Patent Publication Number: US-8126287-B2

Title: Camera with image enhancement functions

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a Continuation of application Ser. No. 10/541,967, entitled “CAMERA WITH IMAGE ENHANCEMENT FUNCTIONS,” filed Jun. 23, 2006 now U.S. Pat. No. 7,627,193, which is a national sage filing of PCT/IL04/00040, filed Jan. 15, 2004, which claims priority to U.S. Provisional Appln. 60/440,561, filed Jan. 16, 2003, the entire contents of which are hereby incorporated by reference as if fully set forth herein. The applicants hereby rescind any disclaimer of claim scope in the parent application(s) or the prosecution history thereof and advise the USPTO that the claims in this application may be broader than any claim in the parent application(s). 
    
    
     FIELD OF THE INVENTION 
     The present invention relates generally to digital imaging, and specifically to methods and devices for enhancing image quality in digital cameras, as well as designing optics for 10 such cameras. 
     BACKGROUND OF THE INVENTION 
     The inherent resolution of digital cameras known in the art, such as cameras based on CCD or CMOS sensor arrays, is generally limited by the resolution of the sensor array itself. In order to avoid degrading resolution still further, the objective optics used in such cameras are typically designed so that the optical point spread function (PSF) has an extent smaller than the pitch of the array. In other words, the optics are designed so that a bright point in the object plane is imaged to a spot no more than one pixel in size on the sensor array. 
     The PSF of an optical system may vary from the ideal due to focal variations and aberrations. A number of methods are known in the art for measuring and compensating for such PSF deviations by digital image processing. For example, U.S. Pat. No. 6,154,574, whose disclosure is incorporated herein by reference, describes a method for digitally focusing an out-of-focus image in an image processing system. A mean step response is obtained by dividing a defocused image into sub-images, and calculating step responses with respect to the edge direction in each sub-image. The mean step response is used. in calculating PSF coefficients, which are applied in turn to determine an image restoration transfer function. An in-focus image is obtained by multiplying this function by the out-of-focus image in the frequency domain. 
     As another example, U.S. Pat. No. 6,567,570, whose disclosure is incorporated herein by reference, describes an image scanner, which uses targets within the scanner to make internal 30 measurements of the PSF. These measurements are used in computing convolution kernels, which are applied to images captured by the scanner in order to partially compensate for imperfections of the scanner lens system. 
     It is also possible to add a special-purpose blur to an image so as to create invariance to certain optical aberrations. Signal processing is then used to remove the blur. A technique of this sort is described by Kubala et al., in “Reducing Complexity in Computational Imaging Systems,”  Optics Express  11 (2003), pages 2102-2108, which is incorporated herein by reference. The authors refer to this technique as “Wavefront Coding.” A special a spheric optical element is used to create the blur in the image. This optical element may be a separate stand-alone element, or it may be integrated into one or more of the lenses in the optical system. Optical designs and methods of image processing based on Wavefront Coding of this sort are described, for example, in U.S. Pat. No. 5,748,371 and in U.S. Patent Application Publications US 2002/0118457 A1, US 2003/0057353 A1 and US 2003/0169944 A1, whose disclosures are incorporated herein by reference. 
     SUMMARY OF THE INVENTION 
     Some embodiments of the present invention provide methods and devices for improving the image quality of digital imaging systems. These methods and devices are useful particularly in creating digital cameras with simple, low-cost lenses (or even no lenses) that produce high-quality images. The novel principles of optical design and image processing that are taught by the present invention, however, are also applicable to digital imaging systems of other types. 
     In some embodiments of the present invention, an electronic imaging camera comprises an image sensing array. The camera typically comprises objective optics, which image an object onto the array with a PSF of extent substantially greater than the pitch of the sensing array even at optimal focus of the optics. (In one embodiment, the “objective optics” simply comprise an aperture—in the nature of a large “pinhole”—with no lens at all.) As a result, the images captured by the array itself are blurred. An image processor, which may be integrated in the camera, applies a deblurring function—typically in the form of a deconvolution filter—to the signal output by the array in order to generate an output image with reduced blur. This blur reduction is made possible by novel methods of image processing described hereinbelow. These methods make it possible to design and use camera optics with a poor inherent PSF, while restoring the electronic image generated by the sensing array to give an acceptable output image. 
     Typically, the objective optics are cylindrically symmetrical about the optical axis. In other words, if the optics are rotated about the axis, the image formed on the sensor array—and therefore the deblurring function that must be applied to the image—is substantially unchanged. By contrast, systems based on Wavefront Coding, as described in the publications cited in the Background of the Invention, intentionally use optics whose cylindrical symmetry is broken. Thus, these systems require exotic optical elements and are sensitive to variations in rotational alignment of the optics. 
     Because defect-free detector arrays are very costly, most electronic imaging cameras for consumer and industrial applications have a certain number of bad pixels, which do not produce a useful light signal. In some embodiments of the present invention, statistical analysis of images captured by the camera is used to discover these bad pixels in the sensing array. Typically, the bad pixels are identified in images captured by the camera as those whose pixel values differ from those of neighboring pixels by more than a threshold amount, which is determined by the PSF of the camera optics. Before deblurring the image (by deconvolution filtering), the values of the bad pixels are replaced with substitute values, so as to neutralize the influence of the bad pixels on the restored image. An iterative process of creating further blurred images and then deblurring the images is used to generate restored images of improved quality. In this process, the pixel values in the vicinity of the known bad pixel locations are modified at each iteration, in order to reduce propagation of artifacts due to the bad pixels. The input image may be surrounded by a frame of dummy pixels for the purposes of the deblurring operations, wherein the dummy pixels are treated as bad pixels. 
     Although the deblurring methods of the present invention may be used in cameras with substantially any sort of objective optics, in some embodiments of the present invention, the optics are designed by an iterative process, which takes into account the deblurring capabilities of the camera. For this purpose, an initial optical design is generated, and the PSF of the design is calculated based on the aberrations and tolerances of the optical design. A representative digital image, characterized by this PSF, is computed, and a deblurring function is determined in order to enhance the PSF of the image, i.e., to reduce the extent of the PSF. The design of the optical system is then modified so as to reduce the extent of the enhanced PSF. 
     In order to modify the optical design, the joint effects of variations in the optical design and concomitant changes in the enhanced PSF may be estimated. This estimate is used in determining how the optical design should be changed. The process of estimating the PSF enhancement and modifying the optical design accordingly is typically repeated iteratively until the design converges to within a predetermined tolerance. This process optimizes the overall performance of the camera, while permitting the use of low-cost optics with relatively high manufacturing tolerances and a reduced number of optical elements. 
     There is therefore provided, in accordance with an embodiment of the present invention, imaging apparatus, including: 
     an array of optical sensing elements, characterized by a pitch, which is adapted to generate a signal in response to optical radiation that is incident on the elements; and 
     objective optics, which have an optical axis and are characterized by a cylindrical symmetry about the axis, and which are arranged to focus the optical radiation from an object onto the array with a point spread function (PSF) having an extent greater than twice the pitch of the array at an optimal focus of the objective optics. 
     In a disclosed embodiment, the extent of the PSF is at least three times the pitch. 
     Typically, the apparatus includes an image processor, which is coupled to process the signal from the array responsively to the PSF so as to generate an electronic image of the object while reducing a blur of the electronic image. In disclosed embodiments, the signal from the array corresponds to an input image including pixels having respective pixel values, and the image processor is adapted to reduce the blur by applying a deconvolution filter, based on the PSF, to the input image. The image processor may be adapted to identify one or more bad elements in the array and to replace the pixel values of the pixels in a vicinity of the bad elements in the input image with substitute values so as to produce a corrected blurred image, and to apply the deconvolution filter to the corrected blurred image so as to generate a restored image. 
     In some embodiments, the objective optics include one or more lenses, having optical characteristics that are selected, responsively to the PSF, so as to optimize a quality of the electronic image following application of the deconvolution filter. 
     Additionally or alternatively, the image processor is adapted to modify the deconvolution filter that is applied to the input image responsively to a change in a focal relationship between the object and the array. In one embodiment, the image processor is adapted to select a macro deconvolution filter when the object is less than a predetermined minimum distance from the objective optics. 
     There is also provided, in accordance with an embodiment of the present invention, imaging apparatus, including: 
     an array of optical sensing elements, characterized by a pitch, which is adapted to generate a signal in response to optical radiation that is incident on the elements; and 
     a camera body, containing the array and having an aperture formed therein having a transverse dimension substantially greater than the pitch of the array, wherein the aperture and array are arranged so that the optical radiation from an object passes through the aperture and is incident on the array substantially without intervention of an optical focusing element. 
     In a disclosed embodiment, the apparatus includes a mechanism, coupled to at least one of the array and the aperture, for varying a distance between the array and the aperture so as to adjust a field of view of the device. 
