image data The scalar measure value for each fractional Fourier transform power may be stored in memory It is understood that the above system amounts to a negative-feedback control or adaptive control system with a fixed or adaptive observer. As such, it is understood that alternate means of realizing this automated adjustment can be applied by those skilled in the art. It is also clear to one skilled in the art that various means of interactive human intervention may be introduced into this automatic system to handle problem cases or as a full replacement for the automated system. In general, the corrective fractional Fourier transform operation can be accomplished by any one or combination of optical, numerical computer, or digital signal processing methods as known to those familiar with the art, recognizing yet other methods may also be possible. Optical methods may give effectively exact implementations of the fractional Fourier transforms, or in some instances, approximate implementations of the transforms. For a pixilated image, numerical or other signal processing methods may give exact implementations through use of the discrete version of the fractional Fourier transform [10]. Additional computation methods that are possible include one or more of: -
- dropping the leading scalar complex-valued phase term (which typically has little or no effect on the image);
- decomposing the fractional Fourier transform as a premultiplication by a “phase chirp” e
^{icz2}, taking a conventional Fourier transform with appropriately scaled variables, and multiplying the result by another “phase chirp;” and changing coordinate systems to Wigner form:$\begin{array}{cc}\left\{\frac{\left(x+y\right)}{w},\frac{\left(x-y\right)}{w}\right\}& \left(13\right)\end{array}$ If desired, any of these just-described computation methods can be used with the approximating methods described below.
Other embodiments provide approximation methods for realizing the corrective fractional Fourier transform operation. For a non-pixilated image, numerical or other signal processing methods can give approximations through: -
- finite-order discrete approximations of the integral representation;
- finite-term discrete approximations by means of the Hermite expansion representation; and
- the discrete version of the fractional Fourier transform [10].
Classical approximation methods [11, 12] may be used in the latter two cases to accommodate particular engineering, quality, or cost considerations. In the case of Hermite expansions, the number of included terms may be determined by analyzing the Hermite expansion of the image data, should this be tractable. In general, there will be some value in situations where the Hermite function expansion of the image looses amplitude as the order of the Hermite functions increases. Hermite function orders with zero or near-zero amplitudes may be neglected entirely from the fractional Fourier computation due to the eigenfunction role of the Hermite functions in the fractional Fourier transform operator. One method for realizing finite-order discrete approximations of the integral representation would be to employ a localized perturbation or Taylor series expansion of the integral representation. In principal, this approach typically requires some mathematical care in order for the operator to act as a reflection operator (i.e., inversion of each horizontal direction and vertical direction as with the lens law) since the kernel behaves as a generalized function (delta function), and hence the integral representation of the fractional Fourier transform operator resembles a singular integral. In a compound lens or other composite optical system, the reflection operator may be replaced with the identity operator, which also involves virtually identical delta functions and singular integrals as is known to those familiar in the art. However, this situation is fairly easy to handle as a first or second-order Taylor series expansion. The required first, second, and any higher-order derivatives of the fractional Fourier transform integral operator are readily and accurately obtained symbolically using available mathematical software programs, such as Mathematica or MathLab, with symbolic differential calculus capabilities. In most cases, the zero-order term in the expansion will be the simple reflection or identity operator. The resulting expansion may then be numerically approximated using conventional methods. Another method for realizing finite-order discrete approximations of the integral representation would be to employ the infinitesimal generator of the fractional Fourier transform, that is, the derivative of the fractional Fourier transform with respect to the power of the transform. This is readily computed by differentiating the Hermite function expansion of the fractional Fourier transform, and use of the derivative rule for Hermite functions. Depending on the representation used [5, 14, 15], the infinitesimal generator may be formed as a linear combination of the Hamiltonian operator H and the identity operator I; for the form of the integral representation used earlier, this would be:
