Patent Publication Number: US-2015070562-A1

Title: Generalized assorted pixel camera systems and methods

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application is a continuation of U.S. patent application Ser. No. 12/736,333, filed May 11, 2011, which is the United States National Phase Application under 35 U.S.C. §371 of International Application No. PCT/US2009/038510, which claims the benefit of U.S. Provisional Patent Application No. 61/072,301, filed Mar. 28, 2008 and U.S. Provisional Patent Application No. 61/194,725, filed Sep. 30, 2008. Each of the above-referenced patent applications is hereby incorporated by reference herein in its entirety. 
    
    
     NOTICE CONCERNING COLOR DRAWINGS 
     It is noted that the patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawings will be provided by the Office upon request and payment of the necessary fee. 
     Nonetheless, because some readers will not have the color drawings available, the description will also endeavor to describe the drawings and the images they depict in a color-neutral manner, which may create apparent redundancies of description. 
     COPYRIGHT NOTICE 
     A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever. 
     TECHNICAL FIELD 
     The disclosed subject matter relates to generalized assorted pixel camera systems and methods. 
     BACKGROUND 
     Most digital cameras and camcorders have a single image sensor, such as a charge coupled device (CCD) image sensor or a complementary metal-oxide semiconductor (CMOS) image sensor. These image sensors use a color filter array or mosaic, which is an assortment of different spectral filters, formed in front of the CCD or CMOS image sensor for color acquisition. 
     A commonly-used color filter array or mosaic is the Bayer mosaic shown in  FIG. 1 . As shown, the Bayer mosaic includes color filters of the three primary colors red (R), green (G), and blue (B), where the green (G) color filters are arranged in a checkerboard pattern and the red (R) and blue (B) color filters are arranged in line sequence. One reason tri-chromatic filter arrays are used is that tri-chromatic sensing is near-sufficient in terms of colorimetric color reproducibility. It is also commonly assumed that this pixel assortment is the only practical approach for sensing color information with a semiconductor image sensor. However, the Bayer mosaic is limited in its capacity because it provides a limited amount of spectral information. That is, the Bayer mosaic provides spectral information for the three colors red (R), green (G), and blue (B). In addition, while interpolation and other techniques are available to fill in missing spectral information, these approaches typically provide a resulting image showing color aliasing and other artifacts. For example,  FIG. 2  shows the differences between a ground truth image  210  and an image  220 , which suffers from color aliasing and other artifacts resulting from a bicubic interpolation applied to signals captured using the Bayer mosaic. Dashed region  222  identifies the portion of image  220  that is affected by color aliasing and other artifacts. 
     In recent years, new image sensing technologies have emerged that use pixel assortments to enhance image sensing capabilities. For high dynamic range (HDR) imaging, a mosaic of neutral density filters with difference transmittances has been used. This approach to high sensitivity imaging builds upon the standard Bayer mosaic by using panchromatic pixels that collect a significantly larger proportion of incident radiation. 
     Despite these advances, the previously described mosaics and camera systems have limitations. For example, these mosaics and camera systems are used to generate one specific type of output image. 
     Accordingly, it is desirable to provide generalized assorted pixel camera systems and methods that overcome these and other deficiencies of the prior art. 
     SUMMARY 
     In accordance with various embodiments, generalized assorted pixel camera mechanisms are provided. In some embodiments, generalized assorted pixel camera systems and methods are provided that use a color filter array or mosaic with a rich assortment of color filters, such as the one shown in  FIG. 4 . A color filter array is used for an imaging or camera system in which one of a plurality of filters having different color separation characteristics (or colors) is bonded to each pixel. Each of the color filters in the color filter array can enhance a particular attribute of image quality. These attributes include, for example, color reproduction, spectral resolution, dynamic range, and sensitivity. By using the information captured by each of the filters in the color filter array, these generalized assorted pixel camera mechanisms allow a user to create a variety of image types (e.g., a monochrome image, a high dynamic range (HDR) monochrome image, a tri-chromatic (RGB) image, a HDR RGB image, and/or a multispectral image) from a single captured image. 
     In some embodiments, these mechanisms can provide an approach for determining the spatial and spectral layout of the color filter array, such as the one shown in  FIG. 4 . For example, generalized assorted pixel camera systems and methods are provided that use a cost or error approach to balance variables relating to colorimetric and spectral color reproduction, dynamic range, and signal-to-noise ratio (SNR). 
     In some embodiments, these mechanisms can provide a demosaicing approach for reconstructing the variety of image types. For example, generalized assorted pixel camera systems and methods are provided that include submicron pixels and anti-aliasing approaches for reconstructing under-sampled channels. In particular, information from particular filters is used to remove aliasing from the information captured by the remaining filters. 
     It should be noted that these mechanisms can be used in a variety of applications. For example, these mechanisms for enhancing spatial and spectral layout of a color filter array can be used in a generalized assorted pixel camera system. The camera system can capture a single image and, using the information from each of the filters in the color filter array, to balance or trade-off spectral resolution, dynamic range, and spatial resolution for generating images of multiple image types. These image types can include, for example, a monochrome image, a high dynamic range (HDR) monochrome image, a tri-chromatic (RGB) image, a HDR RGB image, and/or a multispectral image) from a single captured image. 
