Patent Publication Number: US-8111937-B2

Title: Optimized image processing for wavefront coded imaging systems

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a divisional application of, and claims priority to, commonly owned and U.S. patent application Ser. No. 10/376,924 filed 27 Feb. 2003, now U.S. Pat. No. 7,379,613 which claims priority to U.S. Provisional Patent Application No. 60/360,147 filed 27 Feb. 2002. Both of the above-identified patent applications are incorporated herein by reference. 
    
    
     BACKGROUND 
     Wavefront coding is a set of specialized aspheric optics, detection and signal processing. In the prior art, the signal processing is dictated by the aspheric optics such that the spatial blur caused by wavefront coding is removed. The aspheric optics “code” the wavefront so that the sampled image is relatively insensitive to misfocus-related aberrations. The intermediate image from the coded wavefront (i.e., the image sampled by the detector) is not sharp and clear. Signal processing processes data from the detector to remove the spatial blur caused by the aspheric optics, to “decode” wavefront coding. 
     By way of example,  FIG. 1  shows one prior art wavefront coded optical imaging system  100 . System  100  includes optics  101  and a wavefront coded aspheric optical element  110  that cooperate to form an intermediate image at a detector  120 . Element  110  operates to code the wavefront within system  100 . A data stream  125  from detector  120  is processed by image processing section  140  to decode the wavefront and produce a final image  141 . A digital filter (a “filter kernel”)  130  is implemented by image processing section  140 . Filter kernel  130  consists of filter taps, or “weights,” that are real or integer values with a maximum dynamic range due to the limits of processing hardware and software, for example 128-bit extended floating-point arithmetic, real number mathematics, and scaling/truncation of integer values. Scaling may for example include general series of partial products computations and accumulations. 
     In the prior art, optical element  110  and detector  120  do not necessarily match the available processing capability of image processing section  140 ; accordingly, image degradation may be noted within final image  141 . If optical element  110  and detector  120  are matched to the available processing capability, it may require a very complex and costly hardware implementation in terms of detector capability, optical design of element  110 , and/or computer processing architectures associated with image processing section  140 . 
     SUMMARY OF THE INVENTION 
     As described herein below, wavefront coding imaging systems are disclosed with jointly optimized aspheric optics and electronics. In one example, the optics and electronics are jointly optimized for desired optical and/or mechanical characteristics, such as to control aberrations, minimize physical lens count, loosen mechanical tolerances, etc. In another example, the optics and electronics are jointly optimized with targeted emphasis on electronic parameters, so as to reduce the amount of silicon and/or memory required in hardware processing; this serves to optimize image quality in the presence of non-ideal detectors and/or to optimize image quality with a fixed hardware processing solution (e.g., a low cost solution). 
     An emphasis on optimizing electronic processing is for example important in imaging applications that involve high unit quantities and dedicated hardware processing. Examples of such applications include miniature cameras for cell phones, video conferencing, and personal communication devices. By jointly optimizing the optics and mechanics with the electronic parameters, high quality imaging systems are made that are inexpensive in terms of physical components, assembly, and image processing. 
     In certain aspects therefore, wavefront coded optical imaging systems are disclosed with optimized image processing. In one aspect, the image processing is optimized for hardware implementations in which digital filters have filter tap values that are a specialized, reduced set of integers. These integers in turn affect the processing hardware to reduce complexity, size and/or cost. Examples of such reduced set filter tap values include “power of two” values, sums and differences of power of two values, and filters where differences of adjacent filter values are powers of two. The restrictions associated with these example reduced sets operate to reduce the number of multiplications (or generalized partial products summations) associated with image processing. 
     The filter tap values may associate with the spatial location of the filter kernel. The dynamic range of the filter tap values may also be a function of the spatial position relative to positioning of the optical elements and wavefront coding element. In one example, values near the geometric center of the filter kernel have a larger dynamic range and values near the edge of the filter kernel have a smaller dynamic range, useful when the wavefront coding response tends to converge toward small values as distance from the kernal center increases. The filter tap values may also be a function of multi-color imaging channel characteristics since different wavelength bands respond differently when processed by a detector or a human eye. 
     By way of example, two easy to describe optical forms of wavefront coded optics that are suitable for system optimization herein include weighted sums of separable powers, p(x,y)=Sum ai[sign(x)|x|^bi+sign(y)|y|^bi], and the cosinusoidal forms p(r,theta)=Sum ai r^bi*cos(ci*theta+phii). 
     In one aspect, an image processing method is provided, including the steps of: wavefront coding a wavefront that forms an optical image; converting the optical image to a data stream; and processing the data stream with a reduced set filter kernel to reverse effects of wavefront coding and generate a final image. 
     The step of processing may include the step of utilizing a filter kernel that is complementary to an MTF of the optical image. 
     The steps of wavefront coding, converting and processing may occur such that the MTF is spatially correlated to mathematical processing of the data stream with the reduced set filter kernel. 
     In one aspect, the method includes the step of formulating the reduced set filter kernel to an MTF and/or PSF of the optical image resulting from phase modification of the wavefront through constant profile path optics. 
     The method may include the step of processing the data with a reduced set filter kernel consisting of a plurality or regions wherein at least one of the regions has zero values. 
     The step of wavefront coding may include the step of wavefront coding the wavefront such that a PSF of the optical image spatially correlates to the regions of the reduced set filter kernel. 
     In another aspect, the method includes the step of modifying one of the wavefront coding, converting and processing steps and then optimizing and repeating (in a design loop) one other of the wavefront coding, converting and processing steps. For example, the step of processing may include utilizing a reduced set filter kernel with a weighted matrix. 
     In another aspect, an image processing method is provided, including the steps of: wavefront coding a wavefront that forms an optical image; converting the optical image to a data stream; and processing the data stream with a color-specific filter kernel to reverse effects of wavefront coding and generate a final image. 
     In another aspect, an image processing method is provided, including the steps of: wavefront coding a wavefront that forms an optical image; converting the optical image to a data stream; colorspace converting the data stream; separating spatial information and color information of the colorspace converted data stream into one or more separate channels; deblurring one or both of the spatial information and the color information; recombining the channels to recombine deblurred spatial information with deblurred color information; and colorspace converting the recombined deblurred spatial and color information to generate an output image. The method may include a first step of filtering noise from the data stream. In one aspect, the method generates an MTF of the optical image such that spatial correlation exists with the first step of filtering noise. A second step of filtering noise may occur by processing the recombined deblurred spatial and color information. This second step of filtering noise may include the step of utilizing a complement of an MTF of the optical image. 
     In another aspect, an optical imaging system is provided for generating an optical image. A wavefront coding element codes a wavefront forming the optical image. A detector converts the optical image to a data stream. An image processor processes the data stream with a reduced set filter kernel to reverse effects of wavefront coding and generate a final image. The filter kernel may be spatially complementary to a PSF of the optical image. A spatial frequency domain version of the filter kernel may be complementary to an MTF of the optical image. 
     In another aspect, an electronic device is provided, including a camera having (a) a wavefront coding element that phase modifies a wavefront that forms an optical image within the camera, (b) a detector for converting the optical image to a data stream, and (c) an image processor for processing the data stream with a reduced set filter kernel to reverse effects of wavefront coding and generate a final image. The electronic device is for example one of a cell phone and a teleconferencing apparatus. 
     In another aspect, an image processing method includes the steps of: wavefront coding a wavefront with constant profile path optics that forms an optical image; converting the optical image to a data stream; and processing the data stream to reverse effects of wavefront coding and generate a final image. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows a prior art wavefront coded optical imaging system; 
         FIG. 2  shows one wavefront coded optical imaging system with optimized image processing; 
         FIG. 3  shows one other wavefront coded optical imaging system with optimized image processing; 
         FIG. 4  shows one other wavefront coded optical imaging system with optimized image processing; 
         FIG. 5  schematically illustrates one filter kernel and associated image processing section; 
         FIG. 6A  shows a general set filter kernel and  FIG. 6B  shows one reduced set filter kernel; 
         FIG. 7A  shows one generalized logic ALU for a general set filter kernel;  FIG. 7B  and  FIG. 7C  show alternative specialized logic for a reduced set filter kernel; 
         FIG. 8  shows one wavefront coded optical imaging system with an optimized filter kernel; 
         FIG. 9A  shows one wavefront coded optical imaging system with optimized image processing and filter kernel using diagonal reconstruction; 
         FIG. 9B  shows one wavefront coded optical imaging system with optimized image processing and filter kernel using circular reconstruction; 
         FIG. 10  shows one wavefront coded optical imaging system with color-optimized image processing and filter kernel; 
         FIG. 11A  shows one wavefront coded optical imaging system with optimized image processing with single-shift differential taps and a gradient filter kernel; 
         FIG. 11B  shows one wavefront coded optical imaging system with optimized image processing with scaling taps and a scaling filter kernel; 
         FIG. 11C  shows one wavefront coded optical imaging system with optimized image processing with scaling-accumulating taps and a distributive-property filter kernel; 
         FIG. 11D  shows one wavefront coded optical imaging system with optimized image processing with accumulating-scaling taps and a distributive-property filter kernel; 
         FIG. 12  illustratively shows a final image processed by the system of  FIG. 1  and a final image processed by the system of  FIG. 2 ; 
         FIG. 13  illustratively shows a final image processed by the system of  FIG. 1  and a final image processed by the system of  FIG. 2  employing a filter as in  FIG. 11 ; and 
         FIG. 14  shows frequency response curves for the filter kernels used in  FIG. 12  and  FIG. 13 . 
         FIG. 15  shows one optical imaging system with optimized color image processing; 
         FIG. 16A  shows one optical imaging system with optimized color image processing; 
         FIG. 16B  illustrates one correlation and color component analysis associated with  FIG. 16A ; 
         FIG. 17-FIG .  23  demonstrate image components through various stages of processing within  FIG. 16A ; 
         FIG. 24  shows a block diagram of one electronic device employing characteristics of certain wavefront coded imaging systems; 
         FIG. 25  illustrates select optimization trade-offs in a schematic diagram, between system components that are jointly optimized in a design phase to construct the optical imaging system; 
         FIG. 26  illustrates select profiles for constant profile path optics; 
         FIG. 27  illustrates select profiles for constant profile path optics; 
         FIG. 28  illustrates select profiles for constant profile path optics; 
         FIG. 29  illustrates select profiles for constant profile path optics; 
         FIG. 30  shows a surface profile, resulting MTF, along and cross path surface forms for one profile of  FIG. 16 ; 
         FIG. 31  shows one other surface profile, MTF, and along and cross path surface forms; 
         FIG. 32  shows one other surface profile, MTF, and along and cross path surface forms; 
         FIG. 33  shows one other surface profile, MTF, and along and cross path surface forms relating to one profile of  FIG. 26 ; 
         FIG. 34  shows sampled PSFs from the example of  FIG. 32  and without wavefront coding; 
         FIG. 35  shows sampled PSFs from the example of  FIG. 32  and with wavefront coding; 
         FIG. 36  and  FIG. 37  show and compare cross sections through sampled PSFs from  FIG. 34  and  FIG. 35 ; 
         FIG. 38  shows an example of one 2D digital filter; 
         FIG. 39  shows a processed PSF through the filter of  FIG. 38 ; 
         FIG. 40  illustrates an amount of rank power for the in-focus sampled PSF and the digital filter of  FIG. 35  and  FIG. 38 , respectively; 
         FIG. 41  shows corresponding MTFs before and after filtering and in the spatial frequency domain; and 
         FIG. 42  shows one design process for optimizing optics, detector, wavefront coding optics, digital filter (filter kernel), and/or image processing hardware. 
     