In cases where the discrete version of the fractional Fourier transform [10] is implemented, the transform may be approximated. Pairs of standard two-dimensional matrices, one for each dimension of the image, can be used. As with the continuous case, various types of analogous series approximations, such as those above, can be used. Alternatively, it is noted that because of the commutative group property of the fractional Fourier transform, the matrix/tensor representations, or in some realizations even the integrals cited above may be approximated by precomputing one or more fixed step sizes and applying these respectively, iteratively, or in mixed succession to the image data. One exemplary embodiment utilizing a pre-computation technique may be where the fractional Fourier transform represents pre-computed, positive and negative values of a small power, for example 0.01. Negative power deviations of increasing power can be had by iteratively applying the pre-computed −0.01 power fractional Fourier transform; for example, the power −0.05 would be realized by applying the pre-computed −0.01 power fractional Fourier transform five times. In some cases of adaptive system realizations, it may be advantageous to discard some of the resulting image data from previous power calculations. This may be accomplished by backing up to a slightly less negative power by applying the +0.01 power fractional Fourier transform to a last stored, resulting image. As a second example of this pre-computation method, pre-computed fractional Fourier transform powers obtained from values of the series 2 It is noted that any of the aforementioned systems and methods may be adapted for use on portions of an image rather than the entire image. This permits corrections of localized optical aberrations. In complicated optical aberration situations, more than one portion of an image may be processed in this manner, with differing corrective operations made for each portion of the image. It is further noted that the systems and methods described herein may also be applied to conventional lens-based optical image processing systems, to systems with other types of elements obeying fractional Fourier optical models, as well as to widely ranging environments such as integrated optics, optical computing systems, particle beam systems, electron microscopes, radiation accelerators, and astronomical observation methods, among others. Commercial products and services application are widespread. For example, the present invention may be incorporated into film processing machines, desktop photo editing software, photo editing web sites, VCRs, camcorders, desktop video editing systems, video surveillance systems, video conferencing systems, as well as in other types of products and service facilities. Four exemplary consumer-based applications are now considered. 1. One particular consumer-based application is in the correction of camera misfocus in chemical or digital photography. Here the invention may be used to process the image optically or digitally, or some combination thereof, to correct the misfocus effect and create an improved image which is then used to produce a new chemical photograph or digital image data file. In this application area, the invention can be incorporated into film processing machines, desktop photo editing software, photo editing web sites, and the like. 2. Another possible consumer-based application is the correction of video camcorder misfocus. Camcorder misfocus typically results from user error, design defects such as a poorly designed zoom lens, or because an autofocus function is autoranging on the wrong part of the scene being recorded. Non-varying misfocus can be corrected for each image with the same correction parameters. In the case of zoom lens misfocus, each frame or portion of the video may require differing correction parameters. In this application area, the invention can be incorporated into VCRs, camcorders, video editing systems, video processing machines, desktop video editing software, and video editing web sites, among others. 3. Another commercial application involves the correction of image misfocus experienced in remote video cameras utilizing digital signal processing. Particular examples include video conference cameras or security cameras. In these scenarios, the video camera focus cannot be adequately or accessibly adjusted, and the video signal may in fact be compressed. 4. Video compression may involve motion compensation operations that were performed on the unfocused video image. Typical applications utilizing video compression include, for example, video conferencing, video mail, and web-based video-on-demand, to name a few. In these particular types of applications, the invention may be employed at the video receiver, or at some pre-processing stage prior to delivering the signal to the video receiver. If the video compression introduces a limited number of artifacts, misfocus correction is accomplished as presented herein. However, if the video compression introduces a higher number of artifacts, the signal processing involved with the invention may greatly benefit from working closely with the video decompression signal processing. One particular implementation is where misfocus corrections are first applied to a full video frame image. Then, for some interval of time, misfocus correction is only applied to the changing regions of the video image. A specific example may be where large portions of a misfocused background can be corrected once, and then reused in those same regions in subsequent video frames. 5. The misfocus correction techniques described herein are directly applicable to electron microscopy systems and applications. For example, electron microscope optics employ the wave properties of electrons to create a coherent optics environment that obeys the Fourier optics structures as coherent light (see, for example, John C. H. Spence, High-Resolution Electron Microscopy, third edition, 2003, Chapters 2-4, pp. 15-88). Electron beams found in electron microscopes have the same geometric, optical physics characteristics generally found in coherent light, and the same mathematical quadratic phase structure as indicated in Levi [1] Section 19.2 for coherent light, which is the basis of the fractional Fourier transform in optical systems (see, for example, John C. H. Spence High-Resolution Electron Microscopy, third edition, 2003, Chapter 3, formula 3.9, pg. 55). Most photographic