     In accordance with some embodiments, a color filter array is provided, the array comprising: a plurality of primary filters and a plurality of secondary filters, wherein each filter has a particular spectral response and each filter is formed on a corresponding pixel of a plurality of pixels; and wherein each of the plurality of primary filters and the plurality of secondary filters enhances an attribute of image quality and wherein the information obtained using the plurality of primary filters and the plurality of secondary filters is used to balance spatial resolution and image quality for generating an image of a plurality of image types. 
     In accordance with some embodiments, a method for generating images is provided, the method comprising: providing a color filter array, the color filter array comprising: a plurality of primary filters and a plurality of secondary filters, wherein each filter has a particular spectral response and each filter is formed on a corresponding pixel of a plurality of pixels; and wherein each of the plurality of primary filters and the plurality of secondary filters enhances an attribute of image quality and wherein the information obtained using the plurality of primary filters and the plurality of secondary filters is used to balance spatial resolution and image quality for generating an image of a plurality of image types; capturing an image using the color filter array, wherein information from the plurality of primary filters and the plurality of secondary filters corresponding to the image is obtained; and generating the image in a plurality of image types using the information from the plurality of primary filters and the plurality of secondary filters. 
     In accordance with some embodiments, a camera system is provided, the system comprising: a color filter array, the color filter array comprising: a plurality of primary filters and a plurality of secondary filters, wherein each filter has a particular spectral response and each filter is formed on a corresponding pixel of a plurality of pixels; and wherein each of the plurality of primary filters and the plurality of secondary filters enhances an attribute of image quality and wherein the information obtained using the plurality of primary filters and the plurality of secondary filters is used to balance spatial resolution and image quality for generating an image of a plurality of image types. 
     In some embodiments, an image processing system is provided, the system comprising: a processor that is configured to: receive information corresponding to an image from a color filter array, wherein the color filter array includes a plurality of primary filters and a plurality of secondary filters, wherein each filter has a particular spectral response and each filter is formed on a corresponding pixel of a plurality of pixels and wherein each of the plurality of primary filters and the plurality of secondary filters enhances an attribute of image quality and wherein the information obtained using the plurality of primary filters and the plurality of secondary filters is used to balance spatial resolution and image quality for generating an image of a plurality of image types; and generate the image in a plurality of image types using the information from the plurality of primary filters and the plurality of secondary filters. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates an example of a Bayer mosaic. 
         FIG. 2  illustrates the differences between a ground truth image and an image showing color aliasing and other artifacts resulting from a bicubic interpolation applied to signals captured using the Bayer mosaic of  FIG. 1  in accordance with some embodiments of the disclosed subject matter. 
         FIG. 3  illustrates the Modulation Transfer Function (MTF) calculated for various pixel sizes in accordance with some embodiments of the disclosed subject matter. 
         FIG. 4  illustrates a color filter array or arrangement in accordance with some embodiments of the disclosed subject matter. 
         FIG. 5  illustrates the Nyquist or usable frequency region of the color filter array shown in  FIG. 4  and the optical resolution limit for submicron pixels in accordance with some embodiments of the disclosed subject matter. 
         FIG. 6  illustrates the spectral responses of the seven enhanced filters (e.g., filters a, b, c, d, e, f, and g) in the color filter array of  FIG. 4  in accordance with some embodiments of the disclosed subject matter. 
         FIG. 7  is a schematic diagram of a system for creating multiple image types from a single captured image using a camera system with a color filter array, such as the one shown in  FIG. 4 , in accordance with some embodiments of the disclosed subject matter. 
         FIG. 8  illustrates examples of low exposure RGB images calculated from the secondary filters through the anti-aliasing approach in accordance with some embodiments of the disclosed subject matter. 
         FIG. 9  illustrates an original image of a Circular Zone Plate (CZP) and multiple images generated from a single captured image of the original image in accordance with some embodiments of the disclosed subject matter. 
         FIG. 10  illustrates the Modulation Transfer Function (MTF) calculated for the generated images shown in  FIG. 9  in accordance with some embodiments of the disclosed subject matter. 
         FIGS. 11 and 12  illustrate additional examples of images generated from a single captured image in accordance with some embodiments of the disclosed subject matter. 
         FIG. 13  illustrates an 8×8 color filter array that includes five different color filters (Green (G), Red (R), Blue (B), Yellow (Y), and Emerald (E)), where each color filter has two exposures (a bright exposure and a dark exposure), in accordance with some embodiments of the disclosed subject matter. 
         FIG. 14  illustrates the Nyquist or usable frequency region of the color filter array shown in  FIG. 13  in accordance with some embodiments of the disclosed subject matter. 
         FIG. 15  illustrates a 9×9 color filter array that includes five different color filters (Green (G), Red (R), Blue (B), Yellow (Y), and Emerald (E)) in accordance with some embodiments of the disclosed subject matter. 
         FIG. 16  illustrates the Nyquist or usable frequency region of the color filter array shown in  FIG. 15  in accordance with some embodiments of the disclosed subject matter. 
         FIG. 17  illustrates a directional smoothing approach that can be used to reduce aliasing effects for images generated using the color filter arrays shown in  FIGS. 13 and 15  in accordance with some embodiments of the disclosed subject matter. 
         FIG. 18  illustrate additional examples of images generated from a single captured image using a camera system having one of the color filter arrays shown in  FIGS. 13 and 15  in accordance with some embodiments of the disclosed subject matter. 
     
