Abstract:
A method of optical data processing, comprising:
       providing a first data set to be optically transformed using a transform;   combining a reference data set with said first data set to generate coherent light, encoding a combined data set;   optically and coherently transforming said light that encodes the combined data set, into coherent light that encodes a transformed combined data set;   obtaining a transformed reference data set by determining the effect said optical transform has on light encoding said reference data set; and   extracting a second data set that represents a transform of said first data set, from an intensity portion of light encoding said transformed combined data set, using said transformed reference data set to extract a phase of at least one element of said second data set.

Description:
RELATED APPLICATIONS 
     This application is a continuation of U.S. application Ser. No. 09/979,182 filed Feb. 25, 2002, now abandoned which is a U.S. national filing of PCT Application No. PCT/IL00/00284, filed May 19, 2000. This application is also a continuation in part of PCT application No. PCT/IL99/00479, filed Sep. 5, 1999, now U.S. application Ser. No. 09/926,547, filed on Mar. 5, 2002 now U.S. Pat. No. 7,194,139, the disclosures of which are incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     The invention relates to optical methods and apparatus for performing computations and in particular to transforming a first data set into a second data set by a linear transformation and determining the phase of data elements in the second data set. 
     BACKGROUND OF THE INVENTION 
     Optical data processing can often be used to process data more rapidly and efficiently than conventional computational methods. In particular, optical methods can be used to perform linear transformations of data sets rapidly and efficiently. 
     For example, it is well known that converging lenses can be used to substantially “instantaneously” transform a first image into a second image that is a Fourier transform of the first image. It is to be noted that the Fourier transform is a relationship between the complex amplitudes of light in the images and not between the intensities of light in the images. The same is generally true with respect to other transformations of images, the transformation is a transformation of complex amplitudes of light and not intensities of light. It is therefore to be understood that when a second image is said to be a Fourier, or other, transform of a first image, what is meant is that the spatial pattern of the complex amplitude of light in the second image is the Fourier, or other, transform of the spatial pattern of the complex amplitude of light in the first image. 
     If the first image is coded with data, the second image is coded with data that is the Fourier transform of the data in the first image. A suitable optical processor can therefore provide substantial advantages in comparison to a conventional data processor when a spectral analysis of a data set is desired. However, a Fourier transform of a data set in general involves complex numbers, even if the data set comprises only real numbers. Therefore, in order to properly detect an “optical” Fourier transform of a data set, phase as well as intensity of light of an image representing the Fourier transform must be detected. While this can be accomplished, most light detectors are generally sensitive only to light intensity and are not responsive to phase. 
     It is therefore generally more convenient to determine values for data represented by an image from only the intensity of light in the image. Consequently, it is usually advantageous to process data optically using methods that generate only real numbers from the data. 
     For example, it is often preferable to optically process data coded in an image in accordance with a cosine transform to perform a spectral analysis of the data rather than a Fourier transform. The cosine transform of a real data set generates real values. However, whereas a cosine transform of a real data set does not generate complex numbers it does, usually, generate both positive and negative numbers. Therefore, while most of the information in an optical cosine transform of an image can be acquired from measurements of intensity of light in the image, sign information is not preserved in the intensity measurements. As a result, an optical processor that transforms an input image into an output image that represents the cosine transform of the input image requires a means for determining which of the numbers represented by the output image are positive and which are negative. 
     K. W. Wong et al, in an article entitled “Optical cosine transform using microlens array and phase-conjugate mirror ”, Jpn J. Appl. Phys. vol. 31, 1672-1676, the disclosure of which is incorporated herein by reference, describes a method of distinguishing positive and negative data in a cosine transform of an image. 
     The problem of distinguishing the sign of numbers represented by an image when only the intensity of light in the image is measured is of course not limited to the case of data optically generated by a cosine transform. The problem affects all real linear transforms, such as for example the sine and discrete sine transforms and the Hartley transform, when the transforms are executed optically and only their intensities are sensed, if they generate both positive and negative values from a real data set. 