     Typically, the transverse dimension of the aperture is at least 50% greater than the pitch. Optionally, the transverse dimension of the aperture is at least twice the pitch or even at least three times the pitch. 
     There is additionally provided, in accordance with an embodiment of the present invention, imaging apparatus, including: 
     objective optics, which are arranged to focus optical radiation from an object onto a focal plane with a characteristic point spread function (PSF); 
     an array of optical sensing elements, which is positioned at the focal plane and is adapted to generate a signal in response to the optical radiation that is incident on the elements, the signal corresponding to an electronic image of the object including pixels having respective pixel values; 
     an image processor, which is coupled to receive the signal from the array and is adapted to identify one or more bad elements in the array and, responsively to the PSF, to modify the pixel values of the pixels in a vicinity of the bad elements in the input image so as to produce a corrected blurred image while reducing propagation of artifacts due to the bad elements, and to apply a deconvolution filter, based on the PSF, to the corrected blurred image so as to generate a restored output image. 
     There is further provided, in accordance with an embodiment of the present invention, an image processor, including at least one integrated circuit device, which is adapted to receive a blurred input image of an object originating from an array of light detecting elements that includes one or more bad elements and is characterized by a point spread function (PSF), and responsively to the PSF, to modify values of pixels in a vicinity of the bad elements in the blurred input image, so as to produce a corrected blurred image while reducing propagation of artifacts due to the bad elements, and to apply a deconvolution filter to the corrected blurred image so as to generate a restored image. 
     There is moreover provided, in accordance with an embodiment of the present invention, a method for producing an optical system, including: 
     producing a design of the optical system; 
     computing an optical point spread function (PSF) of the optical system according to the design; 
     determining an enhanced PSF that can be achieved by application of a given image restoration function to an image produced by the optical system according to the design; and 
     modifying the design of the optical system to reduce an extent of the enhanced PSF. 
     In some embodiments, producing the design includes assigning respective merit scores to each of a plurality of optical aberrations, determining the aberrations that are characteristic of the design, computing a design score responsively to the aberrations and the respective merit scores, and optimizing the design responsively to the design score. Typically, assigning the respective merit scores includes determining respective effects of each of the aberrations on the PSF, and calculating the respective merit scores responsively to the respective effects. Additionally or alternatively, determining the aberrations includes generating a Zernike polynomial expansion of the aberrations, and computing the design score includes calculating a weighted sum of the respective merit scores responsively to coefficients of the polynomial expansion. 
     In disclosed embodiments, determining the enhanced PSF includes determining coefficients of a deconvolution filter responsively to the computed optical PSF, and applying the deconvolution filter to a representative image that is characterized by the computed optical PSF. 
     Additionally or a alternatively, modifying the design includes estimating effects of variations in the design on the enhanced PSF, and modifying the design of the optical system responsively to the estimated effects. In a disclosed embodiment, determining the enhanced PSF includes computing an image quality score responsively to the enhanced PSF, and estimating the effects of the variations includes finding a dependence of the image quality score on the variations. Typically, computing the image quality score includes determining respective local point spread functions at a plurality of points in an image field of the optical system, and determining the image quality score responsively to the local point spread functions. Determining the image quality score may include finding respective local scores at the plurality of the points, and combining the local scores to generate the image quality score. 
     In one embodiment, determining the enhanced PSF includes calculating a weighted PSF by taking a weighted sum of the local point spread functions, and estimating the effects of the variations includes calculating the effects of the variations on the a weighted PSF. 
     In another embodiment, computing the image quality score includes finding a resolution score, which is indicative of a resolution of an image produced with the enhanced PSF, and finding an artifact score, which is indicative of a level of artifacts in the image, and combining the resolution score and the artifact score to determine the image quality score. 
     Typically, computing the image quality score includes determining the image quality score responsively to aberrations of the optical system, and finding the dependence of the image quality score includes calculating a gradient of the score with respect to changes in the aberrations. In a disclosed embodiment, finding the dependence of the image quality score includes determining a response matrix, indicative of the changes in the aberrations that will result as a function of specific modifications to the design of the optical system, and modifying the design includes determining a modification to be applied to the design responsively to the gradient and the response matrix. 
     In some embodiments, the method includes iteratively repeating, with respect to the modified design of the optical system, computing the optical PSF, determining the enhanced PSF, and modifying the design. Typically, iteratively repeating includes performing iterations through compiling the optical PSF, determining the enhanced PSF, and modifying the design until a modification to the design that is indicated by one of the iterations is less than a predetermined tolerance that is to be applied in manufacturing the optical system. 
     Typically, the optical system includes one or more optical elements, and the variations in the design include a change in at least one of a curvature of an optical surface of one of the elements, a thickness of one of the elements and a spacing between the elements. In a disclosed embodiment, the one or more optical elements include at least one aspheric element having at least one surface defined by a polynomial function, and modifying the design includes optimizing coefficients of the polynomial function. 
     There is furthermore provided, in accordance with an embodiment of the present invention, an optical system, including one or more optical elements, which are produced and assembled in accordance with a design that is optimized by computing an optical point spread function (PSF) of the optical system according to the design, determining an enhanced PSF that can be achieved by application of a given image restoration function to an image produced by the optical system according to the design, and modifying the design of the optical system to reduce an extent of the enhanced PSF. 
     There is also provided, in accordance with an embodiment of the present invention, a computer software product, including a computer-readable medium in which program instructions are stored, which instructions, when read by the computer, cause the computer to produce a design of the optical system, to compute an optical point spread function (PSF) of the optical system according to the design, to determine an enhanced PSF that can be achieved by application of a given image restoration function to an image produced by the optical system according to the design, and to modify the design of the optical system so as to reduce an extent of the enhanced PSF. 
     There is additionally provided, in accordance with an embodiment of the present invention, a method for electronic imaging, including: 
     generating a signal, using an array of optical sensing elements that is characterized by a pitch, in response to optical radiation that is incident on the elements; and 
     focusing the optical radiation from an object onto the array using objective optics, which have an optical axis and are characterized by a cylindrical symmetry about the axis, to form an optical image with a point spread function (PSF) having an extent greater than twice the pitch of the array. 
     In disclosed embodiments, the method includes processing the signal from the array responsively to the PSF so as to generate an electronic image of the object while reducing a blur of the electronic image. Typically, the signal from the array corresponds to an input image including pixels having respective pixel values, and processing the signal includes applying a deconvolution filter, based on the PSF, to the input image. In one embodiment, applying the deconvolution filter includes modifying the deconvolution filter responsively to a change in a focal relationship between the object and the array. For example, modifying the deconvolution filter may include selecting a macro deconvolution filter when the object is less than a predetermined minimum distance from the objective optics. 
     There is further provided, in accordance with an embodiment of the present invention, a method for image processing, including: 
     identifying one or more bad elements in an array of light detecting elements; 
     capturing a blurred input image, characterized by a point spread function (PSF), of an object using the array; 
     modifying, responsively to the PSF, values of pixels in a vicinity of the bad elements in the blurred input image, so as to produce a corrected blurred image while reducing propagation of artifacts due to the bad elements; and 
     applying a deconvolution filter, based on the PSF, to the corrected blurred image so as to generate a restored image. 
     Typically, the method includes performing iteratively modifying the values of the pixels in the vicinity of the bad elements in the restored image, so as to produce a new blurred image, and applying the deconvolution filter to the new blurred image so as to generate a new restored image. 
     In a disclosed embodiment, the method includes adding a frame of dummy pixels surrounding the input image, wherein the dummy pixels are treated similarly to the pixels associated with the bad elements for purposes of producing the corrected blurred image and generating the restored image. 
     In some embodiments, modifying the values includes determining substitute values with which to replace the pixel values in the vicinity of the bad pixels responsively to the PSF. 
     In one of these embodiments, determining the substitute values includes setting the substitute values using a correction map M[x] substantially of a form given by 
               M   ⁡     [   x   ]       =     {           (       1       M   ′     ⁡     [   x   ]         -   1     )               M   ′     ⁡     [   x   ]       ≠   0             0               M   ′     ⁡     [   x   ]       =   0     ,                   
wherein x is a pixel coordinate, and M′[x] is a convolved map given by a convolution of a map of the bad elements with a conjugate of the PSF.
 