and electronic image capture, storage, and production technologies are only designed to operate with image amplitude information, regardless as to whether the phase of the light is phase coherent (as is the case with lasers) or phase noncoherent (as generally found in most light sources). In sharply focused images involving noncoherent light formed by classical geometric optics, this lack of phase information is essentially of no consequence in many applications. In representing the spatial distribution of light, the phase coefficient of the basis functions can be important; as an example, FIG. 3.6, p. 62 of Digital Image Processing—Concepts, Algorithms, and Scientific Applications, by Bernd Jahne, Springer-Verlag, New York, 1991 [20] shows the effect of loss and modification of basis function phase information and the resulting distortion in the image. Note in this case the phase information of the light in the original or reproduced image differs from the phase information applied to basis functions used for representing the image. In using fractional powers of the Fourier transform to represent optical operations, the fractional Fourier transform reorganizes the spatial distribution of an image and the phase information as well. Here the basis functions serve to represent the spatial distribution of light in a physical system and the phase of the complex coefficients multiplying each of the basis functions mathematically result from the fractional Fourier transform operation. In the calculation that leads to the fractional Fourier transform representation of a lens, complex-valued coefficients arise from the explicit accounting for phase shifts of light that occurs as it travels through the optical lens (see Goodman [2], pages 77-96, and Levi [1], pages 779-784). Thus, when correcting misfocused images using fractional powers of the Fourier transform, the need may arise for the reconstruction of relative phase information that was lost by photographic and electronic image capture, storage, and production technologies that only capture and process image amplitude information. In general, reconstruction of lost phase information has not previously been accomplished with much success, but some embodiments of the invention leverage specific properties of both the fractional Fourier transform and an ideal correction condition. More specifically, what is provided—for each given value of the focus correction parameter—is the calculation of an associated reconstruction of the relative phase information. Typically, the associated reconstruction will be inaccurate unless the given value of the focus correction parameter is one that will indeed correct the focus of the original misfocused image. This particular aspect of the invention provides for the calculation of an associated reconstruction of relative phase information by using the algebraic group property of the fractional Fourier transform to back calculate the lost relative phase conditions that would have existed, if that given specific focus correction setting resulted in a correctly focused image. For convergence of human or machine iterations towards an optimal or near optimal focus correction, the system may also leverage the continuity of variation of the phase reconstruction as the focus correction parameter is varied in the iterations. To facilitate an understanding of the phase reconstruction aspect of the invention, it is helpful to briefly summarize the some of the image misfocus correction aspects of the invention. This summary will be made with reference to the various optical set-ups depicted in FIGS. 4-8, and is intended to provide observational details and examples of where and how the relative phase reconstruction may be calculated (FIG. 9) and applied (FIG. 10). FIG. 4 shows a general optical environment involving sources of radiating light FIG. 5 is an optical environment similar to that depicted in FIG. 4, but the FIG. 5 environment includes only a single point light source FIG. 5 also shows directionally modified rays Reference is now made to FIGS. 6-8, which disclose techniques for mathematical focus correction and provides a basis for understanding the phase correction aspect of the present invention. For clarity, the term “lens” will be used to refer to optical element FIG. 6 provides an example of image information flow in accordance with some embodiments of the present invention. As depicted in block For a monochrome image, the light amplitude values are typically represented as scalar quantities, while color images typically involve vector quantities such as RBG values, YUV values, and the like. In some instances, the digital file may have been subjected to file processes such as compression, decompression, color model transformations, or other data modification processes to be rendered in the form of an array of light amplitude values The array, or in some instances, arrays of light amplitude values FIG. 7 shows an optical environment having nonfocused planes. This figure shows that the power of the fractional Fourier transform operator increases as the separation distance between optical lens operation By mathematical extension, as described in [5], a long misfocused image plane In terms of geometric optics, misfocus present in short misfocused image plane FIG. 8 is a more detailed view of image planes
Simple geometry yields the following inequality relationships: δ _{0} ^{S}<δ_{0} ^{F}<δ_{0} ^{L } (21) δ _{1} ^{S}<δ_{1} ^{F}<δ_{1} ^{L } (22) δ _{2} ^{S}<δ_{2} ^{F}<δ_{2} ^{L } (23) For a given wavelength λ, the phase shift ψ created by a distance-of-travel variation δ is given by the following: ψ=2πδ/λ (24) so the variation in separation distance between the focus image plane 504 and the misfocus image planes 505, 506 is seen to introduce phase shifts along each ray.