     SUMMARY OF THE INVENTION 
     An aspect of some embodiments of the present invention relates to providing a method for determining the sign of data encoded in an output image of a linear optical processor using measurements of intensity of light in the output image, hereinafter referred to as a “data output image”. The data output image is assumed to be generated by the processor responsive to an input image, a “data input image”, encoded with input data that is real. The input data is either all positive or all negative. For clarity of presentation it is assumed that the input data is all positive. 
     According to an aspect of some embodiments of the present invention, a reference input image is defined for the optical processor. Magnitude and phase of amplitude of a “reference” output image generated by the processor responsive to the input reference image are used to determine the sign of data represented by the data output image. 
     The operation of a linear optical processor may be described by the equation F(u,v)=O(u,v:x,y)f(x,y). In the equation f(x,y) is a complex amplitude of light in an input image, i.e. a data input image, that represents input data, which data input image is located on an input plane of the processor, and x and y are coordinates of the input plane. Similarly, F(u,v) is a complex amplitude of light in a data output image that the processor generates responsive to f(x,y). The data output image is located on an output plane of the processor having position coordinates u and v corresponding respectively to position coordinates x and y of the input plane. Intensity of light in the data input image is equal to |f(x,y)| 2  and intensity of light in the data output image is equal to |F(u,v)| 2 . 
     O(u,v:x,y) represents any continuous or discrete linear operator that transforms a first real data set into a second real data set. O(u,v:x,y) may for example represent the continuous or discrete sine or cosine transform or the Hartley transform. For continuous linear transformations u, v, x and y are continuous and multiplication in the equation representing operation of the processor represents integration over the x, y coordinates. For discrete linear operators u, v, x, and y are discrete coordinates and multiplication represents summation over the x, y coordinates. 
     Since, in accordance with embodiments of the present invention, the input data is assumed to be real and positive, the phase of f(x,y) is constant and input data is represented by the magnitude of f(x,y). F(u,v) also represents a real data set. However F(u,v) may have both positive and negative data. Data having positive values are represented by values of F(u,v) having a same first phase. Data having negative values are represented by values of F(u,v) having a same second phase 180° different from the first phase. 
     Let the reference input image and its corresponding reference output image be represented by r(x,y) and R(u,v). Both r(x,y) R(u,v), and intensity of light in the reference output image |R(u,v)| 2  are known. It is to be noted that it is possible to define and synthesize any predefined reference function r(x,y) and use it for sign reconstruction in accordance with embodiments of the present invention. Whereas descriptions of the present invention assume that r(x,y) is real the invention is not limited to the reference image being real. Magnitude and phase of R(u,v) are known from the transform that the optical processor executes and can be checked experimentally using methods known in the art. Preferably, r(x,y) is real. Therefore R(u,v) preferably corresponds to a real data set. In some embodiments of the present invention R(u,v) is a real data set comprising values all of which have a same sign. In some embodiments of the present invention the data set comprises one of or a combination of positive, negative and complex values. 
     In accordance with an embodiment of the present invention, to determine both the magnitude and sign of F(u,v) the intensity of the data output image |F(u,v| 2  is measured. In addition, in accordance with an embodiment of the present invention, a combined input image c(x,y)=f(x,y)+r(x,y) are processed by the processor to provide a combined output image C(u,v)=F(u,v)+R(u,v). Intensity of light in the combined output image, which is equal to |C(u,v)| 2 =|F(u,v)| 2 +|R(u,v)| 2 +2F(u,v)R(u,v), is measured. Since |F(u,v)| 2 , |R(u,v)| 2  and R(u,v) are known, the sign of F(u,v) can be determined from the “interference” term 2F(u,v)R(u,v). 
     It is to be noted that not only sign of F(u,v) can be determined from |C(u,v)| 2 , |F(u,v)| 2 , |R(u,v)| 2  and R(u,v). In general, (|C(u,v)| 2 −|F(u,v)| 2 −|R(u,v)| 2 )/2R(u,v) provides a magnitude and a phase for F(u,v). In some cases the phase is known to within an ambiguity, for example, a symmetry ambiguity or a 180°. In some embodiments of the invention the ambiguity is removed and the phase extracted by determining a combined image C(u,v) for two or more different reference images r(x,y). The phase can be extracted for example by solving for F(u,v) using the two combined and reference images. 