     Additionally or alternatively, identifying the one or more bad elements includes determining differences between the values of the pixels, and comparing the differences to a maximum difference dependent upon the PSF. 
     The present invention will be more fully understood from the following detailed description of some embodiments thereof, taken together with the drawings in which: 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a schematic side view of a lensless camera, in accordance with an embodiment of the present invention; 
         FIG. 1B  is a schematic detail view of the camera of  FIG. 1A , illustrating a point spread function of the camera; 
         FIG. 2  is a schematic detail view of the camera of  FIG. 1A , illustrating an optical zoom operation, in accordance with an embodiment of the present invention; 
         FIG. 3  is a schematic side view of a camera, in accordance with another embodiment of the present invention; 
         FIG. 4  is a schematic frontal view of a matrix of pixels in the image plane of a sensor array used in the camera of  FIG. 3 ; 
         FIG. 5  is a flow chart that schematically illustrates a method for identifying bad pixels in images captured by an electronic imaging camera, in accordance with an embodiment of the present invention; 
         FIG. 6  is a flow chart that schematically illustrates a method for deblurring images, in accordance with an embodiment of the present invention; 
         FIG. 7  is a flow chart that schematically illustrates a method for optical design, in accordance with an embodiment of the present invention; 
         FIG. 8  is a flow chart that schematically illustrates a method for computing an image quality score, for use in conjunction with the method of  FIG. 7 ; 
         FIG. 9  is a schematic side view of objective optics designed using the method of  FIG. 7 , in accordance with an embodiment of the present invention; 
         FIGS. 10A and 10B  are schematic plots of the PSF of the objective optics shown in  FIG. 9 ; 
         FIGS. 11A and 11B  are schematic plots of a deconvolution filter (DCF) kernel that is used to deblur images formed by the optics of  FIG. 9 , in accordance with an embodiment of the present invention; and 
         FIGS. 12A and 12B  are schematic plots of an enhanced PSF that is obtained by deblurring images using the DCF illustrated in  FIGS. 11A and 11B , in accordance with an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     Definitions 
     The following is a non-exhaustive list of technical terms that are used in the present patent application and in the claims. Although these terms are used herein in accordance with the plain meaning accorded the terms in the art, they are listed below for the convenience of the reader in understanding the following description and the claims.
         Pitch of a detector array refers to the center-to-center distance between elements of the array.   Cylindrical symmetry describes a structure, such as a simple or compound lens, which has an optical axis such that the structure is invariant under rotation about the optical axis for any and all angles of rotation.   Point spread function (PSF), expressed herein as h, is the impulse response of an optical system in the spatial domain, i.e., the image formed by the system of a bright point object against a dark background.   Extent of the PSF is the full width at half maximum (FWHM) of the PSF.   Optical transfer function (OTF) is the two-dimensional Fourier transform of the PSF to the frequency domain. Because of the ease with which a PSF may be transformed into an OTF, and vice versa, computation of the OTF is considered to be equivalent to computation of the PSF for the purposes of the present invention.   Optical radiation refers to electromagnetic radiation in any of the visible, infrared and ultraviolet regions of the spectrum.       