Further, for π/2>θ Referring again to FIGS. 6 and 7, an example of how a misfocused image In general, the fractional Fourier transform operation creates results that are complex-valued. In the case of the discrete fractional Fourier transform operation, as used herein, this operation may be implemented as, or is equivalent to, a generalized, complex-valued array multiplication on the array image of light amplitudes (e.g., φ). In the signal domain, complex-valued multiplication of a light amplitude array element, ν FIG. 9 shows a series of formulas that may be used in accordance with the present invention. As indicated in block For example, block Referring still to FIG. 9, block As previously noted, conventional photographic and electronic image capture, storage, and production technologies typically only process or use image amplitude information, and were phase information is not required or desired. In these types of systems, the relative phase information created within the original misfocused optical path is lost since amplitude information is the only image information that is conveyed. This particular scenario is depicted in block In accordance with some embodiments, missing phase information may be reintroduced by absorbing it within the math correction stage, as shown block In the case where γ is close enough to be effectively equal to ε, the phase correction will effectively be equal to the value necessary to restore the lost relative phase information. Note that this expression depends only on γ, and thus phase correction may be obtained by systematically iterating γ towards the unknown value of ε, which is associated with the misfocused image. Thus the iteration, computation, manual adjustment, and automatic optimization systems, methods, and strategies of non-phase applications of image misfocus correction may be applied in essentially the same fashion as the phase correcting applications of image misfocus correction by simply substituting F FIG. 10 provides an example of image information flow in accordance with some embodiments of the invention. This embodiment is similar to FIG. 6 is many respects, but the technique shown in FIG. 10 further includes phase restoration component Next, the calculation of the phase-restored mathematical correction is considered. Leveraging two-group antislavery properties of the fractional Fourier transform operation, the additional computation can be made relatively small. In the original eigenfunction/eigenvector series definitions for both the continuous and discrete forms of the fractional Fourier transform of power α, the nth eigenfunction/eigenvectors are multiplied by:
Also, because the nth Hermite function h Further, since the Hermite functions and discrete Fourier transform eigenvectors are real-valued, the complex conjugate can be taken on the entire term, not just the exponential, as shown by:
The relative phase-restored mathematical correction can thus be calculated directly, for example, by the following exemplary algorithm or its mathematical or logistic equivalents: - 1. For a given value of γ, compute F
^{γ }using the Fourier transform eigenvectors in an ordered similarity transformation matrix; - 2. For the odd-indexed eigenvectors, either reverse the order or the sign of its terms to get a modified similarity transformation;
- 3. Compute the complete resulting matrix calculations as would be done to obtain a fractional Fourier transform, but using this modified similarity transformation;
- 4. Calculate the complex conjugate of the result of operation (3) to get the phase restoration, (Φ(F
^{γ}))*; and - 5. Calculate the array product of the operation (1) and operation (4) to form the phase-restored focus correction (Φ(F
^{γ}))*F^{γ}.
As an example of a mathematical or logistic equivalent to the just described series of operations, note the commonality of the calculations in operations (1) and (3), differing only in how the odd-indexed eigenvectors are handled in the calculation, and in one version, only by a sign change. An example of a mathematical or logistic equivalent to the above exemplary technique would be: - 1. For a given value of γ, partially compute F
^{γ }using only the even-indexed Fourier transform eigenvectors; - 2. Next, partially compute the remainder of F
^{γ }using only the odd-indexed Fourier transform eigenvectors; - 3. Add the results of operation (1) and (2) to get F
^{γ} - 4. Subtract the result of operation (2) from the result of operation (1) to obtain a portion of the phase restoration;
- 5. Calculate the complex conjugate of the result of operation (4) to obtain the phase restoration (Φ(F
^{γ}))*; and - 6. Calculate the array product of operations (1) and (4) to form (Φ(F
^{γ}))*F^{γ}.