     In some embodiments of the present invention the reference image is chosen so that |R(u,v)|≧|F(u,v)| for all values of u and v for which R(u,v) and F(u,v) have opposite signs. For these embodiments of the present invention only the combined input image c(x,y)=f(x,y)+r(x,y) is processed by the processor to determine both the magnitude and sign of F(u,v). If the intensity of light in the combined image minus the intensity light in the reference image at a point (u,v) in the output plane of the processor is greater than zero, the signs F(u,v) and R(u,v) are the same at the point. If on the other hand the difference is less than zero, the signs of F(u,v) and R(u,v) are opposite. Since the sign of R(u,v) is known, the sign of F(u,v) is known. The magnitude of F(u,v) at the point can be determined from the intensity |C(u,v)| 2  and the known magnitude and sign of R(u,v) by solving a quadratic equation. 
     An aspect of some embodiments of the present invention relates to providing an improved method for generating a cosine transform of an “input” image using an optical processor that generates a Fourier transformed output image from an input image. 
     In accordance with an embodiment of the present invention, a first Fourier image that is a Fourier transform of the input image is generated by the optical processor and the intensity of the Fourier image measured and stored. A second Fourier image is generated by the optical processor from the input image plus a known first reference image and the intensity of the second Fourier image is measured and stored. The input image is parity transformed to generate a second input image, referred to as a “parity image”. A third Fourier image, which is a Fourier transform of the parity image is generated and its intensity measured and stored. A fourth Fourier image is generated which is a Fourier transform of the parity image plus a known second reference image. The intensities of the four Fourier images and the amplitudes of the known reference images are used to determine the cosine transform of the input image. In some embodiments of the present invention the first and second reference images are the same. 
     There is thus provided in accordance with an exemplary embodiment of the invention, a method of optical data processing, comprising:
         providing a first data set to be optically transformed using a transform;   combining a reference data set with said first data set to generate a combined data set;   optically transforming said combined data set into a transformed combined data set; and   extracting a second data set that represents a transform of said first data set, from an amplitude portion of said transformed combined data set, using said reference image to extract a phase of at least one element of said second data set. Optionally, said transformed combined data set is detected using a power detector. Alternatively or additionally, said transformed combined data set is encoded using incoherent light.       

     In an exemplary embodiment of the invention, said transformed combined data set is a discrete data set. Alternatively or additionally, said first data set comprises a one-dimensional data set. Alternatively, said first data set comprises a two-dimensional data set. Optionally, said first data set comprises an image. 
     In an exemplary embodiment of the invention, said first data set comprises at least one positive value. Alternatively or additionally, said first data set comprises at least one negative value. Alternatively or additionally, said first data set comprises at least one complex value. 
     In an exemplary embodiment of the invention, extracting comprises extracting using electronic processing. 
     In an exemplary embodiment of the invention, combining a reference data set comprises adding at least one additional value to an existing element of said first data set. Alternatively or additionally, combining a reference data set comprises replacing at least one existing element of said first data set with an element from a second data set. Optionally, the method comprises compensating for an effect of said replaced value after said extraction. Optionally, said compensating comprises compensating using electronic processing. 
     In an exemplary embodiment of the invention, combining a reference data set comprises adding at least one additional value alongside existing elements of said first data set. Optionally, said at least one additional value is arranged at a corner of a matrix layout of said first data set. 
     In an exemplary embodiment of the invention, the method comprises selecting said reference image to create a desired offset in said transformed combined data set. Optionally, said selecting takes into account system imperfections. Alternatively or additionally, said offset is substantially uniform. Alternatively, said offset is substantially non-uniform. 
     In an exemplary embodiment of the invention, said reference data is at least one delta-function. Optionally, said reference data comprises a plurality of delta-functions. Alternatively or additionally, said at least one delta function has an amplitude substantially greater than that of any of the data elements of said first data set. 
     In an exemplary embodiment of the invention, said at least one delta function has an amplitude substantially greater than that of any of the data elements of said first data set that have a certain phase. 
     In an exemplary embodiment of the invention, said at least one delta function has an amplitude substantially greater than an amplitude of a component of any of the data elements of said first data set that fit in a certain phase range. 