    
    
     DETAILED DESCRIPTION OF ILLUSTRATED EMBODIMENTS 
       FIG. 2  shows one wavefront coded optical imaging system  200  with optimized image processing. System  200  includes optics  201  and a wavefront coded aspheric optical element  210  that cooperate to form an intermediate image at a detector  220 . Element  210  operates to code the wavefront within system  200 ; by way of example, element  210  is a phase mask that modifies phase of the wavefront. A data stream  225  from detector  220  is processed by image processing section  240  to decode the wavefront and produce a final image  241 . System  200  has a filter kernel  230  with a reduced set of filter kernel values that optimally match the particular image processing implementation of image processing section  240 . Filter kernel  230  may be constructed and arranged with the reduced set of kernel values such that a final image  241  of system  200  is substantially equivalent to final image  141  of system  100 . As described in more detail below, certain examples of the reduced set of kernel values (i.e., tap or weight values) suitable for use with filter kernel  230  include: (i) powers of two, (ii) sums and differences of powers of two, (iii) a difference between adjacent taps restricted to a power of two, and (iv) taps with a range of values specified by spatial location or color of the image being processed. 
       FIG. 3  shows another wavefront coded optical imaging system  300  with optimized image processing. System  300  includes optics  301  and a wavefront coded aspheric optical element  310  that cooperate to form an intermediate image at a detector  320 . Element  310  operates to code the wavefront within system  300 . A data stream  325  from detector  320  is processed by image processing section  340  to decode the wavefront and produce a final image  341 . Image processing section  340  processes data stream  325  using a filter kernel  330 . Detector  320 , data stream  325 , filter kernel  330  and image processing section  340  are constructed are arranged to optimize system  300  in terms of complexity, size, and/or cost. Final image  341  may be substantially equivalent to final image  141 ,  FIG. 1 . As described in more detail below, detector  320  and data stream  325  may include diagonal readouts for latter diagonal reconstruction, and/or multi-channel readouts for color filter array detectors. Diagonal reconstruction is useful when performing processing on color filter arrays since many color interpolation algorithms operate on diagonal components in the image. Color-specific filter kernels and processing offer reduced cost and improved performance as opposed to processing all colors equally. Filter kernel  330  may be the same as filter kernel  230  of  FIG. 2 ; though those skilled in the art appreciate that filter kernel  230  may include effects of color and directional (e.g., diagonal) convolution, such as described below. 
       FIG. 4  shows another wavefront coded optical imaging system  400  with optimized image processing. System  400  includes optics  401  and a wavefront coded aspheric optical element  410  that cooperate to form an intermediate image at a detector  420 . Element  410  operates to code the wavefront within system  400 . A data stream  425  from detector  420  is processed by image processing section  440  to decode the wavefront and produce a final image  441 . Image processing section  440  processes data stream  425  using a filter kernel  430 . Image processing section  440  may be optimized such that final image  441  is substantially equivalent to final image  141 ,  FIG. 1 . Optics  401  and image processing section  440  are jointly optimized for alternative reconstruction algorithms, such as diagonal processing or processing with a reduced set of kernel values, that facilitate higher performance at lower cost, size, etc. 
       FIG. 5  demonstrates one embodiment of a filter kernel  530  and an image processing section  540 ; kernel  530  and section  540  may for example be used within certain implementations of systems  200 - 400  above (e.g., to operate as filter kernel  430  and image processing section  440  of  FIG. 4 ). As shown, a detector  520  captures an intermediate image of the optical system, similar to detector  220 ,  320 ,  420  above; data stream  525  is input to image processing section  540 , similar to data stream  225 ,  325 ,  425  above. Filter kernel  530  is a reduced set filter kernel with values restricted to absolute values that are a power of two (including zero), such as {−2 N , . . . , −4, −2, 1, 0, 1, 2, 4, . . . , 2 N }. As described in more detail below, image processing section  540  implements the multiplication for filter kernel  530  through only shifting, as illustrated; only this operation is used since filter kernel  530  is limited to powers of two. In one embodiment, filter kernel  530  may be represented by the exponent or equivalently the shift for that coefficient. 
     More particularly, digital filtering is a sum of products for both linear and non-linear filtering. In the example of  FIG. 5 , filter kernel  530  is applied to the intermediate blurred image sampled by detector  520 . Consider filter kernel  530  and the intermediate blurred image as separate two-dimensional objects, with filter kernel  530  centered on a particular image pixel output from detector  520 . At every kernel overlapping with an image pixel, a product of the associated filter kernel value and image pixel is made; these products are then summed. For linear filtering, this sum is the final filtered image pixel. To complete the convolution, filter kernel  530  is then similarly centered over every other pixel in the entire image to produce a like value for each pixel. For linear filtering with an N×N two-dimensional filter kernel, therefore, each filtered image pixel has N 2  products and N 2 −1 sums of these products. Wavefront coded optical imaging systems  200 ,  300 ,  400  may thus be optimized with respective optics and filter kernels matched to particular hardware implementations of the image processing section so as to reduce the number of multipliers, sums and related implementation costs. Note that a kernel value of 0 requires only storage, so controlling sparseness within a kernel also reduces implementation cost. 
       FIG. 6A  and  FIG. 6B  show, respectively, a schematic diagram of a general set filter kernel and a schematic diagram of a reduced set filter kernel (based on power of two values). In  FIG. 6A , the general set filter kernel is not limited to coefficients that are powers of two; it thus requires a generalized arithmetic logic unit (“ALU”)  550  to perform the product of each filter coefficient (i.e., the kernel value) and the image pixel, as shown. The generalized ALU  550  performs an extended series of intermediate logic (known as partial products  552 ) and accumulations of the intermediate results (at accumulators  554 ) to form the final product (pixel out). In  FIG. 6B , on the other hand, since the reduced set filter kernel is implemented with power of two kernel values, the arithmetic logic used to form the product may be a shifter  560 . The amount of bits shifted for each image pixel is determined by the exponent of the power of two filter value. If for example the reduced set filter kernel consists of a sum of two power of two kernel values, then only two shifters and one adder are used in the arithmetic logic. Accumulations of all shifters  560  result in the final product (pixel out). The implementation of  FIG. 6B  is thus simplified as compared to the generalized ALU  550  of  FIG. 6A  since the shifting and adding logic is predefined as a shift-add sequence, and not as a generalized partial products summation. 
     Those skilled in the art appreciate that the arithmetic logic of  FIG. 6B  may favorably compare to other multiplication or generalized scaling implementations. One example is integer-based look-up-table (LUT) multiplication. Short bit-depth ALUs often implement scalar operands within the LUT to improve speed and simplify instruction sets since a series of partial product summations may consume more time and register space as compared to LUT resources. Like the general partial products accumulator multiplier, the LUT multiplier thus allocates resources for generalized scaling operations. The arithmetic logic of  FIG. 6B , on the other hand, employs a reduced set of scaling operations through bit-shifting logic using power of two coefficients. The arithmetic logic of  FIG. 6B  is therefore many times smaller and faster than the generalized ALU  550  needed to process generalized filter kernel values. 
       FIG. 7A ,  FIG. 7B  and  FIG. 7C  show further detail of the arithmetic logic in  FIG. 6A  and  FIG. 6B , to compare generalized and specialized arithmetic logical functions used at each tap in a linear filter.  FIG. 7A  schematically illustrates generalized ALU  550  and  FIG. 7B  and  FIG. 7C  schematically illustrate alternate configurations of specialized ALU  560 . In this example, consider a generalized filter set kernel and a reduced set filter kernel that range over the positive integer values between 1 and 128. The generalized set filter kernel can have all values between 1 and 128. The reduced set filter kernel can only have the values in this range that are a power of two, i.e.: 1, 2, 4, 8, 16, 32, 64, and 128. Now consider the product of an image pixel p with a generalized set filter kernel with a value of 7. Using generalized ALU  550 , a total of three shifts are required, (4+2+1)xp, or as in  FIG. 7A , Q=3. These three shifts are needed to form partial products with factors 4 (shift of two), then 2 (shift of one), and then 1 (no shift). Implementation of a generalized set filter kernel with a value of 127 thus requires seven shifts and seven intermediate sums. 
     In comparison, using the reduced set filter kernel with power of two values, ALU  560 A has just one shift. Even a more sophisticated implementation of generalized ALU  550 , such as a vector machine or pipelined device, would likely require more resources than such a single shift. 
     Another specialized ALU  560 B of  FIG. 7C  depicts a shift-subtract sequence for a known non-power of two coefficient. Continuing the previous example, if the coefficient is  127  for a pixel value p, then the sum of powers of two implementation of 127xp is: 64+32+16+8+4+2+1=127. In ALU  560 B, therefore, the logic subtracts and enables the summation of negative powers of two to proceed as 128−1=127, requiring only shift and one subtraction to produce the final result (pixel out). 
       FIG. 8  describes one wavefront coded optical imaging system  800  in terms of one reduced set filter kernel. System  800  includes optics  801  and a wavefront coded aspheric optical element  810  that cooperate to form an intermediate image at a detector  820 . Element  810  operates to code the wavefront within system  800 . A data stream  825  from detector  820  is processed by image processing section  840  to decode the wavefront and produce a final image  841 . Image processing section  840  processes data steam  825  using a filter kernel  830 . By reducing the set of filter kernels in filter kernel  830 , as a function of value and of spatial location, the hardware implementation of image processing section  840  may be optimized to optics  801 , wavefront coding aspheric optical element  810  and detector  820 . Regions A, B, C and D of filter kernel  830  illustrate the reduced geometric set. For this example, values of filter kernel  830  are restricted to zero value in regions A and D, to a moderate range of values (or dynamic range) in region B, and to a large range of values in region C. All particular kernel values within the prescribed ranges are efficiently implemented from power of two values, sums and differences of power of two values, etc. 