In many situations, partially computing two parts of one similarity transformation, as described in the second exemplary algorithm, could be far more efficient than performing two full similarity transformation calculations, as described in the first exemplary algorithm. One skilled in the art will recognize many possible variations with differing advantages, and that these advantages may also vary with differing computational architectures and processor languages. Where relative phase-restoration is required or desired in mathematical focus correction using the fractional Fourier transform, phase restoration element It is to be realized that in image misfocus correction applications which do not account for phase restoration, pre-computed values of F Each of the various techniques for computing the phase-restored focus correction may include differing methods for implementing pre-computed phase-restorations. For example, in comparing the first and second exemplary algorithms, predominated values may be made and stored for any of: - First example algorithm operation (5) or its equivalent second example algorithm operation (6);
- First example algorithm operation (4) or its equivalent second example algorithm operation (5); and
- Second example algorithm operations (1) and (2) with additional completing computations provided as needed.
Again, it is noted that these phase restoration techniques can apply to any situation involving fractional Fourier transform optics, including electron microscopy processes and the global or localized correction of misfocus from electron microscopy images lacking phase information. Localized phase-restored misfocus correction using the techniques disclosed herein may be particularly useful in three-dimensional, electron microscopy and tomography where a wide field is involved in at least one dimension of imaging. It is also noted that the various techniques disclosed herein may be adapted for use on portions of an image rather than the entire image. This permits corrections of localized optical aberrations. In complicated optical aberration situations, more than one portion may be processed in this manner, in general with differing corrective operations made for each portion of the image. It is also possible to structure computations of the fractional Fourier transform operating on a sample image array or sampled function vector so that portions of the computation may be reused in subsequent computations. This is demonstrated in the case of a similarity transformation representation of the fractional Fourier transform in FIG. 11. The general approach applies to vectors, matrices, and tensors of various dimensions, other types of multiplicative decompositional representations of the fractional Fourier transform, and other types of operators. In particular the approach illustrated in FIG. 11 may be directly applied to any diagonalizable linear matrix or tensor, not just the fractional Fourier transform. FIG. 11 illustrates the action of a diagonalizable matrix, tensor, or linear operator on an underlying vector, matrix, tensor, or function In particular, to obtain the α power of the chosen diagonalizable matrix, tensor, or linear operator It is possible to reuse parts of calculations made utilizing this structure in various application settings. In a first exemplary application setting, the image is constant throughout but the power α takes on various values, as in an iteration over values of α in an optimization loop or in response to a user-adjusted focus control. In this first exemplary application setting, the product FIG. 12 is a flowchart showing exemplary operations for approximating the evolution of images propagating through a physical medium, in accordance with embodiments of the invention. This approximation may be achieved by calculating a fractional power of a numerical operator, which is defined by the physical medium and includes a diagonalizable numerical linear operator raised to a power (α). In block Next, in block A second exemplary application is one in which parts of previous calculations made using the structure of FIG. 11 are reused. In this embodiment, the power α is constant through-out but the image U FIG. 13 is a flowchart showing exemplary operations for approximating the evolution of images propagating through a physical medium, in accordance with alternative embodiments of the invention. This approximation may be achieved by calculating a fractional power of a numerical operator, which is defined by the physical medium and includes a diagonalizable numerical linear operator raised to a power (α) having any one of a plurality of values. In block Next, in block FIG. 14 comparatively summarizes the general calculations of each of the two described exemplary embodiments in terms of the structure and elements of FIG. 11. Image-specific calculations involving matrix or tensor multiplications can be carried out in an isolated step The discrete fractional Fourier transform is often described as being based on the conventional definition of the classical discrete Fourier transform matrix. Because the classical discrete Fourier transform matrix has elements with harmonically-related periodic behavior, there is a shift invariance as to how the transform is positioned with respect to the frequency-indexed sample space and the time-indexed sample space. The classical discrete Fourier transform matrix typically starts with its