     Deblurring of Camera Images 
       FIG. 1A  is a schematic side view of an electronic imaging camera  20 , in accordance with an embodiment of the present invention. Camera  20  comprises an image sensor array  22 , such as a CCD or CMOS image sensor, comprising a matrix of detector elements  24 , as is known in the art. Optical radiation, typically visible light, from an object is incident on array  22  through an aperture  26 , which serves as the objective optic of the camera. A motion device  28  may be operated to translate array  22  towards and away from aperture  26 , for purposes of an optical zoom operation, as described below. Alternatively, the motion device may be coupled to translate aperture  26 , rather than array  22 , or to translate both the aperture and the array. 
       FIG. 1B  is a schematic, detail view of aperture  26  and array  22 . In a conventional electronic imaging camera, a lens is used to focus an image of an object onto the sensor array, so as to produce a sharp image, wherein each point on the object is imaged to a point on the sensor plane. In camera  20 , however, substantially no objective lens is used, so that the image incident on array  22  is necessarily blurred. Light emitted from a point  40  on the object will pass through aperture  26  to form a spot  42  on array  22 . For conceptual simplicity (ignoring diffraction and other optical effects), for object point  40  much farther from aperture  26  than the aperture is from array  22 , it can be stated that the extent of spot  42  is roughly equal to the diameter of aperture  26 . Typically, the aperture diameter is substantially greater than the pitch of array  22 , for example, twice or four times the pitch, or even greater. Thus, spot  42  covers a group of at least several mutually-adjacent pixels. It may equivalently be stated that the point spread function (PSF) of the image formed on array  22  has a characteristic extent that is roughly equal to the diameter of aperture  26 . 
     Returning now to  FIG. 1A , an image processor  30  processes the blurred input image generated by array  22  to produce a restored output image  32 . Image processor  30  typically comprises one or more dedicated integrated circuit chips, which are contained inside camera  20 . Alternatively or additionally, the image processor may comprise a programmable processor, with suitable software, and may be located either inside or outside the camera, i.e., the processor could be in a separate computer into which images from the camera are loaded. Processor  30  may be used not only in lensless camera  20 , but also, mutatis mutandis, as an integral part or add-in element in other types of digital cameras and other imaging devices. Output image  32  may be displayed on a video monitor  34 , for example, or it may alternatively be printed or stored in a memory. 
       FIG. 2  is a schematic, detail view of aperture  26  and array  22 , illustrating an optical zoom function of camera  20 , in accordance with an embodiment of the present invention. Array  22  is translated by motion device  28  ( FIG. 1 ) left/right along the horizontal axis in the figure, toward and away from aperture  26 . When the array is relatively far from the aperture, it receives light from objects within a relatively narrow field of view, of angular half-width α. When the array is moved forward, toward aperture  26 , to a position marked as  22 ′, it sees a wider field of view, of angular half-width a′. This effect is equivalent to the optical zoom effect obtained by translating the objective lens (or lenses) in a conventional camera. 
       FIG. 3  is a schematic side view of an electronic imaging camera  44 , in accordance with another embodiment of the present invention. Camera  44  is similar in construction and operation to camera  20 , as described above, except that camera  44  comprises refractive objective optics  46 , typically made up of one or more lenses. There may also be one or more filters  48  in the optical, path of the camera, such as an infrared-blocking filter. Typically, to reduce manufacturing costs, optics  46  comprise relatively few refractive elements, which are designed with relatively loose tolerances. As a result, the extent of the PSF of the optics is substantially greater than the pitch of array  22 . Typically, the PSF extent is at least twice the pitch and may be greater than three times the pitch or even four times the pitch, in order to relax the design constraints on optics  46  and thus reduce the cost of camera  44 . Note that the PSF may vary if the object distance or magnification (zoom) of the camera is changed. (The change in PSF in this case is more complex than the simple zoom function illustrated in  FIG. 2 .) Processor  30  may be programmed to compensate for such variations. 
     Image processor  30  deblurs the output image of camera  20  or  44  by applying a deconvolution filter (DCF) to the image, after first removing noise from the blurred image. The DCF may be regarded as a regularized inverse of the complex PSF h, such as: 
                   DCF   =       h   *         hh   *     +   ɛ               (   1   )               
wherein ε is a small constant. The processing method applied by processor  30  is described in detail in the above-mentioned U.S. patent application Ser. No. 09/778,849.
 
     Further refinements to the technique described in that application, which may be applied by processor  30 , are described in greater detail hereinbelow. For example, because the PSF h may vary over the image field of the camera, the deblurring process may be applied separately to different image segments, typically using a different DCF in each segment. The deblurred segments are then stitched together to generate a final, restored image. Deblurring of images using a DCF is the method of image restoration that is used in at least some embodiments of the present invention that are described herein. The DCF may be applied in either the spatial domain or the frequency domain. Equivalent, alternative techniques of image restoration will be apparent to those skilled in the art upon reading the above-mentioned U.S. patent application and the present patent application, and are considered to be within the scope of the present invention. 
       FIG. 4  is a schematic frontal view of a matrix  50  of pixels  54  captured by array  22 , for processing by processor  30  in camera  20  or  44 , in accordance with an embodiment of the present invention. Matrix  50  comprises an image area  52 , made up of pixels  54 , having values corresponding to the signals that are output by the corresponding elements  24  of array  22 . For the most part, within the operating regime of array  22 , the signals (and therefore the pixel values) are proportional to the intensity of light incident on the corresponding detector elements, with a fixed proportionality. For some pixels, however, such as a deviant pixel  56 , the proportionality deviates from the norm, providing either higher or lower sensitivity than expected. Other pixels, such as a defective pixel  57 , may provide unusable outputs, either because the corresponding elements  24  of array  22  are abnormally noisy or because they have very low sensitivity (or no sensitivity) to light. Such defective pixels are referred to herein as bad pixels. Note that “bad pixels” in the image may also arise from overexposed and underexposed pixels in the image, such as “hot spots” or “black spots” in the image, and that groups of mutually-adjacent bad pixels may cause bad areas in the image. 
     Spot  42  is effectively the image of a point source in the object plane of the camera. The blurring caused by the PSF of the camera, exemplified by spot  42  shown in  FIG. 4 , is equivalent in effect to low-pass filtering of the image, so that each pixel value is effectively averaged with the values of its neighbors. Therefore, the PSF imposes a limit on the possible difference in values of neighboring pixels, which is substantially less than the total dynamic range (for example, 255:1) of array  22 . Differences between neighboring pixels that are in excess of this limit are indicative of defects in elements  24  of the array, due to deviant or bad pixels. 
       FIG. 5  is a flow chart that schematically illustrates a method for identifying such bad and deviant pixels, in accordance with an embodiment of the present invention. The method is described hereinbelow with reference to camera  44 , although it may similarly be applied to camera  20  or to other types of electronic imaging devices. As a starting point for this method, the applicable PSF is ascertained, at  60 . The PSF may be measured, using optical methods known in the art, or it may alternatively be calculated, based on the known characteristics of optics  46  and other salient characteristics of array  22  and camera  20 . The PSF is used to determine the maximal permissible difference between values of neighboring pixels, at  62 . This maximal difference may be determined, for example, by calculating the values of neighboring pixels, subject to the known PSF, in the area of a sharp edge of maximal contrast between bright and dark features in the object plane. 
     Camera  44  is operated in a test mode, in order to collect image statistics, at  64 . In the test mode, the camera may be scanned, for example, over different image types and areas, and a sequence of corresponding images may be collected. Alternatively, the image statistics may be collected during normal operation of the camera. The values of neighboring pixels over the sequence of images are then compared, at  66 , in order to identify pixels that are incompatible with their neighbors. Two types of incompatibility are identified, at  68 , depending on the pattern of deviation between a given pixel and its neighbors:
         Consistent deviation, characteristic of deviant pixels  56  in which the average value of the given pixel is higher or lower than its neighbors by a consistent proportionality factor. In this case, processor  30  determines a calibration factor, at  70 , to be applied to the value of this pixel output by array  22  in order to adjust for the deviant sensitivity of the corresponding detector element  24 .   Deviation without a consistent proportionality factor, characteristic of bad pixels  57 , which are considered to give no useful information. Processor  30  marks these pixels for special processing treatment, at  72 . Handling of these bad pixels by processor  30  is described below with reference to  FIG. 6 .
 
An advantage of this approach for handling bad and deviant pixels is that it allows camera  44  to use an array  22  with a relatively large number of deviant and/or defective elements  24 , while still producing images of good quality.
       