    
    
     DETAILED DESCRIPTION 
     In accordance with various embodiments, generalized assorted pixel camera mechanisms are provided. In some embodiments, generalized assorted pixel camera systems and methods are provided that use a color filter array or mosaic with a rich assortment of color filters, such as the one shown in  FIG. 4 . A color filter array is used for an imaging or camera system in which one of a plurality of filters having different color separation characteristics (or colors) is bonded to each pixel. Each of the color filters in the color filter array can enhance a particular attribute of image quality. These attributes include, for example, color reproduction, spectral resolution, dynamic range, and sensitivity. By using the information captured by each of the filters in the color filter array, these generalized assorted pixel camera mechanisms allow a user to create a variety of image types (e.g., a monochrome image, a high dynamic range (HDR) monochrome image, a tri-chromatic (RGB) image, a HDR RGB image, and/or a multispectral image) from a single captured image. 
     In some embodiments, these mechanisms can provide an approach for determining the spatial and spectral layout of the color filter array, such as the one shown in  FIG. 4 . For example, generalized assorted pixel camera systems and methods are provided that use a cost or error approach to balance variables relating to colorimetric and spectral color reproduction, dynamic range, and signal-to-noise ratio (SNR). 
     In some embodiments, these mechanisms can provide a demosaicing approach for reconstructing the variety of image types. For example, generalized assorted pixel camera systems and methods are provided that include submicron pixels and anti-aliasing approaches for reconstructing under-sampled channels. In particular, information from particular filters is used to remove aliasing from the information captured by the remaining filters. 
     It should be noted that these mechanisms can be used in a variety of applications. For example, these mechanisms for enhancing spatial and spectral layout of a color filter array can be used in a generalized assorted pixel camera system. The camera system can capture a single image and, using the information from each of the filters in the color filter array, balance or trade-off spectral resolution, dynamic range, and spatial resolution for generating images of multiple image types. These image types can include, for example, a monochrome image, a high dynamic range (HDR) monochrome image, a tri-chromatic (RGB) image, a HDR RGB image, and/or a multispectral image) from a single captured image. 
     In some embodiments, generalized assorted pixel camera mechanisms with an image sensor having submicron pixels are provided. Generally speaking, it has been determined that the resolution performance of an imaging sensor with submicron pixels exceeds the optical resolution limit. 
     To fabricate such a camera system, it should be noted that the resolution of an optical imaging system can be limited by multiple factors, such as diffraction and aberration. While aberrations can be corrected during lens design, diffraction is a limitation that cannot be avoided. The two-dimensional diffraction pattern of a lens with a circular aperture is generally referred to as the Airy disk, where the width of the Airy disk determines the maximum resolution limit of the system. This is generally defined as: 
         I (θ)= I   0 {2 J   1 ( z )/ z}   2 ,
 