first-row, first-column element as a constant, i.e., zero frequency (or in some applications, the lowest-frequency sample point), largely as a matter of convenience since the family of periodic behaviors of the elements and time/frequency sample spaces (i.e., periodic in time via application assumption, period in frequency via aliasing phenomena) are shift invariant. The periodicity structure of underlying discrete Fourier transform basis functions (complex exponentials, or equivalently, sine and cosine functions) facilitate this elegant shift invariance in the matter of arbitrary positioning of the indices defining the classical discrete Fourier transform matrix. Thus the discrete classical Fourier transform is defined with its native-zero and frequency-zero at the far edge of the native-index range and frequency-index range. An example of this is the matrix depicted in FIG. 15. In this figure, the left-most column and top-most row, both having all entries with a value of 1, denote the native-zero and frequency-zero assignments to the far edge of the native-index range and frequency-index range (noting e However, the continuous fractional Fourier transform operates using an entirely different basis function alignment. The continuous fractional Fourier transform is defined with its native-zero and frequency-zero at the center of the native-variable range and frequency-variable range. This is inherited from the corresponding native-zero and frequency-zero centering of the continuous classical Fourier transform in the fractionalization process. This situation differs profoundly from the discrete classical Fourier transform and a discrete fractional Fourier transform defined from it (which, as described above, is defined with its native-zero and frequency-zero at the far edge of the native-index range and frequency-index range). In particular, with respect to performing image propagation modeling with a fractionalization of a classical discrete Fourier transform, it is further noted that the native-zero and frequency-zero centering that corresponds to the continuous fractional Fourier transform naturally matches the modeling of the optics of lenses or other quadratic phase medium. In these optical systems, the phase of the light or particle beam varies as a function of the distance from the center of the lens. This aspect is illustrated in FIGS. 16A through 16C. Turning now to FIGS. 16A through 16C, light or high-energy particles are shown radiating spherically from a point in source plane FIG. 16A is a side view of source point FIG. 16b is a side view of second source point Similarly, FIG. 16C is a side view of third source point FIG. 16D is a top view of image The fractional Fourier transform model, at least in the case of coherent light or high-energy particle beams, can be thought of as performing the phase accounting as the image evolves through the propagation path. Thus, the classical continuous Fourier transform and its fractionalization match the modeled optics in sharing the notion of a shared zero origin for all images and lenses, while the classical discrete Fourier transform and its fractionalization do not because of the above-described offset in zero origin to the far edge of the transform index range. It is understood that in the foregoing discussion with respect to lenses applies equally to almost any type of quadratic-index media, such as GRIN fiber. The basis functions used in defining the continuous fractional Fourier transform are the Hermite functions, which are not periodic despite their wiggling behavior—the n Thus, brute-force application of the same fractionalization approach used in the continuous case to the classical discrete Fourier transform matrix (which does not position time and frequency center at zero as does the continuous case), could create undesirable artifacts resulting from the non-symmetric definition. It is, in effect, similar to defining the continuous fractional Fourier transform by the fractionalization of a “one-sided” continuous classical Fourier transform whose range of integration is from zero to positive infinity rather than from negative infinity to positive infinity. This can be expected to have different results. For example, although it can be shown that pairs of Hermite functions which are both of odd order or both of even order are indeed orthogonal on the half-line, pairs of Hermite functions which are one of odd order and one of even order are not orthogonal on the half-line. As a result, fractionalization of a “one-sided” continuous classical Fourier transform whose range of integration is from zero to positive infinity would not have the full collection of Hermite functions as its basis and hence its fractionalization would have different properties than that of the “two-sided” continuous fractional Fourier transform. However, it is the fractionalization of the “two-sided” continuous classical Fourier transform that matches the optics of lenses and other quadratic phase medium. The fractionalization of a differently defined transform could well indeed not match the optics of lenses and other quadratic phase medium. Hence, the brute-force application of the same fractionalization approach to the classical discrete Fourier transform matrix (which does not position time and frequency center at zero), could for some implementations be expected to create artifacts resulting from a non-symmetric definition. Some studies