     In an exemplary embodiment of the invention, said at least one delta function has an amplitude not greater than that of any of the data elements of said first data set. 
     Optionally, said amplitudes are measured as amplitudes of transform elements. 
     In an exemplary embodiment of the invention, combining comprises combining electronically and generating a combined modulated light beam. 
     Alternatively, combining comprises combining optically. Optionally, combining comprises creating said reference image optically. Optionally, said reference image is created using a refractive optical element. Alternatively, said reference image is created using a dedicated light source. 
     In an exemplary embodiment of the invention, said transform is a Fourier-derived transform. 
     In an exemplary embodiment of the invention, said transform is a DCT transform. 
     In an exemplary embodiment of the invention, extracting a phase comprises extracting only a sign. 
    
    
     
       BRIEF DESCRIPTION OF FIGURES 
       A description of exemplary embodiments of the present invention follows. In the figures, identical structures, elements or parts that appear in more than one figure are generally labeled with the same numeral in all the figures in which they appear. Dimensions of components and features shown in the figures are chosen for convenience and clarity of presentation and are not necessarily shown to scale. The figures are listed below. 
         FIG. 1  schematically shows an optical processor generating a Fourier transform of an image according to prior art; 
         FIG. 2  schematically shows the optical processor shown in  FIG. 1  generating a cosine transform of an image in accordance with prior art; 
         FIGS. 3A and 3B  schematically show an optical processor generating a cosine transform of an image in accordance with an embodiment of the present invention; 
         FIG. 4A  schematically shows an optical processor that generates a reference image that is a delta function, in accordance with an embodiment of the present invention; 
         FIG. 4B  schematically shows a lens system for generating a delta function reference image, in accordance with an embodiment of the present invention; and 
         FIGS. 5A-5D  schematically illustrate a method of generating a cosine transform of an image, in accordance with an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     In the following discussion an embodiment of the present invention is described in which a real linear transform performed by an optical processor is a cosine transform. The optical processor uses the Fourier transform properties of converging lenses whereby a converging lens transforms an image into its Fourier transform, to generate a cosine transform of an image. The Fourier transform properties of lenses are described in “Introduction to Fourier Optics” by J. W. Goodman, McGraw Hill-Hill Companies, second edition Copyright 1996. 
       FIG. 1  schematically shows an optical processor  20  that functions to transform images into their Fourier transforms according to prior art. Optical processor  20  comprises a converging lens  22 , an input plane  24  and an output plane  26 . Input and output planes  24  and  26  are coincident with focal planes of lens  22 . It is well known that lens  22  can be used to generate an image on output plane  26  that is a Fourier transform of an image on input plane  24 . 
     For example, assume that a spatial light modulator  30  having pixels  32  is located at input plane  24  and that the spatial light modulator is illuminated with collimated coherent light, represented by wavy arrows  34 , from a suitable light source. Pixels  32  have transmittances as a function of position that are proportional to a desired function. Spatial light modulator  30  may, for example, be a photographic transparency, a printed half tone image, a liquid crystal array or a multiple quantum well (MQW) modulator. In  FIG. 1 , by way of example, the transmittances are determined so that when spatial light modulator  30  is illuminated by light  34  a happy face  36  is formed at input plane  24 . Lens  22  will form an image (not shown) on output plane  26  that is the Fourier transform of the happy face  36  on input plane  24 . 
     Given a function f(x,y), the Fourier transform of the function (1/4)[f(x,y)+f(−x,y)+f(x,−y)+f(−x,−y)] is the cosine transform of f(x,y). Each ofthe functions in the square brackets is a parity transform, or a one dimensional reflection in the x or y axis, of the other functions in the brackets. It is therefore seen that the cosine transform of a two dimensional function can be generated by Fourier transforming all possible parity transforms of the function. 
       FIG. 2  illustrates how optical processor  20  shown in  FIG. 1  can be used to generate a cosine transform of an image  40  in accordance with prior art by Fourier transforming all of the image&#39;s parity transforms. Image  40  may, by way of example, be an 8 by 8 block of pixels from an image that is to be compressed according to the JPEG standard using a discrete cosine transform. 