     It is also possible for processor  30  to identify and compensate for pixels that are not inherently defective, but give “bad” output signals in a particular image. For example, when camera  44  captures a scene having an excessively bright point of light, the corresponding pixel in the image may be saturated, and blooming may cause the saturation to spread to neighboring pixels. Before deblurring the image, processor  30  may recognize these effects and may add the pixels at which they occur to its current map of bad pixels. 
       FIG. 6  is a flow chart that schematically illustrates a method applied by processor  30  in generating deblurred output image  32 , in accordance with an embodiment of the present invention. The method is based, inter alia, on creating a map D[x] of bad pixels, as described above, at  80 . Here x is a generalized vector coordinate in the spatial (pixel) domain. D[x] has the value 0 for bad pixels, and 1 elsewhere. 
     For the purposes of this method, a “blurred” map of the bad pixels is created, by convolving map D with the conjugate PSF h*, at  82 . In the spatial frequency domain, the Fourier transform of the convolved map is given by:
 
 {tilde over (M)}′[q]={tilde over (h)}*[q]·{tilde over (D)}[q]   (2)
 
wherein {tilde over (h)}*[q] and {tilde over (D)}[q] are the Fourier transforms of the PSF and the bad pixel map, respectively, and q is a generalized vector coordinate in the transform (frequency) domain. The new correction map M is found by retransforming M′ to the spatial domain and setting:
 
                     M   ⁡     [   x   ]       =     {           (       1       M   ′     ⁡     [   x   ]         -   1     )               M   ′     ⁡     [   x   ]       ≠   0             0             M   ′     ⁡     [   x   ]       =   0                     (   3   )               
Alternatively, M′ may be computed in the spatial domain, rather than transforming the maps back and forth by Fourier transformation.
 
     Note that in the vicinity of bad pixels, the convolved map values M′[x] may be much less than one, so that the corresponding correction map values M[x] may be very large. The inventors have found that the use of such a correction map, in the manner described below, gives much better results than conventional methods of interpolation, in terms of faithful reconstruction of the correct image with minimal artifacts. These artifacts stem, inter alia, from the non-localized nature of the PSF and of the DCF that must be used to compensate for it. When simple interpolation is used to smooth over the areas of bad pixels, the resulting image is typically either blurred (at least in the interpolated area) or is marred by noticeable artifacts. The correction map provided by some embodiments of the present invention, on the other hand, replaces the values of the bad pixels while preserving the underlying spatial frequency structure of the surrounding good pixels. Thus, propagation of artifacts from the bad pixels to the other pixels in their vicinity is reduced substantially. Alternatively, the correction map may be determined using other formulas that are similar in behavior to that defined in equation (3), though differing in algebraic form. Such alternative formulas are considered to be within the scope of the present invention. In any case, the bad pixel replacement technique described herein permits array  22  to include a relatively large number of bad pixels without causing a substantial adverse effect in the output images from the camera. Thus, the cost of the camera may be reduced by using inexpensive, low-quality arrays with relatively large numbers of bad pixels. 
     Processor  30  receives an input image from array  22  (which is blurred by the PSF of optics  46 ), at  84 . The image includes bad pixels, at locations that were marked at  80 . Processor replaces the values of the bad pixels, at  86 , with estimated correct values. The estimated values may be determined, for example, by interpolation from the values of adjacent good pixels. The good pixels in the image remain unchanged, except to the extent that it is necessary to adjust their values by the calibration factors found at  70  ( FIG. 5 ). 
     In addition, at  86 , processor  30  adds a frame  58  around the edges of image area  52 . The purpose of this frame is to provide for the inherent periodicity of the Fourier transform that is to be applied in deblurring the image. The width of frame  58  is, for example, roughly equal to the radius of the PSF. The pixels in frame  58  are treated as “bad” pixels for the purposes of the method of  FIG. 6 . The initial values of these pixels are obtained by interpolating between the values of the “good” pixels at the adjacent right and left edges or lower and upper edges of area  52 . The frame pixels thus add a periodic structure to the image, which is useful in suppressing artifacts that could otherwise be introduced due to the image edges when transforming the image between the spatial and transform domains. 
     The corrected, framed, blurred image obtained at  86 , B 0 , is convolved with the DCF (equation 1) to generate an initial deblurred image r 0 , at  88 . Formally, in simplified terms, r 0 [x]=DCF*B 0 [x], wherein the operator “*” represents convolution. In practice, the operation is typically performed in the Fourier transform domain, and is accompanied by noise reduction steps, as described in the above-mentioned U.S. patent application Ser. No. 09/778,849. 
     Processor  30  then evaluates r 0  to determine whether additional processing is desired in order to improve the image quality, at  90 . The basis for the decision at  90  may be the amount of processing time that has elapsed, or the number of iterations that have been carried out, or it may be a measure of the image quality. These decision criteria are described further hereinbelow. If processor  30  determines that processing of the current image has been completed, the image is output, at  92 , and the process terminates. Typically, frame  58  is removed before outputting the image. Alternatively, part or all of the frame may be left in the output image, since it may contain information (due to the large PSF of the camera) that was outside the nominal field of view of the image. 
     Alternatively, processor  30  may continue to process the current image iteratively, in order to generate a sharper, more accurate image of the object. For this purpose, the blurred image B i [x] B 0 ) and the deblurred image r i [x] r 0 ) from the preceding iteration are used in creating a new blurred image B i+1 [x], at  94 , with better values of the bad pixels:
 
 B   i+1   [x ]=( h*r   i   [x ])(1 −D[x ])+ D[x]B   i   [x]   (4)
 
In other words, the previous deblurred image is convolved with the PSF, and the bad pixel map D is used to substitute the pixel values thus obtained into B i+1  at the locations of the bad pixels. The values of the good pixels in B i  remain unchanged.
 
     The new blurred image, B i+1 , with the improved bad pixel values, is used to create an improved deblurred image r i+1 , at  96 . The pixel values in r i+1  can be expressed as follows:
 
 r   i+1   [x ]=DCF* B   i+1   [x]+M[x]h*[x ]*[( B   i   [x]−h*r   i   [x ]) D[x]]   (5)
 
Here again, the convolution operations are typically carried out by transforming the terms of equation (5) to the Fourier transform domain, multiplying, and then transforming back to the spatial domain, as will be apparent to those skilled in the art. The first term of r i ÷ i  simply represents deblurring of the new blurred image, B i+1  by the DCF. The second term is added to remove the effect of the estimated bad pixel values on the values of the good pixels in the new deblurred image. This term uses the correction map values derived at  82 , as expressed by equation (3). In each iteration, this term generates new values for the pixels on which it operates, so as to reproduce more faithfully the actual spatial frequency structure of the correct image.
 