have reported that the discrete fractional Fourier transform defined from just such “brute force” direct diagonallization of the classical discrete Fourier transform report pathologies and non-expected results. It may, then, in some implementations, be advantageous—or even essential—to align the zero-origin of the discrete fractional Fourier transform with the zero-origin of the lens action being modeled. There are a number of ways to define a solution to address this concern. A first class of approaches would be to modify the discrete classical Fourier transform so that its zero-origins are centered with respect to the center of the image prior to fractionalization. This class of approaches would match the discrete transform structure to that of the optics it is used to model. Another class of approaches would be to modify the image so that the optics being modeled matches the zero-origins alignment of the classical discrete Fourier transform matrix, and then proceed with its brute-force fractionalization. Embodiments of the invention provide for either of these approaches, a combination of these approaches, or other approaches which match the zero-origins alignment of a discrete Fourier transform matrix and the image optics being modeled prior to the fractionalization of the discrete Fourier transform matrix. An exemplary embodiment of the first class of approaches would be to shift the classical discrete Fourier transform to a form comprising symmetry around the zero-time index and the zero-frequency index before fractionalization (utilizing the diagonalization and similarity transformation operations). This results in a “centered” classical fractional Fourier transform, and its fractionalization would result in a “centered” discrete fractional Fourier transform. The classical discrete Fourier transform, normalized to be a unitary transformation, can be represented as an L-by-L matrix whose element in row p and column q is:
The unitary-normalized, classical discrete Fourier transform may be simultaneously shifted in both its original and its frequency indices by k units by simply adding or subtracting the offset variable k for each of those indices:
Due to the reflective aliasing of negative frequency components into higher-index frequency samples, the classical discrete Fourier transform is shifted in such a way towards the indices centers would typically not compromise redundancy and diminished bandwidth due to symmetry around zero as might be expected. As an example, consider an exemplary signal comprising a unit-amplitude cosine wave of frequency 30 offset by a constant of Both the classical discrete Fourier transform and the classical continuous Fourier transform naturally respond to the complex exponential representation, specifically
In contrast, FIG. 18B illustrates the “centered” classical discrete Fourier transform acting on the same signal. Here the domain of the sampling and frequency indices range from −100 to +100, specifically {−100, −199, . . . −2, −1, 0, 1, 2, . . . , 99, 100}. The constant term of ¼ appears at zero frequency (frequency point 101 in the sequence), the positive frequency component ½ e This leads to direct interpretations of positive frequency and negative frequency discrete impulses that correspond with positive frequency and negative frequency Dirac delta functions that would appear as the classical continuous Fourier transform. More importantly, this re-configuring of the computational mathematics of the underlying discrete Fourier transform matrix gives a far more analogous fractionalization to that of the continuous fractional Fourier transform than the discrete fractional Fourier transform described in most publications (which are based on the unshifted classical discrete Fourier transform matrix definition). Of course, care must be taken to avoid artifacts created by frequency aliasing effects as would be known to, clear, and readily addressable to one skilled in the well-established art of frequency-domain numerical image processing. For a monochrome rectangular N×M image X(r,s), the unshifted classical discrete Fourier transform result Y(m,n) is, as is well known to one skilled in the art, given by an expression such as:
In the more general case, one can leave the centerings k The contrasting second class of approaches for aligning the center of the numerical transform and that of the images involves adapting the images to the centering of the numerical transforms. An exemplary embodiment of this second class of approaches could begin with the partition of an original image into, for example, four quadrant images separated by a pair of perpendicular lines that intersect at the center of the original image. Referring now to FIG. 19, image Each of the four distinct quadrant parts The brute-force fractionalization of the discrete classical Fourier transform can be applied to each of these to obtain four quadrant transformed images, denoted by transformed images If desired, these edge effects may be resolved, softened, or eliminated by performing additional calculations. For example, a second pair, or more, of perpendicular lines can be used to partition the original image in a manner that differs from that which is shown in FIG. 20 (for example, rotated and/or shifted with respect the original pair). Then, the process shown and described in conjunction with FIG. 20 may then be applied to these distinctly different quadrants as