     Let positions on input plane  24  and spatial light modulator  30  be defined by coordinates along x and y axes indicated on the spatial light modulator and positions on output plane  26  by coordinates along u and v axes indicated on the output plane. Let respective origins  25  and  27  of the x, y coordinates and the u, v coordinates be the intersections of the optic axis (not shown) of lens  22  with input and output planes  24  and  26  respectively. 
     Image  40  is formed on the upper right quadrant of spatial light modulator  32  and reflections  42  and  44  of image  40  in the x and y axes are respectively formed in the lower right and upper left quadrants of the spatial light modulator. A reflection  46  of image  40  along a 45° diagonal (not shown) to the x axis through the origin is formed in the lower left quadrant of spatial light modulator  30 . Let the amplitude of light in image  40  be represented by f(x,y). Let the amplitude of light in the image formed on input plane  24  comprising image  40  and its parity reflections be f′(x,y). Then f′(x,y)=(1/4)[f(x,y)+f(−x,y)+f(x,−y)+f(−x,−y)]. (The decrease in amplitude by 75%, i.e. the factor 1/4, which is not necessary, can of course be achieved by proper control of spatial light modulator  30 ). If the amplitude of light in an image formed on output plane  26  by lens  22  responsive to f′(x,y) is represented by F(u,v) then F(u,v) is the Fourier transform of f′(x,y). Because of the symmetry of the image on input plane  24 , F(u,v) is also the cosine transform of f(x,y). If F.T. represents the operation of the Fourier transform and C.T. represents the operation of the cosine transform then the relationships between F(u,v), f′(x,y) and f(x,y) is expressed by the equation F(u,v)=F.T. {f′(x,y)}=C.T.{f(x,y)}. 
     It is to be noted that f(x,y) and f′(x,y) represent data that is either all positive or all negative. For clarity of presentation data represented by f(x,y) is assumed to be positive. Further, since the cosine transform performed by optical processor  20  is a real linear transform, as noted above, F(u,v) also represents real data. However, F(u,v) may have both positive and negative data. Therefore, the cosine transform of image f(x,y) cannot be determined from the image on output plane  26  by measuring only the intensity |F(u,v)| 2 . 
       FIGS. 3A and 3B  schematically show an optical processor  50  being used to determine the sign and magnitude of the cosine transform F(u,v) of image  40 , i.e. f(x,y), in accordance with an embodiment of the present invention. 
     Optical processor  50  is similar to optical processor  20  and comprises a lens  22 , input and output planes  24  and  26 . At output plane  26 , processor  50  preferably comprises an array  52  of rows and columns of photosensors  54 . Each photosensor  54  generates a signal responsive to an intensity of light in an image on output plane  26  at a position determined by the row and column of array  52  in which the photosensor  54  is located and a pitch of array  52 . Photosensors  52  sample intensity of light at “discrete” positions (u,v) in output plane  26 . Preferably, the number of photosensors  52  is equal to the number of pixels  32  in spatial light modulator  30  and the locations of photosensors  52  are homologous with the locations of pixels  32 . 
     In  FIG. 3A , in accordance with an embodiment of the present invention, spatial light modulator  30  generates a first image at input plane  24  comprising image  40  and its parity reflections  42 ,  44  and  46 . The image is the same as the image comprising image  40  and its reflections shown in  FIG. 2 . Lens  22  forms an image at output plane  26  having amplitude F(u,v). Photosensors  54  generate signals responsive to intensity IF(u,v) of light in the image, where IF=|F(u,v)| 2 , at their respective locations u,v. 
     In  FIG. 3B , in accordance with an embodiment of the present invention, spatial light modulator  30  generates a second “combined” image at input plane  24  that comprises the image shown on the input plane in  FIG. 3A  with the addition of a reference image  60  having a known amplitude r(x,y). Preferably r(x,y) is chosen so that its Fourier transform is real, i.e. it has a symmetry with respect to the origin of axes x and y which results in its Fourier transform being real. By way of example, in  FIG. 3B , reference image  60  is formed by controlling central pixels  61 ,  62 ,  63  and  64  located at the origin of coordinates of input plane  24  to transmit light and appear bright. 