     At this point, processor  30  again assesses the quality of the deblurred image and/or the processing time or iterations elapsed, at  90 , and decides whether to terminate the computation, or continue for another iteration. The iterations may continue in this manner indefinitely, until the image is finally output at  92 . 
     In addition, further image enhancement operations may be applied at  92  before the final image is stored and/or displayed. The further operations are generally directed to reducing noise in the image and/or removing artifacts. For example, a smoothing filter may be applied in order to reduce noise. In order to maintain edge contrast, local image gradients may be determined, and a one-dimensional smoothing filter may then be applied in a direction orthogonal to the local gradient. 
     As another example, “ghost edges,” which sometimes appear as artifacts in the vicinity of actual sharp edges in the image, may also be removed at this stage. For this purpose, an edge image is created by smoothing the deblurred image, while preserving the actual, strong edges in the image. This edge image is considered to be artifact-free (although lacking in all detail except for the strong edges). Blurring and restoration operations are applied to the edge image, resulting in the appearance of the ghost edges. Subtracting the artifact-free edge image from the this latter image gives an image that contains only artifacts (footprints) of the ghost edges. The footprints of the ghost edges can now be subtracted from the original deblurred image in order to eliminate the artifacts. 
     Alternatively or additionally, other methods for image enhancement and artifact reduction may be applied at  92 . Exemplary methods, which may be applied,  mutatis mutandis , by processor  30 , are described in U.S. Pat. Nos. 4,691,366, 5,751,861 and 6,094,467, whose disclosures are incorporated herein by reference. 
     Optical Design Optimized for Image Restoration 
       FIG. 7  is a flow chart that schematically illustrates a method for designing optics  46 , for use in a digital camera with a deblurring function, in accordance with an embodiment of the present invention. The method takes advantage of foreknowledge of the image restoration function to be applied by processor  30 , in order to generate an optical design that is optimized for this function. After generating an initial optical design, the optical PSF of the optical system is computed, and is used in determining an enhanced PSF that can be achieved by application of the deblurring function to images produced by the optical system according to the design. A response matrix is also computed based on the optical design, indicating the effects of variations in the design on the enhanced PSF. The design of the optical system is then modified based on the response matrix to give a new design that will have a better enhanced PSF after image restoration. This process may be repeated iteratively to give successively better designs. Although the method of  FIG. 7  is described hereinbelow with reference to camera  44  and to the deblurring method shown in  FIG. 5 , it may similarly be applied to other cameras and to other types of electronic imaging systems, including systems that use other algorithms for image restoration, as are known in the art. 
     Turning now to the details of the method of  FIG. 7 , as a preliminary stage, before generating the optical design, aberration scores may be defined for use in the optical design, at  100 . This stage is not mandatory, but it is useful in achieving faster convergence of the iterative design method described below. A merit function is assigned to each significant aberration that may characterize optics  46 . For this purpose, the inventors have found it most convenient to express the aberrations in terms of the Zernike polynomials, as are known in the art, for each of the colors red, green and blue individually. The Zernike polynomials are described, for example, by Born &amp; Wolf in  Principles of Optics,  4 th  edition (Pergamon Press, 1970), in section 9.2, pages 464-467, which is incorporated herein by reference. Standard software packages for optical design, such as ZEMAX® (produced by ZEMAX Development Corp., San Diego, Calif.), are capable of computing the Zernike polynomial coefficients for substantially any design that they generate. Values of the merit functions may be provided at  100  in tabular form. Generation of these values is described below with reference to  FIG. 8 . 
     An initial optical design is produced, at  102 . The design is typically generated using optical design software running on a suitable computer workstation. The software may be downloaded to the workstation in electronic form, or it may alternatively be provided on tangible media, such as CD-ROM or DVD. The optical design software may comprise an off-shelf package, such as the above-mentioned ZEMAX software, with additional program components for carrying out the optimization functions described herein. The software accepts as inputs the system requirements that are imposed on optics  46 , such as the number of optical elements, materials, tolerances, focal length, magnification, aperture (F-number) and resolution. As noted above, the optical resolution, expressed in terms of the PSF of the optics, typically gives a focal spot size that is substantially greater than the pitch of array  22 . The overall system resolution is then enhanced by deblurring. 
     The optical design software may use the aberration scores provided at  100  in determining a design score, which is a total of the merit scores taken over all the aberrations of the design. The design score indicates how to trade off one aberration against another in order to generate an initial design that maximizes the total of the merit scores subject to the optical requirements mentioned above. As a result of using the merit scores in this manner, the optical design generated at  102  may not be the one that gives the minimum total blur in the image formed on array  22 . The optical image will, however, be characterized by a combination of aberrations that can typically be deblurred by processor  30  to give a better-quality electronic output image. 
     The optical design software is also used to generate a design change response matrix, at  104 . This matrix reflects the impact that small changes in the optical design parameters will have on the different aberrations of the optical system. The optical design parameters that may be changed include, for example, curvatures of different lens surfaces, as well as thickness and spacing of the lenses. The matrix is a sort of multi-dimensional gradient, having entries (i,j) that represent the ratio R i   j =Δ(aberration[j])/Δ(optical_parameter[i]). 
     The optical design is also used in computing the PSF of the optics and an associated image quality score S, at  106 . The total aberration—and hence the PSF—at any point in the image plane may be obtained from the total wave front aberration, which is calculated simply by summing the values of the Zernike polynomials. As noted above, the Zernike polynomial coefficients are generally available as output from standard optical design programs, such as the above-mentioned ZEMAX. The image quality score computed at  106  typically combines the effects of the PSF on image resolution and on artifacts in the image, and reflects the ability of processor  30  to compensate for these effects using the chosen image restoration (deblurring) algorithm. The PSF typically varies with wavelength and is not uniform over the entire image plane. Therefore, the resolution and artifact scores may be computed at multiple field points in the image plane and then combined, typically by weighted averaging. An exemplary method for calculating the image quality score S is described below with reference to  FIG. 8 . 
     The effects of small changes in the aberrations on the image quality score is determined at  108 . At this stage, small changes are made in each of the aberrations, and the resulting change in the image quality score is computed. This operation gives a score gradient A j =ΔS/Δ(abarration[j]). 
     The results of  104  and  108  are then combined to determine an optimal change that should be applied to one or more of the optical design parameters in order to increase S, at  110 . Using the response matrix R and the score gradient vector  A , whose elements R j   i  and A j  are defined above, the optical design change vector Δ O  is given by:
 
Δ O =( R   t   R ) −1   ·R   t     A     (6)
 
wherein R t  is the Hermitian transpose of R. Each of the elements of Δ O  indicates the changes that should be made to one of the variable design parameters of the current optical design, as generated at  102 .
 
     The magnitude of the design changes indicated by Δ O  is compared to a predetermined tolerance threshold, at  112 . The tolerance threshold may be set arbitrarily, but typically it is determined by the manufacturing tolerances that have been specified for camera  44  and optics  46 . For example, if the spacing between two lenses in the optical system is specified to be accurate to within 0.2 mm, then there is little point in implementing a change in the design spacing of 0.1 mm that may be indicated by Δ O . If the elements of Δ O  are all less than the applicable tolerance thresholds, the design cycle is done, and the design may be output for production, at  114 . 
     Alternatively, as long as one or more elements of Δ O  are found to exceed the tolerance thresholds at  112 , the design process returns to  102 , in order to generate a revised optical design implementing the design change Δ O  computed at  110 . The operations of  104 - 110  are then repeated, as described above. This iterative process may continue until all elements of Δ O  are found to be less than the applicable thresholds at  112 , or until a certain maximum number of iterations have taken place, or until some other criterion or combination of criteria have been satisfied. 
       FIG. 8  is a flow chart that schematically shows details of a method of scoring that is performed at  106 , in accordance with an embodiment of the present invention. As noted above, the image quality score is determined by calculating and combining a resolution score and an artifact score. Individual scores may be computed at multiple points in the image field and at multiple different wavelengths. In one embodiment, the scores are computed for nine points along the positive X-axis, at each of three different wavelengths (red, green and blue). Because the optics are cylindrically symmetrical, the optical characteristics on the positive X-axis are sufficient to characterize the entire field. 
     To find the resolution score, individual X- and Y-direction resolution scores, S x   Res  and S y   Res , are determined for each wavelength at each selected field point, at  120 . Since the field points are taken along the X-axis, S x   Res  represents the resolution in the tangential plane, while S y   Res  represents the resolution in the sagittal plane. In an exemplary embodiment, the scores are calculated at each field point and wavelength as follows:
     1. The PSF is calculated based on the aberrations, typically using the Zernike polynomial representation as described above.   2. The PSF is scaled to the pitch of sensor array  22 .   3. The PSF is applied to generate a representative image. Typically, this representative image is created by blurring an ideal image that is zero everywhere except for the pixel at the origin (the current field point), which has the value 1. In other words, I(x, y)=δ x,y , wherein I(x, y) is the image pixel array, x, y are the integer pixel coordinates, and δ x,y  is the Kronecker delta function. Alternatively, other types of representative images may be used, such as line or grid images.   4. The representative image is deblurred using a deconvolution filter DCF(h), as described above, in order to generate a restored image r(x,y). Here h is the PSF that is used in generating the DCF. Various deconvolution filters may be chosen and used at this stage. For example, h may be taken as the local PSF calculated for this particular field point and wavelength, and the DCF may be computed based on this local value, using the formula given by equation (1) above. Alternatively, a single, optimized DCF may be used for all field points and wavelengths. This same optimized DCF may be applied subsequently by processor  30  in order to deblur images produced by camera  44 . An exemplary method for calculating the optimized DCF is described below.   5. The restored image r(x,y) is transformed to the spatial frequency domain by applying a two-dimensional Fourier transform, to give the transformed image {tilde over (R)}(p,q), wherein (p,q) are the spatial frequency coordinates in the X- and Y-directions, respectively.   6. The score in the X-direction is determined by the slope of a step function in the X-direction that has been blurred and restored, as given by the transformed image:   