     Wavefront coded optical imaging system  800  is thus optimized to the reduced geometric set of filter kernels implemented in hardware (e.g., via filter kernel  830  and image processing section  840 ). Reduced geometric set kernels are particularly important in systems with specialized hardware processors and are used with a variety of different wavefront coded optics and detectors, or used with optics that change characteristics, like wavefront coded zoom optics. By way of example, optics  801  and optical element  810  adjust the depth of field and also adjust scaling within the kernel size. The prescribed dynamic range is therefore achieved by positioning the coefficient values of filter kernel  830  in appropriate spatial location. By designing optics  801  and optical element  810  such that the scaled kernel values lie within regions of adequate dynamic range, image processing section  840  is optimized for such optics  801  and element  810  given the constraints of the largest kernel. While different filter kernels have different values within the geometric regions A-D, the arrangement of values is optimized with respect to the capabilities of image processing section  840 . By way of illustration, the rotation of optical element  810  (e.g., by 45 degrees) would dictate an equivalent rotation of the kernel; this would not be permitted since the B regions would then lie in a processing space with only near-zero coefficients rather than the moderate dynamic range of the B region coefficients. The rotation would only be permitted if the processing space and kernel space were both rotated. 
       FIG. 9A  shows one wavefront coded optical imaging system  900 A that is optimized for its detector and data stream readout. System  900 A includes optics  901 A and a wavefront coded aspheric optical element  910 A that cooperate to form an intermediate image at a detector  920 A. Element  910 A operates to code the wavefront within system  900 A. A data stream  925 A from detector  920 A is processed by image processing section  940 A to decode the wavefront and produce a final image  941 A. Image processing section  940 A processes data stream  925 A using filter kernel  930 A. In this embodiment, detector  920 A and data stream  925 A are not read out in typical row and column format; rather, the output format is for example a diagonal readout. With such a diagonal readout format, image processing section  940 A does not utilize rectilinear processing (e.g., rectangularly separable or rank-N processing) since the image format is not row and column based. If the data format is diagonal, optics  901 A and wavefront coded aspheric optical element  910 A are arranged such that the resulting image of a point object (or its point spread function) contains substantially all information along diagonals in a manner equivalent to the data stream diagonals. The corresponding filter kernel  930 A and image processing section  940 A then optimally act on image data  925 A in a diagonal format, as shown, and either or both diagonals can be processed. In one embodiment, the diagonals are remapped to row-column order and operated on in a rectilinear fashion, and then remapped again in the diagonally ordered output sequence. In one implementation, optics  901 A and optical element  910 A are optimized with image processing section  940 A to provide information in a geometrical orientation that reduces cost. One example is an optical element  910 A that is rotated in a way that corresponds to the diagonal angle of data stream readout  925 A. 
       FIG. 9B  shows one wavefront coded optical imaging system  900 B that is optimized for detector and data stream readout. System  900 B includes optics  901 B and a wavefront coded aspheric optical element  910 B that cooperate to form an intermediate image at a detector  920 B. Element  910 B operates to code the wavefront within system  900 B. A data stream  925 B from detector  920 B is processed by image processing section  940 B to decode the wavefront and produce a final image  941 B. Image processing section  940 B processes data stream  925 B utilizing a filter kernel  930 B. Detector  920 B and data stream  925 B are not of typical rectilinear design and are not read out in typical row and column format. Rather, the output format is, for example, an annular or radial-based readout from a log-polar detector. With this readout format, image processing section  940 B once again does not utilize rectilinear processing (e.g., rectangularly separable or rank-N processing) since the image format is not row and column based. If the data format is circular or annular, optics  901 B and wavefront coded aspheric optical element  910 B are arranged such that the resulting image of a point object (or its point spread function) contains substantially all information along annular or radial regions in a manner equivalent to data stream  925 B. The corresponding filter kernel  930 B and image processing section  940 B then optimally act on image data  925 B in a circular format, as shown. In one embodiment, the concentric rings are remapped to row-column order and operated on in a rectilinear fashion, and then remapped again in the circular ordered output sequence. In one embodiment, optics  901 B and element  910 B may be optimized with image processing section  940 B to provide information in a geometrical orientation that reduces cost. 
       FIG. 10  describes one wavefront coded optical imaging system  1000  optimized for color imaging. System  1000  includes optics  1001  and a wavefront coded aspheric optical element  1010  that cooperate to form an intermediate image at a detector  1020 . Element  1010  operates to code the wavefront within system  1000 . A data stream  1025  from detector  1020  is processed by image processing section  1040  to decode the wavefront and produce a final image  1041 . Image processing section  1040  utilizes a color-specific filter kernel  1030  in processing data stream  1025 . Optics  1001 , aspheric optical element  1010 , detector  1020 , data stream  1025 , filter kernel  1030  and image processing section  1040  cooperate such that each color channel has separate characteristics in producing final output image  1041 . As a function of color, for example, wavefront coded optical imaging system  1000  takes advantage of the fact that a human eye “sees” each color differently. The human eye is most sensitive to green illumination and is least sensitive to blue illumination. Accordingly, the spatial resolution, spatial bandwidth, and dynamic range associated with filter kernel  1030  and image processing section  1040  in the blue channel can be much less than for the green channel—without degrading the perceived image  1041  as viewed by a human. By reducing the requirements as a function of color, the opto-mechanical and opto-electrical implementation of system  1000  is further optimized as compared to treating all color channels equivalently. 
       FIGS. 11A through 11D  show related embodiments of a wavefront coded optical imaging system  1100  that is optimized with specialized filtering (i.e., within an image processing section  1140 ). Each system  1100 A- 1100 D includes optics  1101  and a wavefront coded aspheric optical element  1110  that cooperate to form an intermediate image at a detector  1120 . Element  1110  operates to code the wavefront within system  1100 . A data stream  1125  from detector  1120  is processed by image processing section  1140  to decode the wavefront and produce a final image  1141 . Image processing section  1140  processes data stream  1125  with a filter kernel  1130 . Optics  1101 , aspheric optical element  1110 , detector  1120  and data stream  1125  cooperate such that a filter kernel  1130  utilizes specialized filtering and image processing section  1140  utilizes specialized taps  1145 , providing desired cost savings and/or design characteristics. 
     More particularly, the optical systems of  FIG. 11A-11D  illustrate four separate optimizations that may take into account any given coefficient, or sub-group of coefficients, to best implement a particular configuration. Any combination or combinations of the embodiments in  FIG. 11A through 11D  can be implemented with an efficient image processing section. 
       FIG. 11A  specifically shows one embodiment of a gradient filtering operation. Optics  1101 A, aspheric optical element  1110 A, detector  1120 A and data stream  1125 A cooperate such that a filter kernel  1130 A utilizes gradient filtering and image processing section  1140 A utilizes specialized differential taps  1145 A. The reduced set filter kernel of filter kernel  1130 A is such that the difference between adjacent coefficients supports efficient implementation. For example, filter kernel  1130 A and image processing section  1140 A may be configured such that the difference between the first and second tap, between the second and third tap, between the third and fourth tap, and so on, are all powers of two (or another efficiently implemented value or set of values). If each absolute value difference of the filter kernel  1130 A is a reduced set of power of two values, then the geometry of each implemented filter tap is a differential tap  1145 A. Differential filter tap  1145 A is an efficient implementation because only a single summation is required for any coefficient value in kernel  1130 A. In prior art linear filtering, such as employed by FIR tap delay lines, only the original image pixel value is propagated to the next tap while the previous scaled value is not available to the next stage. Recovery of a previously scaled value or prior result  1146 A in the differential tap  1145 A allows a single addition or subtraction to generate the next value. Since the prior result  1146 A is propagated to the next tap, a savings is further realized in the size and cost of the hardware implementation of image processor section  1140 A. Other combinations of additions and subtractions can be combined in a differential tap  1145 A; a differential tap  1145 A is not limited to a single addition (or subtraction), rather those skilled in the art can recognize that any combination of additions and subtractions can be used in conjunction with carrying forward the prior result  1146 A—it is the carrying-forward action that provides the ability to create an optimal solution, as opposed to the particular chosen design of the differential tap. 
     Numerical values provide an even clearer example for a single-adder differential tap. Consider the coefficient sequence [3, −5, 11, 7, −1, 0]. If the input pixel value is p, then the first tap generates 3p. The next tap generates −5p, or alternatively using 3p from the previous step, generates −8p and adds the two values to obtain 3p−8p=−5p. The coefficient 11 is similarly obtained from the previous −5p by adding 16p. Note that in this case a single shift (p to 16p) and a single addition achieve a scaling factor of 11x, whereas traditional implementation of 11x based on partial products requires 8+2+1, or two shifts and two summations. The example continues with 7p=(11p−4p) and −1p=(7p−8p). In some cases, the difference may be zero, in which case the gradient kernel falls to the detail of the scaling kernel which is described in  FIG. 11B . 
       FIG. 11B  shows an embodiment of a scale-filtering operation. Optics  1101 B, aspheric optical element  1110 B, detector  1120 B and data stream  1125 B cooperate such that a filter kernel  1130 B utilizes scaled filtering and image processing section  1140 B utilizes specialized scaling taps  1145 B. The reduced set filter kernel of filter kernel  1130 B is such that the scale factor between adjacent coefficients supports efficient implementation. For example, filter kernel  1130 B and image processing section  1140 B may be configured such that the scale factor between the first and second tap, between the second and third tap, between the third and forth tap, and so on, are all factors of powers of two (or another efficiently implemented value or set of values). If each scale factor of the adjacent filter kernel coefficient is a reduced set of power of two values, for example, then the geometry of each implemented filter tap is a scaling tap  1145 B and can be implemented with a simple shift on a binary computer (other scale factors may be efficiently implemented by other devices in a more efficient manner than powers of  2 ). Regardless of the scale factor chosen, scaling filter tap  1145 B is an efficient implementation because only the scaled value of an image pixel at any one tap is passed to both the next tap and the kernel accumulator. In prior art linear filtering, such as employed by FIR tap delay lines, the original image pixel value has to propagate to the next tap while the scaled value is also propagated to the accumulator. Storage of two values (the delayed pixel value plus the scaled pixel value) requires two registers. In  FIG. 11B , only one register is needed to store the scaled pixel value (scaled from the previous tap) because of scaled filtering of filter kernel  1130 B. Since the original image pixel information is not propagated to all subsequent taps, a savings is further realized in the size and/or cost of hardware implementation of image processor section  1140 B. An example of a filter kernel where the scalings between adjacent pixels are factors of powers of plus or minus two is [3, −12, 6, 0, −3, 0]. 