     If c(x,y)=(f′(x,y)+r(x,y)) then lens  22  forms an image (not shown) on output plane  26  that is the Fourier transform of c(x,y) and photosensors  54  generate signals responsive to intensity, IC(u,v), of light in the image. If C(u,v) represents the Fourier transform of c(x,y), then the amplitude of light in the image is C(u,v)and IC(u,v)=|C(u,v)| 2 . 
     In accordance with some embodiments of the present invention IF(u,v), IC(u,v) and the known Fourier transform of r(x,y) are used to determine the magnitude and sign of F(u,v) and thereby the cosine transform of f(x,y). 
     C(u,v)=F.T.{c(x,y)}=F.T.{f′(x,y)+r(x,y)}=F.T.{f′(x,y)}+F.T.{r(x,y)}=F(u,v)+R(u,v), where R(u,v) is the known and/or measured Fourier transform of r(x,y). Therefore, IC(u,v)=[|F(u,v)| 2 +|R(u,v)| 2 +2F(u,v)R(u,v)]=IF(u,v)+IR(u,v)+2F(u,v)R(u,v), where IR(u,v)=|R(u,v)| 2 . IR(u,v) can be calculated from the known Fourier transform of r(x,y) or measured experimentally. In some embodiments of the present invention the sign and magnitude of F(u,v) are determined from the equation F(u,v)=[IC(u,v)−IF(u,v)−IR(u,v)]/2R(u,v). 
     In some embodiments of the present invention the magnitude of F(u,v) is determined from the square root of IF(u,v). The sign of F(u,v) can be determined by comparing IF(u,v) and IR(u,v) with IC(u,v). If IF(u,v) &gt;IC(u,v) or IR(u,v) &gt;IC(u,v) then R(u,v) and F(u,v) have opposite sign. Otherwise they have the same sign. Since the sign of R(u,v) is known the sign of F(u,v) is known. 
     Whereas, in  FIGS. 3A and 3B  reference image  60  is a symmetric image located at the center of origin of the (x,y) coordinates other reference images are possible and can be used in the practice of the present invention. For example, pixels  32  at the corners of spatial light modulator  30  can be used to generate useful reference images. In some embodiments of the present invention pixels  32  only in certain regions of spatial light modulator  30  are used to represent data. Pixels that are not needed for data are used, in some embodiments of the present invention, to generate reference images. In some embodiments, some data pixels are canceled or provided elsewhere n the image, for example as pixels in overlapping blocks. In other examples extra pixels are provided for the reference image, for example by inserting one or more rows or columns per block. For example “data” pixels may be restricted to alternate rows or columns of pixels. Or each data pixel may be surrounded by four pixels that are not used for data. In an exemplary embodiment, 9×9 blocks of data are used for an 8×8 block transform, with at least some of the extra pixels being used as a reference image. Alternatively or additionally, the effect of missing pixels may be corrected using an electronic or optical post processing step. 
     It should also be noted that dark pixels, pixels that are “turned off”, that do not transmit light can function to generate reference images. For example, if an image on spatial light modulator  30  has bright pixels at the origin of coordinates (i.e. pixels  61 ,  62 ,  63  and  64  in  FIG. 3B ) a reference image can be generated by “turning off” the pixels. Turning off pixels in an image is of course equivalent to adding a reference image to the image for which light at the turned off pixels has a phase opposite to that of the light in the image. 
     In some embodiments of the present invention, reference image f(x,y) is chosen so that |R(u,v)|≧|F(u,v)| for all values of u and v for which R(u,v) and F(u,v) have opposite signs. For these embodiments of the present invention it is not necessary to determine IF(u,v) and only the operation shown in  FIG. 3B  in which IC(u,v) is measured is required to determine the magnitude and phase of F(u,v). If at a point (u,v), IC(u,v)−IR(u,v) &gt;0 then the signs F(u,v) and R(u,v) are the same at the point otherwise the signs are opposite. The magnitude of F(u,v) at the point can be determined from IC(u,v) by solving the quadratic equation IC(u,v)=[|F(u,v)| 2 +|R(u,v)| 2 +2F(u,v)R(u,v)] for F(u,v). 