               s   s   Res     =           ∑     p   =   0       p   =     p     M   ⁢           ⁢   ax           ⁢            R   ~     ⁡     (     p   ,   0     )                  p     M   ⁢           ⁢   ax       +   1       .           
Here pMax is the Nyquist frequency in the X-direction.
     7. The score in the Y-direction is similarly determined by the slope of a step function in the Y-direction that has been blurred and restored:   

               S   y   Res     =           ∑     q   =   0       q   =     q     M   ⁢           ⁢   ax           ⁢            R   ~     ⁡     (     0   ,   q     )                  q     M   ⁢           ⁢   ax       +   1       .           
In this case, qMax is the Nyquist frequency in the Y-direction.
 
     The individual X- and Y-resolution scores at all the selected field points are combined to give a total resolution score, at  124 . Typically, the average is weighted in such a way as to give greater emphasis to the bad (low) individual resolution scores. For example, the total resolution score may be a product of exponents of the inverses of individual resolution scores (see equation (7) below). Alternative mechanisms for weighted averaging will be apparent to those skilled in the art. As a further alternative, the total score may simply be set equal to the worst of the individual scores. 
     Individual artifact scores, S x   Art  and S y   Art , are similarly computed for each field point and wavelength, at  122 . The individual scores may be calculated as follows:
     1. The extent in the X-direction of the artifact created by the DCF is determined by applying a one-dimensional Fourier transform to {tilde over (R)}(p,0), to give the spatial domain function   

                 R   x     1   ⁢   D       ⁡     (   x   )       .         
This function is typically characterized by a central peak, surrounded by side lobes. The extent of the artifact in the X-direction, W x , is given by the distance of the outer boundary of the highest-order side lobe from the origin:
 
                 W   x     =     Max   ⁢     {     x   |              R   x     1   ⁢   D       ⁡     (   x   )            &gt;     ɛ   Art         }         ,         
wherein ε Art  is some threshold value, typically on the order of 1%.
     2. The amplitude of the artifact created by the DCF is determined by finding the inner boundary of the highest-order side lobe:   

                 Z   x   Art     =     Min   ⁢     {     x   |              R   x     1   ⁢   D       ⁡     (   x   )            &lt;   0       }         ,         
and then finding the maximal amplitude outside this boundary:
 
     
       
         
           
             
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                        
                       
                         
                           R 
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                           x 
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                       x 
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                         Z 
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                         Art 
                       
                     
                   
                   } 
                 
                 . 
               
             
           
         
       
         
         3. The artifact score in the X-direction is given by the product: S x   Art =W x ·A x .
 
The Y-direction artifact score is computed in like manner.
 
       
    
     The total artifact score is computed by combining the individual scores, typically as a weighted average, at  126 . In this case, the weighting typically emphasizes the higher individual scores, which point to more severe artifacts. Alternatively, as at  124 , the total artifact score may be set equal to the worst individual score. The total image quality score S is then determined as an average of the resolution and artifact scores, at  128 . For example, equation (7) below combines the operations of  124 ,  126  and  128 : 
                   S   =     exp   (       ∑     ω   ,   f       ⁢     (       S   x   Art     +     S   Y   Art     +     λ   /     S   x   Res       +     λ   /     S   y   Res         )       )             (   7   )               
Here the summation is taken over all field points f and wavelengths ω. λ is an empirically-derived constant, which may be chosen so as to give greater emphasis to either improving resolution or reducing artifacts in the output image, depending on system requirements.
 
     As noted above, any suitable DCF may be used in computing the image quality scores and gradients. Although a different DCF may be used for each different field point and wavelength, as described above, it is simpler to implement processor  30  using a single, uniform DCF over the entire field, or at least a DCF with relatively few variations over the field. Therefore, in optimizing the optical design, in accordance with the method of  FIGS. 7 and 8 , it is desirable to use a uniform PSF, as well, so that the optical design will reflect the actual image enhancement that the deblurring algorithm will provide. 
     In one embodiment, a weighted PSF, referred to herein as h Opt , is used as the uniform PSF for the purposes of the optical design process. The weighted PSF provides an optimized estimate of the PSF in the sense that the corresponding DCF has the greatest probability of providing optimal deblurring in the greatest number of units of camera  44  that are actually produced. h Opt  is calculated for this purpose by weighted averaging of individual PSFs over the image field. The weighted PSF may also take into account variations that may occur in the individual PSF values due to manufacturing tolerances in optics  46 , in sensor array  22 , and possibly in other elements of camera  44 . For example, hop may include contributions from several different individual PSFs calculated for the same field point and wavelength, using different optical parameters as defined by the bounds of the system tolerances. Each individual PSF, h i , is multiplied in the averaging process by a weight W i , such that the weights sum to one. The weights may be chosen so as to give greater importance to the field points that are closer to the center of the field and/or to favor one wavelength (typically green) over the others. When multiple PSFs are calculated based on different optical parameters, the weights may also reflect the probability of manufacturing variances within the tolerance range. 
     Under these conditions, it can be shown that the optimal DCF, which minimizes the probable deviation of the deblurred image relative to an ideal image, is given (in the frequency domain) by: 
                       DCF   opt     ⁡     (     p   ,   q     )       =         ∑   i     ⁢       W   i     ⁢     h   i   *             ∑   j     ⁢       W   j     ⁢     h   j   *     ⁢     h   j                   (   8   )               
Comparing this result to the naïve definition DCF=1/h gives the result:
 
                       h   opt     ⁡     (     p   ,   q     )       =         ∑   j     ⁢       W   j     ⁢     h   j     ⁢     h   j   *             ∑   i     ⁢     W   ⁢           ⁢     h   i   *                   (   9   )               
This “optimized” PSF may be used as h in the methods of  FIGS. 7 and 8 , and the corresponding DCF opt  may be used by processor  30 .
 