       FIG. 11C  shows an embodiment of a distributive-arithmetic property filtering operation. Optics  1101 C, aspheric optical element  1110 C, detector  1120 C and data stream  1125 C cooperate such that a filter kernel  1130 C utilizes a distributive property of arithmetic filtering and image processing section  1140 C utilizes specialized distributive-arithmetic taps  1145 C. The tap  1145 C performs accumulation of scaled values. The reduced set filter kernel of filter kernel  1130 C is such that the distribution of all coefficients supports efficient implementation. For example, filter kernel  1130 C and image processing section  1140 C may be configured such that there are only two scale factors or multipliers that are implemented, providing five distinct coefficients available for reconstruction (5 by considering positive and negative values of the two coefficients, plus the zero coefficient). Scaling filter tap  1145 C supports efficient implementation because a single tap can be “overclocked” to service many pixels. Certain implementations may prefer to overlock a multiplier as in  1145 C, or some may to prefer to overclock an accumulator as in  1145 D,  FIG. 11D . In prior art linear filtering, such as employed by FIR tap delay lines, a separate and distinct operator was required for each pixel and coefficient. 
       FIG. 11D  shows an embodiment of a distributive-arithmetic property filtering operation. Optics  1101 D, aspheric optical element  1110 D, detector  1120 D and data stream  1125 D cooperate such that a filter kernel  1130 D utilizes distributive-arithmetic filtering and image processing section  1140 D utilizes specialized distributive-arithmetic taps  1145 D. The tap  1145 D performs scaling of accumulated values. The reduced set filter kernel of filter kernel  1130 D is such that the distribution of all coefficients supports efficient implementation. For example, filter kernel  1130 D and image processing section  1140 D may be configured such that there are only two scale factors or multipliers that are implemented, providing 5 distinct coefficients available for reconstruction (5 by considering positive and negative values of the two coefficients, plus the zero coefficient). As in  FIG. 11C , scaling filter tap  1145 D supports efficient implementation because a single tap can be “overclocked” to service many pixels. Certain implementations may to prefer to overclock an accumulator as in  1145 D. In prior art linear filtering, such as employed by FIR tap delay lines, a separate and distinct operator was required for each pixel and coefficient. 
     One method of designing certain of the wavefront coded aspheric optical elements (e.g., distinct elements  1110  of  FIG. 11A-D ) and certain of the filter kernels (e.g., distinct kernels  1130  of  FIG. 11A-D ) utilizes a plurality of weight matrices that target image processing. Other system design goals, such as image quality, sensitivity to aberrations, etc., may also have associated weight matrices. As part of the design and optimization process, the weight matrices may be optimally joined to achieve the selected goals. Since the optics and the signal processing can be jointly optimized, many types of solutions are reachable that are otherwise difficult or impossible to recognize if, for example, only one of the optics or signal processing is optimized. 
     By way of example, one weight matrix for a generalized integer filter (e.g., utilized by logic of  FIG. 6A ) has values of zero for all possible integer values. In contrast, a weight matrix for a power of two reduced set filter kernel (e.g., utilized by logic of  FIG. 6B ) has values of zero only for integers that are a power of two. The integer zero has a zero weight value since this value is trivial to implement. Other integers have higher weight values. When weight values are for integers that are not a power of two, but are above zero, the optimization between optics, filter kernel, image processing, and final image may occur to trade cost to performance by setting appropriate weight values above zero. 
     In one example, in a reduced set filter kernel utilizing sums of powers of two, the weight matrix may have zero value for integers that are a power of two, a larger value for integers that are constructed from the sum or difference of two power of two values (such as 6=4+2 or 7=8−1), and still larger values for integers that are constructed from three power of two values (such as 11=8+2+1, or 11=8+4−1). When both positive and negative values of power of two are used, the weight for integers such as seven are reduced, since 7=8−1. By way of contrast, the general implementation (e.g.,  FIG. 6A ) requires three summations, i.e., 7=4+2+1. 
     The weight matrix can also be used to optimize a gradient filter (e.g., filter  1130 A,  FIG. 11A ), a scaling filter (e.g., filter  1130 B,  FIG. 11B ), and distributive property kernels. Adjacent integer values may be given a weight based on their difference or gradient, or scale factor. In one example, every pair of integer values is assigned a zero value if their difference or scale is a power of two or otherwise a large value. The pairing of coefficients also depends on the readout of pixels since the readout mode determines the adjacency rules for neighboring pixels and since both the readout and kernel orientation are included in the weight matrix. Those skilled in the art appreciate that other gradient kernel filters may assign small values when the difference of the integers is a sum of powers of two, and so on. 
       FIG. 12  illustratively shows a final image  1201  processed by system  100  of  FIG. 1  and a final image  1202  processed by system  200  of  FIG. 2 . In this example, both systems  100 ,  200  have the same optics  101 ,  201  and detector  120 ,  220 , respectively, but have different filter kernels and image processing sections (implementing respective one dimensional kernel filters within rectangularly separable processing). Image  1201  was formed with a filter kernel  130  that is a generalized integer filter. Image  1202  was formed with a filter kernel  230  that is a reduced set filter kernal utilizing power of two values. Both images  1201 ,  1202  are similar, but image  1202  has improved contrast, as shown. Moreover, in comparison, reduced set filter kernel  230  has a smaller and less costly implementation then the generalized integer filter of filter kernel  130 . 
       FIG. 13  illustratively shows a final image  1301  processed by system  100  of  FIG. 1  and a final image  1302  processed by system  200  of  FIG. 2 . As in  FIG. 12 , both systems  100 ,  200  have equivalent optics and detectors  101 ,  201  and  120 ,  220 , respectively, but have different filter kernels and image processing sections (implementing respective one dimensional filter kernels within rectangularly separable processing). System  200  employs a filter kernel  1130 A,  FIG. 11A , as filter kernel  230 ,  FIG. 2 . System  100  employs a generalized integer filter as filter kernel  130 ; thus image  1301  was formed with the generalized integer filter. Both images  1301 ,  1302  are similar, except that image  1302  has lower contrast. Since contrast is also one possible imaging goal, the reduced contrast can be set to meet this goal. Moreover, the reduced set gradient filter kernel used to process image  1302  can be smaller and less costly to implement then the generalized integer filter. 
       FIG. 14  shows the frequency response for the three digital filters used in  FIG. 12  and  FIG. 13 . Each filter has a slightly different frequency response and a different implementation. Certain imaging applications prefer one frequency response while another may prefer a different frequency response. The particular reduced set power of two frequency response (e.g., employed in filter kernel  230 ,  FIG. 2 ) has the highest contrast. The generalized integer frequency response has the next highest contrast, and the reduced set gradient power of two filter kernel  1130  has the lowest frequency response in this example. This need not be the general case. Accordingly, depending on the application, the appropriate response is chosen; if the filter kernal also employs a reduced set of filter kernel values, then the implementation may also be made with reduced cost and complexity. 
     With further regard to  FIG. 10 , optimizing image processing for color may have certain advantages.  FIG. 15  thus illustrates one other optical imaging system with optimized color image processing. System  1500  includes optics  1501  and a wavefront coded aspheric optical element  1510  that cooperate to form an intermediate image at a detector  1520 . Element  1510  operates to code the wavefront within system  1500 . A data stream  1525  from detector  1520  is then processed by a series of processing blocks  1522 ,  1524 ,  1530 ,  1540 ,  1552 ,  1554 ,  1560  to decode the wavefront and produce a final image  1570 . Block  1522  and block  1524  operate to preprocess data stream  1525  for noise reduction. In particular, FPN block  1522  corrects for fixed pattern noise (e.g., pixel gain and bias, and nonlinearity in response) of detector  1510 ; pre-filtering block  1524  utilizes knowledge of wavefront coded optical element  1510  to further reduce noise from data stream  1525 . Color conversion block  1530  converts RGB color components (from data stream  1525 ) to a new colorspace. Blur &amp; filtering block  1540  removes blur from the new colorspace images by filtering one or more of the new colorspace channels. Block  1552  and block  1554  operate to post-process data from block  1540 , for further noise reduction. In particular, SC Block  1552  filters noise within each single channel of data using knowledge of digital filtering within block  1540 ; MC Block  1554  filters noise from multiple channels of data using knowledge of digital filtering within block  1540 . Prior to image  1570 , another color conversion block  1560  converts the colorspace image components back to RGB color components. 
       FIG. 16A  schematically illustrates one color processing system  1600  that produces a final three-color image  1660  from a color filter array detector  1602 . System  1600  employs optics  1601  (with one or more wavefront coded optical elements or surfaces) to code the wavefront of system  1600  and to produce an intermediate optical image at detector  1602 . Wavefront coding of optics  1601  thus forms a blurred image on detector  1602 . This intermediate image is then processed by NRP and colorspace conversion block  1620 . The noise reduction processing (NRP) of block  1620  for example functions to remove detector non-linearity and additive noise; while the colorspace conversion of block  1620  functions to remove spatial correlation between composite images to reduce the amount of silicon and/or memory required for blur removal processing (in blocks  1642 ,  1644 ). Data from block  1620  is also split into two channels: a spatial image channel  1632  and one or more color channels  1634 . Spatial image channel  1632  has more spatial detail than color channels  1634 . Accordingly, the dominant spatial channel  1632  has the majority of blur removal within process block  1642 . The color channels have substantially less blur removal processing within process block  1644 . After blur removal processes, channels  1632  and  1634  are again combined and are processed within noise reduction processing and colorspace conversion block  1650 . This further removes image noise accentuated by blur removal; and colorspace conversion transforms image into RGB for final image  1660 . 