       FIG. 4A  schematically shows a side view of an optical processor  70 , in accordance with an embodiment of the present invention, that generates a reference field for which |R(u,v)|&gt;|F(u,v)| for all values of u and v for which R(u,v) and F(u,v) have opposite signs. 
     Optical processor  70  comprises a “Fourier” lens  22  having an output plane  26  coincident with a focal plane of lens  22 , a spatial light modulator  72  and a “beam partitioner”  74 . A detector array  76  is located at output plane  26  and measures intensity of light at the output plane. Spatial light modulator  72  defines an input plane for Fourier lens  22  and may be located at substantially any position to the left of output plane  26 . In optical processor  70  spatial light modulator  72  is located by way of example adjacent to lens  22 . 
     Beam partitioner  74  preferably receives an incident beam  78  of coherent collimated light generated by an appropriate source (not shown) and focuses a portion of the light to a point  80  and transmits a portion of the light as a transmitted beam of light  82  parallel to the incident beam. Light from transmitted beam  82  illuminates and is transmitted through spatial light modulator  72  and is focused by lens  22  to form a Fourier transform F(u,v) of a transmittance pattern f(x,y) formed on the spatial light modulator. It is assumed that the transmittance pattern has an appropriate symmetry so that the Fourier transform is a cosine transform of a desired image. 
     Point  80  functions substantially as a point source of light and provides a reference image r(x,y) for f(x,y) that is substantially a delta function Aδ(x,y), where A is proportional to an intensity of light focused to point  80 . A Fourier image, R(u,v), of light from point  80  is also formed on output plane  26  by lens  22 . Since r(x,y) is substantially a delta function, R(u,v) is substantially constant and equal to A. 
     The magnitude of F(u,v) at a point (u,v) is of course proportional to the intensity of light in transmitted beam  82 . In accordance with an embodiment of the present invention beam partitioner  74  is designed so that the relative portions of light focused to point  80  and transmitted in transmitted beam  82  beam are such that A=|R(u,v| is greater than |F(u,v)| for all values of u and v for which R(u,v) and F(u,v) have opposite signs. 
     In some embodiments of the present invention beam partitioner  74  is a diffractive optical element such as a Fresnel zone plate having reduced efficiency. In some embodiments of the present invention, beam partitioner  74  comprises an optical system  90  of a type shown in a side view in  FIG. 4B . Optical system  90  comprises a positive lens  92  and a weak negative lens  94 . Positive lens  94  is preferably coated with an antireflective coating using methods known in the art to minimize reflections. Weak negative lens  92  is treated so that at its surfaces light is reflected with a reflectivity α. Light from incident beam  78 , represented by arrowed lines  96 , that is transmitted through both positive lens  92  and negative lens  94  without reflections is focused to produce the point reference light source A.delta(x,y) at point  80 . If the intensity of light in light beam  78  is “I” the amount of light focused to point  80  is substantially equal to I(1−.alpha.).sup.2. Light that undergoes internal reflection twice in negative lens  94  is transmitted as transmitted beam of light  82  substantially parallel to incident beam  78 . The amount of energy in transmitted beam  82  is substantially equal to I(1−.alpha.).sup.2.alpha.sup.2. The ratio of energy focused to point  80  to that contained in transmitted beam  78  is therefore equal to I/.alpha.sup.2. 
     In accordance with an embodiment of the present invention R can be chosen so that A =|R(u,v| is greater than |F(u,v)| for all values of u and v for which R(u,v) and F(u,v) have opposite signs. 
     Given a function f(x,y) it can be shown that the cosine transform C.T.f(x,y)=1/2[ReF.T.{f(x,y)}+ReF.T.{f(x,−y)}]=1/2[ReF p (u,v)+ReF m (u,v)] where Re indicates the real part of a complex number and F p (u,v) and F m (u,v) are the Fourier transforms of f(x,y) and f(x,−y) respectively. 