     where I 0  is the intensity in the center of the Airy diffraction pattern, J 1  is the Bessel function of the first kind of order one, and θ is the angle of observation (i.e., the angle between the axis of the circular aperture and the line between the aperture center and observation point). It should be noted that z=πq/λN, where q is the radial distance from the optical axis in the observation plane, λ is the wavelength of the incident light, and N is the f-number of the system. In the case of an ideal lens, this diffraction pattern is the Point Spread Function (PSF) for an in-focus image and the Fourier transformation of the PSF is used to characterize the resolution of an optical imaging system. This quantity is generally referred to as the Modulation Transfer Function (MTF). The MTF of such an imaging system can be calculated directly from the wavelength λ of incident light and the f-number N. This is denoted by MTF opt  (λ, N)=F(I(θ)), where F(•) denotes the Fourier transformation. 
     It should be noted that pixels generally have a rectangular shape and their finite size contributes to the resolution characteristics of the imaging system. The Modulation Transfer Function (MTF) of an image sensor can be approximated as the Fourier transformation of a rectangular function, which is described by MTF sensor (p)=F(s(t)). The rectangular function s(t) can be expressed as: 
     
       
         
           
             
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     where p is the pixel size and ζ is an aperture ratio, which is generally assumed to be 1 due to the use of on-chip microlenses. 
     It should also be noted that the total fundamental optical resolution limit of a camera system (e.g., including the lens and the sensor) can be described in the frequency domain as MTF=MTF opt  (λ, N)·MTF sensor (p). To calculate this, the values of λ=555 nm (which generally corresponds to the peak of the sensitivity of the human eye) and N=f/5.6 (which is a pupil size generally used in, for example, consumer photography) are used. With these values, the fundamental MTF is determined by pixel size p. 
     The MTF for various pixel sizes is shown in  FIG. 3 . As shown, the MTF is calculated for pixel sizes of 0.70 μm (represented by the leftmost curve  305 ), 1.00 μm (represented by curve  310 ), 1.25 μm (represented by curve  315 ), and 1.75 μm (represented by the rightmost curve  320 ). The MTF with a pixel size p=1.00 μm is about 0.1 at 0.25 fs, where fs is the image sensor&#39;s sampling frequency. It should be noted that the human eye cannot recognize contrast when the MTF is less than 0.1. It should also be noted that the optical resolution limit of an image sensor with p=1.00 μm pixel size is half of the image sensor&#39;s Nyquist frequency. Accordingly,  FIG. 3  shows that the resolution performance of a sensor with submicron pixels exceeds the optical resolution limit. 
     In some embodiments, generalized assorted pixel camera systems and methods are provided that use a color filter array or mosaic with a rich assortment of color filters. Again, as shown in  FIG. 3 , for a 1.0 μm pixel size, the combined MTF, due to diffraction from the lens aperture and averaging by pixels, leads to an optical resolution limit of about one-quarter of the sampling frequency fs, where fs=1/Δs and Δs is the sampling pitch. To exploit this, an exemplary color filter array or arrangement  400  in accordance with some embodiments is shown in  FIG. 4 . As described previously, a color filter array is used for an imaging or camera system in which one of a plurality of filters having different color separation characteristics (or colors) is bonded to each pixel. It should be noted that the term “color” generally refers to a filter or a pixel value of that color obtained from the filter. As shown in  FIG. 4 , color filter array  400  includes primary filters (i.e., color filters a, b, and c) and secondary filters (i.e., color filters d, e, f, and g). 
     The pixels marked a, b, and c in color filter array  400  (collectively referred to herein as “primary filters”) capture three different spectral images on a rectangular grid with sampling pitch Δs a,b,c =2p. Accordingly, the Nyquist frequency for a, b, and c is fn a,b,c =fS a,b,c /2=fs/4. It should be noted that, due to diffraction, filters a, b, and c do not cause aliasing because the optical resolution limit is one-quarter of the sampling frequency fs. These aliasing-free pixels a, b, and c can be used to reconstruct high resolution images, such as high resolution monochrome images and high resolution RGB images. 
     The pixels marked d, e, f, and g in color filter array  400  (collectively referred to as “secondary filters”) each sample the incident image on rectangular grids through different spectral filters. The sampling pitch for each of the secondary filters is Δs d,e,f,g =4p and the Nyquist frequency is fn d,e,f,g =fs d,e,f,g /2=fs/8. 
     To further illustrate the Nyquist frequencies of color filter array  400 , the Nyquist or usable frequency region  500  is shown in  FIG. 5 . As shown, the Nyquist region of the primary filters is located in a substantially square portion of the frequency space indicated by dashed line  510  and the Nyquist region of the secondary filters is located in a substantially square portion of the frequency space indicated by a dashed line  520 . 
     In addition,  FIG. 5  also illustrates the optical resolution limit for submicron pixels, as described above, that is shown by shaded area  530 . It should be noted that, as the Nyquist frequencies for the secondary filters (indicated by dashed line  520 ) are lower than the optical resolution limit, the possibility of aliasing artifacts is introduced. However, as shown herein, these aliasing artifacts can be removed by using high frequency information from the demosaiced image obtained using the primary filters. 
     Using color filter array  400 , a plurality of image characteristics can be captured simultaneously. It should be noted, however, that there may be a trade-off in the fidelity of each characteristic. For example, monochrome and standard RGB images are reconstructed at high resolution using the primary filters of color filter array  400 . For high dynamic range (HDR) images, the spectral resolution is improved by using the secondary filters and decreasing the spatial resolution. 
     In another example, high spatial resolution can be obtained by sacrificing the dynamic range and the spectrum. That is, a monochrome image has high spatial resolution. By sacrificing the spatial resolution, quality of the spectrum is improved. By further sacrifice of the resolution, dynamic range is expanded in addition to the improvement of the spectrum. 
     In some embodiments, a cost or error function can be used to enhance the filter spectra for the primary and secondary filters. The cost function can incorporates several terms, such as the quality of color reproduction (e.g., for a RGB image), reconstruction of reflectance (e.g., for a multispectral image), and dynamic range (e.g., for a HDR image). 
     The value x m  measured at a pixel in the m th  channel, where m is one of the primary or secondary filters a, b, c, d, e, f, or g, is given by the following equation: 
         x   m =∫ λ     min     λ     max     i (λ) r (λ) c   m (λ) dλ,  
 