       FIG. 17  through  FIG. 23  further illustrate colorspace conversion and blur removal processes for a particular embodiment of system  1600 . Detector  1602  is a detector with a Bayer color filter array.  FIG. 17  shows red, green and blue component images  1700 ,  1702  and  1704 , respectively, of an actual object imaged through system  1600  but without wavefront coding in optics  1601  (i.e., the wavefront coded optics are not present with optics  1601 ). In images  1700 ,  1702 ,  1704 , it is apparent that each color image has a high degree of spatial similarity to each other image, and that many parts of the image are blurred. The blurred images are not correctable without wavefront coding, as described below. 
       FIG. 18  shows raw red, green and blue component images  1800 ,  1802  and  1804 , respectively, of the same object imaged through system  1600  with wavefront coded optics  1601 . Images  1800 ,  1802  and  1804  represent images derived from data stream  1625  and prior to processing by block  1620 . Each component image  1800 ,  1802 ,  1804  is again fairly similar to each other image; and each image is blurred due to aspheric optics  1601 . 
       FIG. 19  shows the color component images of  FIG. 18  after blur removal processing of block  1642 ; image  1900  is image  1800  after block  1642 ; image  1902  is image  1802  after block  1644 ; and image  1904  is image  1804  after block  1644 . For this example, processing block  1644  performed no processing. Again, processing blocks  1620  and  1650  were not used. Each component image  1900 ,  1902  and  1904  is sharp and clear; and the combination of all three component images results in a sharp and clear three color image. 
       FIG. 20  shows component images  2000 ,  2002 ,  2004  after colorspace conversion (block  1620 ) from RGB to YIQ colorspace, before processing of spatial and color channels  1632  and  1634 . Component images  2000 ,  2002 ,  2004  thus represent component images of  FIG. 18  after processing by block  1620 , on channel  1632  and channel  1634 . Notice that Y channel image  2000  is similar to the red and green images  1800 ,  1802  of  FIG. 18 . The I and Q channel images  2002 ,  2004  are however much different from all channels of  FIG. 18 . The YIQ colorspace conversion (block  1620 ) resulted in the Y channel component image  2000  containing the majority of the spatial information. The I channel component image  2002  contains much less spatial information than does the Y channel, although some spatial information is visible. Notice the differences in the intensity scales to the right of each image. The Q channel component image  2004  has very little spatial information at the intensity scale shown for these images. 
       FIG. 21  shows the component YIQ wavefront coded images  2100 ,  2102 ,  2104  after blur removal process  1642  of the spatial channel  1632 . The I and Q channel images  2002 ,  2004  were not filtered in block  1644 , thereby saving processing and memory space; accordingly, images  2102 ,  2104  are identical to images  2002 ,  2004 . Those skilled in the art appreciate that images  2002 ,  2004  could be filtered in block  1644  with low resolution and small bit depth filters. After the filtered YIQ image of  FIG. 21  is transformed back into RGB space (block  1650 ), the final three color image  1660  is sharp and clear with much less processing then required to produce the final image based on  FIG. 19 . 
     The YIQ colorspace is one of many linear and non-linear colorspace conversions available. One method for performing colorspace conversion is to select a single space to represent the entire image. Wavefront coded optics and processors can cooperate such that an optimal, dynamic, colorspace conversion can be achieved that reduces the amount of signal processing and improves image quality. In  FIG. 16A , for example, wavefront coded system  1600  may employ a colorspace process block  1620  that varies spatially across the entire image. Such spatial variation may be denoted as Dynamic Color Space (DCS). This spatially-varying colorspace conversion  1620  performs a single transformation on a region, where the region is dynamic and not constrained or enclosed. For example, an image with blue sky in the upper half and sandy beach in the lower half is well suited to the following split colorspace: a “blue-spatial-information-preserving” colorspace conversion for the upper half of the image and a “speckle-brown-information-preserving” colorspace transformation for the lower half. Such spatially varying regions can be defined as an ordered set of rectilinear blocks, or as randomly assigned pixels, or as dynamically allocated contours that are defined in an optimal fashion as each scene changes. Dynamic block allocation methods in conversion process  1620  can be sensitive to the total computational power and desired throughput and accuracy tradeoffs desired in the imaging system, and can be guided or limited by knowledge of the spatial correlating effects of optics  1601 . 
     In general, DCS can provide combined noise reduction, color-plane array interpolation, and reduced-cost image deblurring. The DCS algorithm is appropriate for a variety of processing approaches within software and hardware based systems. One version of DCS implements dynamic linear colorspace conversion. Other versions of DCS implement dynamic non-linear colorspaces. DCS may be combined or preceded by non-linear colorspace conversions or other transformations, such as HSV, depending on processing system resources and application goal. 
     The general DCS algorithm can be understood through correlation and principal component analysis of  FIG. 16B .  FIG. 16B  fits entirely within the block  1620  in  FIG. 16A . The Bayer sensor  1602  and the output image  1632  are numbered identically to  FIG. 16A  to highlight the connection. Since DCS can be a reversible transform, those skilled in the art recognize that an inverse, either exact or approximate, DCS process occurs entirely within block  1650  of  FIG. 16A . The determination of the principal components (or principal color or colors) from the estimated correlation of colors in multi-channel images of an arbitrary region of pixels with an arbitrary number of color channels (more than one) determines the optimal (in the least-squares sense) colorspace transformation for that particular region. In other words, DCS allows blocks  1620  (and  1650  via the inverse DCS transformation) in  FIG. 16A  to be optimized jointly with all other wavefront coding system parameters. In the Bayer detector case with detector  1602  in  FIG. 16B , a color channel matrix is formed by collecting the sampled color channels into columns of a matrix  1603 A. Various schemes can be used to provide estimates of image color channels at each desired location. Not all pixel locations need to be estimated for the entire region. As an example,  FIG. 16B  shows a zero-order hold method that essentially copies nearby red and blue pixel values to every green pixel location, and generates  8  rows of data in the 8×3 matrix, from 16 pixels in the 4×4 region. A principal component decomposition of the correlation estimate  1603 B of the color channel matrix  1603 A provides the colorspace transformation  1603 C. This is only one such example stacking of values to form a color channel matrix for multi-channel color systems; those skilled in the art appreciate that other arrangements are possible without departing from the scope hereof. 
     An estimate of the principal components  1603 C of the correlation matrix  1603 B forms the optimal colorspace transformation matrix for this particular region and channel selection. Obtaining the spatial intensity image  1632  in  FIG. 16A  is achieved by applying the first principal component to the color channel matrix  1603 A in  FIG. 16B , forming the transformed spatial image  1632  in  FIG. 16A . The deblurring function  1642  in  FIG. 16A  then operates on the spatial image  1632 , providing image reconstruction and simultaneous Bayer-pattern interpolation. Returning to the original colorspace then occurs in block  1650  by re-transforming the reconstructed spatial and color channels using a form of an inverse of the color channel transformation matrices  1603 C determined for each region, producing the final image  1660 . Color channel images  1634  may also be extracted from DCS by using the secondary and tertiary principal components in this example. 
     Upon DCS transformation, it is not however guaranteed that only the spatial channel  1632  is to be processed by the spatial channel blur removal function  1642 . Each block or region that was transformed contains its own unique mapping matrix but all regions share a common spatial image. In many cases, the optimal transformation for a given region does not achieve complete transformation of all spatial information to the spatial image. In such a case, deblurring of the color channels may be made by blur removal process block  1644 . Final image assembly is then performed by “coloring” the dynamic regions using respective matrices of colorspace inversion  1650 . 
       FIG. 22  shows the component images of  FIG. 18  after colorspace conversion with the DCS algorithm. After conversion with the dynamically changing colorspace, essentially all spatial information is contained in channel  1  ( 2200 ). The other two color channels  2  ( 2202 ) and  3  ( 2204 ) have essentially no spatial information relative to the shown intensity scale, such that they may be removed from the final image for noise reduction.  FIG. 23  shows these component images after filtering to remove image blur. Only channel  1  ( 2300 ) was filtered in this example, again due to low resolution in channels  2  ( 2302 ) and  3  ( 2303 ). After conversion from DCS to RGB (process block  1650 ), the final image is again sharp and clear, and with fewer color artifacts as compared to the final image of  FIG. 21  (and with less stringent processing and memory requirements). 
     The wavefront coded imaging systems discussed herein may have certain advantages when used within electronic devices such as cell phones, video conferencing apparatus and miniature cameras.  FIG. 24  shows one such electronic device  2400  to illustrate such advantages. Device  2400  has an optical lens  2402  that forms an image from object space  2404  onto a digital detector  2406  (e.g., a 3-color CMOS array), as shown. Detector  2406  converts the image into a data stream  2408 . A microprocessor  2410  processes data stream  2408  to generate a final image  2412 . 
     Optical lens  2402  is also wavefront coded; one surface  2414  of lens  2402  is for example a wavefront coded aspheric optical element. Accordingly, optical lens  2402  and element  2404  may function as the optics and wavefront coded elements discussed hereinabove (e.g., optics  201 , element  210  of  FIG. 2 ). Microprocessor  2410  is for example a processing section discussed hereinabove (e.g., image processing section  240 ,  FIG. 2 ). Microprocessor  2410  employs a filter kernel in processing data stream  2408 , thereby decoding the wavefront due to phase manipulation by lens  2402  and generating a crisp image  2412 . Image  2412  may for example be displayed on an LCD display  2415  or other display screen. 
     The effects of wavefront coding of optical lens  2402  and post-processing by microprocessor  2410  are for example constructed and arranged to reduce misfocus-related aberrations such as defocus, field curvature, or manufacturing and assembly related misfocus. Accordingly, electronic device  2400  may employ a single (plastic) optical element  2402  without the need for other complex optical elements. 