well. The generated calculations may be cross-faded or pre-emphasized and added to produce a composite image with significantly diminished boundary edge-effect artifacts. Typically, each of the various techniques described herein are invariant of which underlying discrete Fourier transform matrix is fractionalized to define the discrete fractional Fourier transform matrix. Although embodiments of the present invention may be implemented using the exemplary series of operations depicted in the figures, those of ordinary skill in the art will realize that additional or fewer operations may be performed. Moreover, it is to be understood that the order of operations shown in these figures is merely exemplary and that no single order of operation is required. In addition, the various procedures and operations described herein may be implemented in a computer-readable medium using, for example, computer software, hardware, or some combination thereof. For a hardware implementation, the embodiments described herein may be implemented within one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs), processors, controllers, microcontrollers, microprocessors, other electronic units designed to perform the functions described herein, or a combination thereof. For a software implementation, the embodiments described herein may be implemented with modules, such as procedures, functions, and the like, that perform the functions and operations described herein. The software codes can be implemented with a software application written in any suitable programming language and may be stored in a memory unit, such as memory The programming language chosen should be compatible with the computing platform according to which the software application is executed. Examples of suitable programming languages include C and C++. The processor may be a specific or general purpose computer such as a personal computer having an operating system such as DOS, Windows, OS/2 or Linux; Macintosh computers; computers having JAVA OS as the operating system; graphical workstations such as the computers of Sun Microsystems and Silicon Graphics, and other computers having some version of the UNIX operating system such as AIX or SOLARIS of Sun Microsystems; or any other known and available operating system, or any device including, but not limited to, laptops and hand-held computers. While the invention has been described in detail with reference to disclosed embodiments, various modifications within the scope of the invention will be apparent to those of ordinary skill in this technological field. It is to be appreciated that features described with respect to one embodiment typically may be applied to other embodiments. Therefore, the invention properly is to be construed with reference to the claims. The following references are cited herein: - [1] L. Levi, Applied Optics, Volume 2 (Section 19.2), Wiley, New York, 1980;
- [2] J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York, 1968;
- [3] K. Iizuka, Engineering Optics, Second Edition, Springer-Verlag, 1987;
- [4] A. Papoulis, Systems and Transforms with Applications in Optics, Krieger, Malabar, Fla., 1986;
- [5] L. F. Ludwig, “General Thin-Lens Action on Spatial Intensity (Amplitude) Distribution Behaves as Non-Integer Powers of Fourier Transform,” Spatial Light Modulators and Applications Conference, South Lake Tahoe, 1988;
- [6] R. Dorsch, “Fractional Fourier Transformer of Variable Order Based on a Modular Lens System,” in Applied Optics, vol. 34, no. 26, pp. 6016-6020, September 1995;
- [7] E. U. Condon, “Immersion of the Fourier Transform in a Continuous Group of Functional Transforms,” in Proceedings of the National Academy of Science, vol. 23, pp. 158-161, 1937;
- [8] V. Bargmann, “On a Hilbert Space of Analytical Functions and an Associated Integral Transform,” Comm. Pure Appl. Math, Volume 14, 1961, 187-214;
- [9] V. Namias, “The Fractional Order Fourier Transform and its Application to Quantum Mechanics,” in J. of Institute of Mathematics and Applications, vol. 25, pp. 241-265, 1980;
- [10] B. W. Dickinson and D. Steiglitz, “Eigenvectors and Functions of the Discrete Fourier Transform,” in IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-30, no. 1, February 1982;
- [11] F. H. Kerr, “A Distributional Approach to Namias' Fractional Fourier Transforms,” in Proceedings of the Royal Society of Edinburgh, vol. 108A, pp. 133-143, 1983;
- [12] F. H. Kerr, “On Namias' Fractional Fourier Transforms,” in IMA J. of Applied Mathematics, vol. 39, no. 2, pp. 159-175, 1987;
- [13] P. J. Davis, Interpolation and Approximation, Dover, N.Y., 1975;
- [14] N. I. Achieser, Theory of Approximation, Dover, N.Y., 1992;
- [15] G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, N.J., 1989;
- [16] N. N. Lebedev, Special Functions and their Applications, Dover, N.Y., 1965;
- [17] N. Wiener, The Fourier Integral and Certain of Its Applications, (Dover Publications, Inc., New York, 1958) originally Cambridge University Press, Cambridge, England, 1933;
- [18] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Princeton University Press, Princeton, N.J., 1993;
- [19] “Taking the Fuzz out of Photos,” Newsweek, Volume CXV, Number 2, Jan. 8, 1990; and
- [20] Jahne, Bernd, Digital Image Processing—Concepts, Algorithms, and Scientific Applications, Springer-Verlag, New York, 1991.
Patent Citations
Non-Patent Citations
Referenced by
Classifications
Legal Events
Rotate |