     Let c p (x,y)=f(x,y)+A p δ(x,y) and c m (x,y)=f(x,−y)+A m δ(x,y). The Fourier transform, C p (u,v), of c p (x,y) may be written C p (u,v)=[F p (u,v)+A]=[ReF p (u,v)+Im F p (u,v)+A p ], where Im indicates the imaginary part of a complex number and A p  is assumed to be real. Similarly the Fourier transform of c m (x,y) may be written C m (u,v)=[F m (u,v)+A m ]=[ReF m (u,v)+Im F m (u,v)+A m ]. 
     If the “intensities” of the Fourier transforms F p (u,v) and C p (u,v) are written as IF p (u,v) and IC p (u,v) respectively so that IF p (u,v)=|F p (u,v)| 2  and IC p (u,v)=|C p (u,v)| 2  then it can be shown that ReF p (u,v)=[IC p (u,v)−IF p (u,v)−A p   2 ]/2A p . Similarly, ReF m (u,v) =[IC m (u,v)−IF m (u,v)−A m   2 ]/2A m  where IF m (u,v)=|F m (u,v)| 2  and IC m (u,v)=|C m (u,v)| 2 . 
     Therefore the cosine transform of f(x,y) can be determined from the intensities IF p (u,v), IC p (u,v) and A p  and IF m (u,v), IC m (u,v) and A m . It should be noted that whereas a delta function has been added as a reference field for f(x,y) and f(x,−y) in the above calculations, similar results can obtain for other reference functions r(x,y).  FIGS. 5A-5D  illustrate a method, in accordance with an embodiment of the present invention by which the functions IF p (u,v), IC p (u,v) and A p  and IF m (u,v), IC m (u,v) and A m  are evaluated using an optical processor  100  to generate a cosine transform of a function f(x,y). Optical processor  100  is similar to optical processors  50  and  70  and comprises a Fourier lens  22 , a photosensor array  52  at an output plane  26 , which is located at a focal plane of lens  22  and a spatial light modulator  30 . 
     Referring to  FIG. 5A  assume that function f(x,y) is represented by an image  40  formed by spatial light modulator  30 . Optical modulator  100  generates the Fourier transform F(u,v) of f(x,y) and acquires values for IF p (u,v). In  FIG. 5B , a point light source  102  generates a delta function reference A p δ(x,y) image which is added to f(x,y) to form an image c p (x,y)=f(x,y) +A p δ(x,y). Processor  100  Fourier transforms c p (x,y) and acquires IC p (u,v). Point light source may be provided using any methods known in the art. In some embodiments of the present invention point light source is provided by methods and apparatus that are similar to those described in the discussion of  FIGS. 4A and 4B . 
     In  FIG. 5C , spatial light modulator  30  forms an image f(x,−y) and acquires IF m (u,v). In  FIG. 5D  a delta function reference function A m δ(x,y) is added to f(x,−y) and IC m (u,v) is acquired. A suitable processor (not shown) receives the acquired data and uses it to determine ReF p (u,v) and ReF m (u,v) from which the cosine transform of f(x,y) may be determined as shown above. 
     The present application is related to the following four PCT applications, all filed on May 19, 2000: PCT/IL00/00282 published as WO 00/72105, which especially describes matching of discrete and continuous optical elements, PCT/IL00/00285 published as WO 00/72107 which especially describes reflective and incoherent optical processor designs, PCT/IL00/00283 published as WO 00/72104 which especially describes various architectures for non-imaging or diffractive based optical processing, and PCT/IL00/00286 published as WO 00/72108 which especially describes a method of processing by separating a data set into bit-planes and/or using feedback. The disclosures of all of these applications are incorporated herein by reference. 
     In the description and claims of the present application, each of the verbs, “comprise” “include” and “have”, and conjugates thereof, are used to indicate that the object or objects of the verb are not necessarily a complete listing of members, components, elements or parts of the subject or subjects of the verb. 
     The present invention has been described using detailed descriptions of embodiments thereof that are provided by way of example and are not intended to limit the scope of the invention. The described embodiments comprise different features, not all of which are required in all embodiments of the invention. Some embodiments of the present invention utilize only some of the features or possible combinations of the features. Variations of embodiments of the present invention that are described and embodiments of the present invention comprising different combinations of features noted in the described embodiments will occur to persons of the art. The scope of the invention is limited only by the following claims.