     Moreover, by selecting the appropriate filter kernel (e.g., a reduced set filter kernel with power of two kernel values) and the corresponding phase mask (i.e., surface wavefront coded element  2414 ), microprocessor  2410  may operate with reduced processor load and memory requirement. In one example, microprocessor  2410  employs image processing section  540  and kernel  530 ,  FIG. 5 , to reduce processor load and memory requirement. In another example, microprocessor  2410  employs logic architectures of  FIG. 6B  and/or  FIG. 7  to reduce processor load and memory requirement. Likewise, microprocessor  2410  may employ image processing section  840  and kernel  830  to achieve certain other advantages, as discussed in  FIG. 8 . Depending upon design considerations, including color processing goals, microprocessor  2410  may for example employ processing techniques disclosed in connection with  FIG. 10 ,  FIG. 16A  and/or  FIG. 16B . In another example, electronic device  2400  may employ opto-electronic components that implement imaging architectures of  FIG. 9A  or  FIG. 9B , or one of  FIGS. 11A-11D . By reducing processing requirements, electronic device  2400  may be constructed with a less expensive processor or with less memory. Those skilled in the art appreciate that memory savings may similarly translate to other memory devices or cards  2416 , resulting in further savings. 
     As those in the art appreciate, electronic device  2400  may further include other electronics  2420  to incorporate other desired operation and functionality, for example cell phone functionality. 
     It is therefore apparent that certain trade-offs or “optimizations” may be made within the above described wavefront coded optical imaging systems, to achieve image characteristic goals and/or cost goals, for example. By way of example, consider  FIG. 25 , illustrating components in a wavefront coded imaging system. Component optics  2501  is for example optics  201 ,  FIG. 2 . Wavefront coded optical element  2510  is for example wavefront coded aspheric optical element  210 ,  FIG. 2 . Detector component  2520  is for example detector  220 ,  FIG. 2 , while filter kernel  2530  is for example filter kernel  230 ,  FIG. 2 . Image processing section  2540  is for example section  240 ,  FIG. 2 . 
       FIG. 25  illustrates one three-component optimization  2570  utilizing a reduced set filter kernel  2530  (e.g., filter kernel  830 ,  FIG. 8 ); optimization  2570  thus ties together (in design) wavefront coded optics  2510 , filter kernel  2530  and image processing section  2540 . A standard detector  2520  and optics  2501  are not necessarily optimized with optimization  2570 , if desired. In another example, one four-component optimization  2572  utilizes a detector  2520  with a particular readout format (e.g., as in  FIGS. 9A ,  9 B). In optimization  2572 , therefore, wavefront coded optics  2510 , filter kernel  2530  and image processing section  2540  are also tied together (in design). In an example optimizing color images (e.g., as in  FIG. 10 ), one two-component optimization  2574  can include a color-specific detector  2520  tied together (in design) with image processing section  2540 . As those in the art appreciate, optics  2501  may be optimized with any of optimizations  2570 ,  2572 ,  2574  to achieve imaging characteristics that support other optimizations, for example. 
     In certain embodiments herein, the form of optics and optical elements used in the above-described imaging systems are also optimized. Such form is for example to provide low variability to misfocus-like aberrations, high MTFs, and low noise gain values. As described in more detail below, these optics and optical elements may make up, or be part of, imaging systems such as microscopes, endoscopes, telescopes, machine vision systems, miniature cameras, cell phone cameras, video cameras, digital cameras, barcode scanners, biometrics systems, etc. In two easy to describe examples, optical forms of wavefront coded optics suitable for system optimization herein may include weighted sums of separable powers, p(x,y)=Sum ai [sign(x)|x|^bi+sign((y)|y|^bi ], and the cosinusoidal forms p(r,theta)=Sum ai r^bi*cos(ci*theta+phii). As described in more detail below, the optics also may include specialized contour surfaces (sometimes denoted herein as “constant profile path optics”) that provide phase variation within the wavefront coded optical element (e.g., element  210 ,  FIG. 2 ). Again, the wavefront coded optical element and optics (e.g., element  210  and optics  201 , respectively) may be combined as a single optical element or system (e g , utilizing lenses and/or mirrors). In one example, the optics produce an MTF within the optical imaging system with a frequency domain that is complimentary complementary to the kernel filter; for example, high power within the MTF implies low power in filter kernel. The MTF and filter kernel may provide a degree of spatial correlation (e.g., between the filter kernel and wavefront coded aspheric optical element) such as described herein. 
     As noted above, these wavefront coded imaging systems have non-traditional aspheric optics and image processing. One possible goal for such systems is to produce a final image that is substantially insensitive to misfocus-like aberrations. This insensitivity supports (a) a large depth of field or depth of focus, (b) tolerance to fabrication and/or assembly errors that for example create misfocus-like aberrations, and/or (c) optically-generated misfocus-like aberrations (e.g., spherical aberrations, astigmatism, petzval curvature, chromatic aberration, temperature-related misfocus). Internal optics within such imaging systems form images with specialized blur that may also be insensitive to these misfocus-like aberrations. Moreover, the effects of coma can also be reduced within such wavefront coded optical imaging systems. After capture by a detector (e.g., a digital detector  220  capturing the intermediate image to generate a digital data stream  225 ,  FIG. 2 ), image processing may operate to remove the blur associated with the intermediate image to produce a final image that is sharp and clear, and with a high signal to noise ratio (SNR). 
     The design process to enable such wavefront coded imaging systems may be such that the optical system is insensitive to misfocus-like aberrations, the size and shape of the point spread function (PSF) is compact and constant as a function wavelength, field angle, object position, etc, and the resulting MTF has high values. The following constant profile path optics provide efficient designs for use in such systems. 
     Certain constant profile path optics are based on parametrically efficient descriptions (e.g., low number of parameters) of optical form; such form may for example support high performance optical imaging systems with selected imaging and/or cost characteristics. In general, these parametrically efficient forms are not required. That is, the surface height of each part of the aspheric surface can be designated as an independent variable used in design and optimization; however, the number of variables to optimize this general case is extremely large and impractical. Constant profile path optics thus facilitate a powerful and general optical form for use in design and optimization. 
     In one embodiment, constant profile path optics include aspheric optical elements where the surface height is defined along paths and where the functional form, or profile, of the surface is the same along normalized version of the paths. The actual surface height varies from path to path, but the functional form or profile along each path does not. Such surface profiles on an optical element operate to modify phase of a wavefront within the optical system; image effects caused by these phase changes are then reversed in image processing, e.g., through operation of the filter kernel. By way of example, when certain constant profile path optics have modified wavefront phase within the optical imaging system, the resulting MTF at the intermediate image (captured by a detector) correlates to the functional and spatial characteristics of the filter kernel used in image processing. The constant profile path optics and the filter kernel may therefore be complementary to one another to provide the desired imaging characteristics. 
     Four examples  2600 A- 2600 D of constant profile path optics are shown in  FIG. 26 . Consider for example the paths within profile  2600 A. These paths are along square contours of the optics; for this optic form, the normalized surface height is the same over normalized versions of the square contours. The paths of profile  2600 B are pentagon-shaped; the functional form of the surface height along a normalized version of each pentagon is the same. The paths of profiles  2600 C and  2600 D are cross- and star-shaped, respectively. Similar to profiles  2600 A,  2600 B, such functional forms have common normalized surface heights along common paths. 
       FIG. 27  shows other variations  2700 A,  2700 B in constant profile path optics. Profile  2700 B represents wavefront coded optics where the paths traces can have closed contours, such as shown by region # 1 . The paths in profile  2700 A on the other hand trace open contours. In both profiles, region # 1  contains a set of related paths. The functional form of the surface height along a normalized version of each path in region # 1  is the same. The same is true for regions  2 ,  3 ,  4  and  5 . The actual surface height along each path within each region can be different. The functional forms over different regions can be related or not. The optics of profile  2700 B shows a combination of paths that are open and closed; the paths of region # 1  trace closed contours while the paths of the other four regions trace open contours. Accordingly, the functional form of the surface height along each path may be the same for each region, yet the actual surface height can change within the region. 
     As those skilled in the art will appreciate, one of several variations of the path profiles shown in  FIG. 26  and  FIG. 27  may include non-straight line paths. That is, straight line paths such as shown in profiles of  FIG. 26  and  FIG. 27  may take other forms wherein the paths are composed of curved segments. For example, the pentagon shaped form of profile  2700 A split into straight line regions may instead be of circular form with separate regions defining arcs as the paths. 
     The type of path traced in constant profile path optics can also change over different regions, as in  FIG. 28 . The paths in the outer region of profile  2800 A trace closed square contours, while its inner region paths trace closed circular contours. These inner and outer regions can be reversed, as shown by profile  2800 B. Only some of the paths in the outer region can be closed, as shown in  2800 B. 
       FIG. 29  shows two profiles  2900 A,  2900 B where at least one region of the optics is not altered with constant profile path optics. In profiles  2900 A,  2900 B, the paths form contours only in the outer region of the optics; the inner region has no paths and thus has no specialized surface shape. The optics of profiles  2900 A,  2900 B may be thought of as the optics associated with profiles  2600 A,  2600 B, respectively, but with an amplitude of zero applied to the paths in the inner region. 
     The parameters that describe the specialized surfaces of  FIG. 26-FIG .  29  may be considered as a composition of two components: 1) the parameters that describe the profile of the optical surface height along the paths of the particular optic and 2) the parameters that describe the surface height across the paths. The second set of components may for example describe an amplitude change for each path, and between each path, in a given region. If the amplitude of the set of paths in a region is set to zero, then optics as in  FIG. 29  may result. 
     In one embodiment, the mathematical description of a surface of one region of a constant profile path optic is described as:
 