     Returning now to  100  in  FIG. 7 , it was noted earlier that a merit function may be used for scoring the aberrations of the initial optical design. The merit function values are typically supplied to the optical design software in tabular form, and depend on the Zernike polynomial coefficients. To generate the merit function values, each aberration, as defined by the corresponding Zernike polynomial, is converted into a PSF. The form of this PSF reflects the blur that is generated in the optical image due to the particular aberration that the PSF represents. The scores may be computed using the method of  FIG. 8 . Typical X- and Y-resolution scores computed for Zernike aberration  4  (defocus) and aberration  8  (coma) are shown below in Table I: 
                     TABLE I                  ABERRATION MERIT SCORES                             X Resolution   Y Resolution                                     Zernike Aberration 4 Coefficient               0.50   1.00   1.00       0.60   1.00   1.00       0.70   0.72   0.72       0.80   0.52   0.52       0.90   0.45   0.45       1.00   0.43   0.43       1.10   0.40   0.40       1.20   0.34   0.34       1.30   0.28   0.28       1.40   0.30   0.30       1.50   0.38   0.38       1.60   0.44   0.44       1.70   0.48   0.48       1.80   0.48   0.48       1.90   0.46   0.46       2.00   0.42   0.42       2.10   0.36   0.36       2.20   0.32   0.32       2.30   0.30   0.30       2.40   0.28   0.28       2.50   0.28   0.28       2.60   0.28   0.28       2.70   0.27   0.27       2.80   0.25   0.25       2.90   0.24   0.24       3.00   0.25   0.25       Zernike Aberration 8 Coefficient       0.50   1.00   1.00       0.60   0.96   0.99       0.70   0.82   0.97       0.80   0.77   0.96       0.90   0.76   0.92       1.00   0.76   0.88       1.10   0.71   0.83       1.20   0.64   0.80       1.30   0.55   0.78       1.40   0.50   0.79       1.50   0.49   0.77       1.60   0.50   0.73       1.70   0.50   0.69       1.80   0.50   0.66       1.90   0.48   0.65       2.00   0.44   0.64       2.10   0.40   0.64       2.20   0.39   0.63       2.30   0.40   0.62       2.40   0.39   0.58       2.50   0.40   0.57       2.60   0.38   0.55       2.70   0.36   0.55       2.80   0.33   0.54       2.90   0.33   0.54       3.00   0.34   0.54                    
Computation of the merit scores to be applied to the remaining Zernike aberrations will be apparent to those skilled in the art upon studying the foregoing description and examples.
 
     Design Examples 
       FIG. 9  is a schematic side view of optics  46 , designed in accordance with an embodiment of the present invention, using the methods described above. In this embodiment, optics  46  comprise two lenses  130  and  132 , both made from polymethyl methacrylate (PMMA), aligned on an optical axis  134 . The lenses are both cylindrically symmetrical, and are designed for fixed focus with an object plane at infinity, to the left of lens  130 . Lens  130  is spherical, with a diameter of 2.5 mm, center thickness 1.0 mm, and front and rear radii of curvature of 2.12 and 6.71 mm, respectively. Lens  132  is spaced 16 mm behind lens  130  on the optical axis, and is 4.5 mm in diameter and 1.2 mm thick. Filter  48  comprises BK7 glass 0.4 mm thick, and is positioned 0.85 mm behind lens  132  and 0.782 mm in front of the plane of sensor array  22 . 
     Both the front and rear surfaces of lens  132  are aspherical, with even aspheric profiles determined using the optimization method of  FIG. 7 . To determine the aspheric profiles, the coefficients of the functions that define the lens surface profiles are included as parameters in the response matrix. In other words, gradient values R j   i , as defined above, are determined for each aberration in terms of each of the aspheric lens coefficients. Using this method, the following coefficients are determined for the surfaces of lens  132 : 
                     TABLE II                  ASPHERIC LENS COEFFICIENTS                                 Polynomial order   Front surface   Back surface                                             r 2     0   0           r 4     −0.54615   −0.0034125           r 6     0.0205194   −0.0093650           r 8     −0.015842   0.00031717                        
Other types of aspheric profiles may be defined and optimized in like manner.
 
     The inventors have found that the optical design of  FIG. 9 , used in conjunction with the deblurring function described above, gives an output image from camera  44  with modulation transfer function (MTF) that is as good as or better than the image produced by a conventional three-lens objective without deblurring. A sensor array with 1.3 Mpixels was used in both cases. The two-lens design shown in  FIG. 9  is less costly to produce and narrower (shorter front-to-back distance) than the three-lens design would be. 
     Reference is now made to FIGS.  10 A/B,  11 A/B and  12 A/B, which are plots that exemplify the improvement in PSF that is obtained using the optical design and image processing techniques described above, in accordance with an embodiment of the present invention.  FIGS. 10A and 10B  schematically show the PSF of the optics of  FIG. 9  at a point X=2 mm, Y=0 mm (i.e., displaced by 2 mm from the optical axis) in the image plane of the optics, for light of wavelength 587.6 nm. The horizontal axis is in units of pixels (i.e., pitch) of the sensor array. The PSF is computed as the sum of all the significant aberrations of the optics, wherein the aberrations are typically expressed in terms of the Zernike polynomials. The extent of the PSF on both axes is approximately 3 pixels. 
       FIGS. 11A and 11B  show the elements along the X- and Y-axes of the convolution kernel of the DCF, which is computed so as to correct for the blur created by the PSF of  FIGS. 10A and 10B . 
       FIGS. 12A and 12B  show the enhancement of the PSF that is obtained when the DCF of  FIGS. 11A and 11B  is applied to images formed by the optics of  FIG. 9 . The extent of the PSF is now reduced to about 1 pixel in each of the X- and Y-directions. 
     Alternatively, the deblurring capabilities of processor  30  in camera  44  may be used to gain various other design tradeoffs, for example:
         The F-number of optics  46  may be reduced, giving a larger aperture and therefore better light sensitivity. Increasing F-number tends to decrease the camera&#39;s depth of field and thus to broaden the PSF, but these effects may be offset by broadening and adjusting the values of the convolution kernel of the deblurring function of processor  30 .   Resolution of the output image generated by camera  44  may be enhanced by processor  30 , relative to the resolution of a conventional camera with a comparable sensor array and objective optics.       

     Additionally or alternatively, the capabilities of camera  44  (or camera  20 ) may be enhanced by configuring processor  30  to adjust the DCF coefficients in response to changes in focal relationships, i.e., changes in the object distance relative to the image distance of the camera. For example:
         Camera  44  may be capable of switching to macro operation, for imaging objects very close (typically 5-30 cm) to the camera, without opto-mechanical adjustments. In conventional cameras, macro operation requires very fine mechanical adjustments in the positions of the objective lenses. In camera  30 , different object distances may be accommodated by means of changes in the DCF coefficients, which are calculated in advance to compensate for the variation of the PSF with object distance. The DCF coefficients may be stored by processor  30  in a look-up table, and macro operation may be invoked by a software switch.   The field of view of optics  46  may, be increased, for enhanced wide-angle operation. The DCF applied by processor  30  may be adjusted in particular to compensate for degradation of the PSF at the edges of the field.
 
An electronic zoom function may also be implemented, particularly when a lensless camera is used, as illustrated generally by  FIG. 2 . Other applications will be apparent to those skilled in the art and are considered to be within the scope of the present invention.
       

     Although the embodiments described above refer to certain particular types and designs of digital cameras, the principles of the present invention may similarly be applied to substantially any type of optical electronic imaging device. It will thus be appreciated that the embodiments described above are cited by way of example, and that the present invention is not limited to what has been particularly shown and described hereinabove. Rather, the scope of the present invention includes both combinations and subcombinations of the various features described hereinabove, as well as variations and modifications thereof which would occur to persons skilled in the art upon reading the foregoing description and which are not disclosed in the prior art.