 S ( R ,theta, a,b )= C ( a ) D ( b )
 
where the optical surface along each path in the region is parameterized by C(a), and evaluated at D(b) for the particular path in the region. Parameters ‘a’ and ‘b’ define the characteristics of the particular surface. The contributions from C(a) are constant over all paths in a region. The contributions from D(b) change for each path in a region. Parameters C(a) may define the surface along each path and the overall surface modulated between or across the paths in a region of D(b). The optical surface of a constant profile path optics may also be separable in terms along the paths and across the paths for the particular optic.
 
     Consider the optical surface of  2600 A, where the paths define open sides of square contours. This leads to four regions, the left side, right side, top side, and bottom side. One example of a mathematical description of the surface profile along the paths of the four regions is:
 
 C ( a )= a 0 +a 1 x+a 2 x 2 + . . . . |x|&lt; 1
 
Thus a set of parameters ai form a polynomial description of the surface height along each normalized path of the four regions of the optics. In this case, the length of each path in the regions is normalized to unity so that a “stretching” of the fundamental surface profile C(a) is applied to each path. The path length does not have to be considered a normalized length, although it may be useful to do so.
 
     Each path can be modified by a gain or constant term without altering the functional form of the surface along each path. One example of the gain applied across the paths in a region can be mathematically described as
 
 D ( b )= b 0 +b 1(PathNumber)+ b 2(PathNumber)2 +b 3(PathNumber)3+ . . . PathNumber=0, 1, 2, 3, . . .
 
where the PathNumber parameter is a sequence of values that describes the set of paths in a region. The number of paths in a region may be large forming an essentially continuous surface. For example, the path of region # 1  (profile  2700 A) closest to the optical center can be assigned PathNumber=0, the adjacent path slightly farther from the optical center can be assigned PathNumber=1, the next adjacent path can be assigned PathNumber=2, and so on. The overall surface is then mathematically composed of the product of the “along the paths” surface description and the “across the paths” surface description. Mathematically, the surface description in one region may be given by:
 
     S(R,theta,a,b)=C(a)D(b), where the optical surface is now parameterized by the parameter vectors a and b. 
     If the functional form of the surface along the paths is a polynomial of second order, or less, then the second derivative along the normalized paths is a constant. In this special case, the constant profile path optics may for example be denoted as “constant power path optics.” Constant power path optics are particularly effective forms for the wavefront coded optics. In terms of optical parameters, a second order polynomial C(a) can be described as:
 
 C ( a )=thickness+tilt+optical power,
 
where the zero th  order term a 0  describes the amount of thickness, the first order term a 1 x describes the amount of tilt, and the second order term a 2 x 2  describes the amount of optical power. If higher order terms are used, then the amount of optical power, or second derivative, can change along the contour. Higher order terms can also more accurately model optical power since a second order polynomial is an approximation to a sphere.
 
     For certain optical imaging systems, such higher order terms can be important. If the tilt parameter is non-zero, then optical surfaces with discontinuities may be described. Given the high degree of accuracy that free-form surfaces can be fabricated today, surfaces with discontinuities need not be avoided. For certain optical systems, the across-the-paths term D is chosen so that the central region of the optical surface is flat (or nearly flat). Often the slope of D can become quite large near the edge of the surface. The central region may also process optical imaging rays that need not change in order to control misfocus aberrations, as rays outside of the central region typically cause the misfocus effects within imaging systems. 
     Since central rays contribute less to misfocus effects then the outer rays, the shape of the contours can change for the central and outer regions to affect these rays differently. This allows shaping of the in- and out-of-focus PSFs, high MTFs, and customization of the power spectrum for digital filtering. The power spectrum of the digital filter modifies the power spectrum of the additive noise after filtering. Noise reduction techniques after filtering are thus increasingly effective by jointly designing the optics and digital processing for noise reduction and imaging, such as described above. 
     Examples of Constant Profile Path Optics 
     Certain constant profile path optics examples described below follow the form S(R,theta,a,b)=C(a)D(b) where the particular paths used are those of profile  2600 A, with each side of the square contour defining one of four regions.  FIG. 30  shows one surface profile  3000 , its in-focus MTF  3002 , and along the path surface form C(a)  3004  and across the path amplitude D(b)  3006 . A third order polynomial has been used for across the path amplitude  3006  and a second order polynomial has been used for the along the path surface form  3004 . This type of optics has an approximately constant amount of optical power along the normalized square path contours. The surface profile  3000  and in-focus MTF  3002  have been drawn via contours of constant surface and MTF height. Notice that the constant height contours vary from roughly circular to rectangular along the diameter of the surface. The resulting in-focus MTF has non-circular contours that roughly follow a “+” shape. The functional form in waves along the paths and across the path amplitude is shown in the bottom graphs  3004 ,  3006 . About one wave of optical power is used along the sides of the paths (as shown by the minimum of 2.8 and a max of 3.8 waves) with an amplitude variation between −0.1 and −0.4 across the paths. 
     Another example of constant profile path optics is shown in  FIG. 31 . The polynomial form S(R,theta,a,b)=C(a)D(b) is used, with C(a) being a second order constant optical power polynomial and D(b) being a third order polynomial. Notice that the surface profile  3100  and MTF  3102  are very different from those of  FIG. 30 . The MTF  3102  for this example is more compact then MTF  3002  of  FIG. 30 , with asymmetry also less pronounced. About six waves of optical power are used along the sides of the path contours as shown in  3104 , with an increasing amplitude used across the paths having a maximum value of about 4. 
       FIG. 32  shows another example that follows the form of  FIG. 30  and  FIG. 31  with a second order constant optical power polynomial describing the functional form along the paths and a third order polynomial describing the across the path amplitude; except the form of the amplitude function changes to allow a flat center region. The surface profile  3200  has a four-sided nature with a flat center, while its resulting MTF  3202  is roughly a four sided pyramid. Such an MTF shape  3200  is suitable for non-uniform sampling, such as found on the green channel of three color Bayer color filter array detectors (see  FIG. 16-23 ). Less then one wave of optical power is used along the paths  3204  with an amplitude variation from 0 to −9 used across the paths  3206 . 
       FIG. 33  shows one example of a constant profile path optics with pentagon-shaped paths as in profile  2700 A,  FIG. 27 . For this example, the functional form of the surface is the same for each straight line segment within the five open pentagon path regions. A second order polynomial describes the surface along the paths ( 3304 ) while a third order polynomial describes the amplitude across the paths ( 3306 ). The optical surface (profile  3300 ) has ten unequally shaped lobes and a fairly flat central region. The corresponding MTF  3302  has near-circular form about the center of the 2D MTF plane. About 1.5 waves of optical power is used along the paths  3304  with an amplitude of zero to 18 used across the paths  3306 . 
       FIG. 34  shows the sampled PSFs from the example of  FIG. 32  without wavefront coding, and for a range of defocus values. The small squares in each mesh drawing of each PSF represent individual pixel samples (output from the detector). The physical parameters used to generate these PSFs are: 10 micron illumination wavelength; grayscale pixels with 25.4 micron centers and 100% fill factor; working F-number of the optics is 1.24; misfocus aberration coefficient W 20  varies from 0 to 2 waves. Accordingly, it is apparent that without wavefront coding, the sampled PSFs have an unavoidable large change in size due to misfocus effects. 
       FIG. 35  similarly shows the sampled PSFs from the example of  FIG. 32  but with wavefront coding. Notice that the sampled PSFs (for the range of defocus values) have a sharp spike and a broad pedestal that decreases with the distance from the sharp spike. Each of the PSFs is similar and essentially independent of misfocus. The small squares in the mesh drawing again represent individual pixels. 
       FIG. 36  and  FIG. 37  show and compare cross sections through sampled PSFs from  FIG. 34  and  FIG. 35 . The PSFs of  FIG. 36  and  FIG. 37  are shown with five misfocus values evenly spaced between 0 and 2 waves. The sampled PSFs in  FIG. 36  are normalized to constant volume, while the PSFs in  FIG. 37  are normalized for a constant peak value.  FIGS. 36 and 37  thus illustrate changes in the sampled PSFs of the system without wavefront coding and the lack of change in the sampled PSFs with wavefront coding. 
       FIG. 38  shows an example of one 2D digital filter (plotted as an image and based on zero misfocus PSF) that may be used to remove the wavefront coding blur from the sampled PSFs of  FIG. 35 . This digital filter is for example used within image processing (e.g., within section  240 ,  FIG. 2 ) as a filter kernel. Notice that this kernel has high values only near the center of the filter and decreasing values related to the distance from the center. The filter, and the sampled PSFs of  FIG. 35 , are spatially compact. 
     After using the 2D filter of  FIG. 38  on the sampled PSFs of  FIG. 35 , the PSFs of  FIG. 39  result. Notice that these PSFs (representing PSFs with wavefront coding and after filtering within an image processing section) have nearly ideal form and are essentially unchanged over the range of misfocus. 
       FIG. 40  illustrates an amount of power contained in a geometric concept known as “rank” for the in-focus sampled PSF and the digital filter of  FIG. 35  and  FIG. 38 , respectively. Rectangularly separable optics, such as a cubic described by P(x)P(y)=exp(j[X^3+Y^3]), forms a sampled PSF that is closely approximated as p(x)p(y), where the independent variables x and y are horizontal and vertical axes defined on a square grid. This separable nature allows rectangularly separable filtering that is computationally efficient. One drawback of rectangularly separable optics is that the PSF can spatially shift with misfocus. One useful feature of certain constant profile path optics is that they do not cause the PSF to spatially shift. Constant profile path optics may also produce PSFs, MTFs and corresponding digital filters that approximate low rank, to further facilitate efficient processing. The top plot of  FIG. 39  shows that there are only two geometric ranks that have appreciable value for this particular sampled PSF; this system is thus approximated by a rank two system. The corresponding digital filter for this example (the lower plot of  FIG. 39 ) is also low rank and can be made at least as low as the sampled PSF. In practice, the filtering of this example PSF may be accomplished with computationally efficient separable filtering. 
       FIG. 41  shows the corresponding MTFs before and after filtering and in the spatial frequency domain. For comparison purposes, also shown are the MTFs of the identical physical system with no wavefront coding. Both systems are shown with misfocus values from 0 to 2 waves in five steps. The MTFs of the system with no wavefront coding is seen to drastically change with misfocus. The MTFs of the system with the constant profile path optics before filtering has essentially no change with misfocus. The filtered MTFs result from filtering by the 2D digital filter of  FIG. 38 ; such MTFs have high values and degrade only for the largest misfocus value. As appreciated by those skilled in the art, other forms of constant profile path optics may control the MTF profile over larger amounts of misfocus by utilizing more processing capability and/or lower MTFs before filtering. 
     The mathematical form of the contour optics of  FIG. 32  may be as follows. The polynomial description of functional form along the paths in four regions:
 
 C ( x )=0.4661−0.7013 x^ 2 , |x|&lt; 1
 
     The polynomial description of across the path amplitude is:
 
 D ( y )=−1.8182+0.5170 y+ 2.520 y^ 2−10.1659 y^ 3, 0.25 &lt;y&lt; 1=−1.8182, 0 ≦y≦ 0.25.
 
In this example, the form along the path contours is given by an even second-order polynomial; and the amplitude across the paths is given by a third order polynomial that changes form so that the central region is relatively flat. Using higher order polynomials for along the paths and for the across the path amplitude supports higher performance then shown in these examples.
 
     Constant profile path optics may be designed with specialized techniques that allow optimization with non-ideal optics that can have large amounts of aberrations, vignetting, and loose tolerances. These techniques allow optical system design that is not practical by traditional analytical methods.  FIG. 42  illustrates a design process  4200  that illustrates certain of these techniques. 
     In process  4200 , optics are modified (in design) and the loop may repeat, as shown. For example, process  4200  includes step  4202  to design constant profile path optics. With a model for the optical surfaces (step  4202 ), the effects of the lens aberrations can be added (step  4204 ) with information about the digital detector (step  4206 ) in order to accurately simulate the sampled PSF and MTF. These lens aberrations are in general a function of field angle, wavelength, object position and zoom position; the aberrations can also be a function of the particular surface form of the constant profile path optic (step  4202 ). One aberration considered in design process step  4204  is vignetting. While vignetting is often used in design of traditional optics to trade off light gathering for sharpness, vignetting within wavefront coding optics, can lead to a poor match between optical design and the actual optical system. The optical surfaces and aberrations that lead to sampled PSFs can be either simulated through ray-based methods or Fourier Transform methods depending on the speed of the lens being designed and the spatial resolution of the digital detector. Both of these general types of PSF simulation methods are well known to those skilled in the art of optical simulation. 
     After the sampled PSFs and MTFs have been simulated (steps  4204 ,  4206 ), a digital filter is used (step  4208 ) to remove wavefront coding blur. This digital filter can be general and calculated for each iteration in design process  4200  or can be fixed or limited in form. An example of a limited form digital filter is a rectangularly separable filter. If the digital filter is limited in this way, the design of the constant profile path optics may be optimized for minimum rank PSFs and MTFs since the rank of the separable filter is 1. Other limited digital filters are filters where costs are assigned to particular filter values and sequences of filter values in order to optimize the implementation (e.g., hardware) of the image processing section. These costs may be reduced or increased as part of design process  4200 . Further examples of limited digital filters are those that have particular power spectrum characteristics. Since the additive noise is modified by the power spectrum of the digital filter, controlling the characteristics of the filter controls the characteristics of the additive noise after digital filtering. Noise Reduction techniques can be optimized jointly with the characteristics of the noise by limited the digital filter. 
     After the sampled PSFs/MTFs have been filtered (step  4208 ), a quality assessment is performed (step  4210 ). Assessment  4210  is typically system and application specific but may include (a) the quality of the filtered PSFs/MTFs both within and outside of the design range and/or (b) the characteristics of the digital filter (and/or its implementation and/or noise effects). The quality assessment is then used with non-linear optimization to modify the optics and repeat the iteration through process  4200 . The modified optics can include particular surfaces that contain the wavefront coding as well as other surfaces, and/or thickness and distances of elements within the optical system. The parameters of the functional form along the paths and across the path amplitude can be similarly optimized. 
     Changes may be made in the above methods and systems without departing from the scope hereof. It should thus be noted that that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. For example, those skilled in the art should appreciate that although the wavefront coded element is often shown separate from the optics within an imaging system (e.g., element  210  separate from optics  201 ), these components may be combined as a single item or group of items without departing from the scope hereof, for example as shown in  FIG. 24 . The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall there between.