     where i(λ) is the spectral power distribution of the illumination, r(λ) is the spectral reflectance of the scene point, and c m (λ) is the spectral response of the camera&#39;s m th  color channel. When the wavelength λ is sampled at equally-spaced L points, x m  can be described by the following discrete expression: 
     
       
         
           
             
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     Moreover, if the above-mentioned equation is rewritten in matrix form, it can be described as X=C T IR, where X=[x a , x b , . . . x g ] T , C=[c m (λ l )], I is a diagonal matrix made up of the discrete illumination samples i(λ l ), and R=[r(λ l )]. 
     In some embodiments, the color reproduction error corresponding to the primary and secondary filters can be determined. For example, to obtain HDR RGB images, a high exposure RGB image can be reconstructed using the primary filters of color filter array  400  and a lower exposure image can be reconstructed using the secondary filters of color filter array  400 . In some embodiments, the spectral responses of the primary and secondary filters are to yield the highest color reproduction. It should be noted that a variety of color rating indicies can be used to evaluate the color reproduction characteristics of a filter and these indicies can use a cost function that minimizes the difference in the color between the measured color of a reference material and its known color. 
     In some embodiments, to calculate the difference of color, the CIE 1931 XYZ color space (created by the International Commission on Illumination), which is based on direct measurements of human visual perception and serves as the basis of which many other color spaces are defined, can be used. The calculation of sRGB tristimulus values (which are employed in some digital cameras or color monitors) from the CIE XYZ tristimulus values is a linear transformation. The CIE XYZ tristimulus values can be defined as Y=A T IR, where Y represents the true tristimulus values and A is a matrix of CIE XYZ color matching functions [  x   y   z ]. The estimated CIE tristimulus values corresponding to the primary filters Ŷ′ can be expressed as an optimal linear transformation: Ŷ′=T′X′, where X′=[x a , x b , x c ] T . The transformation T′ is determined so as to minimize the color difference: min∥Y−T′X′∥ 2 . Similarly, the estimated CIE tristimulus values corresponding to the secondary filters Ŷ″ can be expressed as Ŷ″=T″X″, where X″=[x d , x e , x f , x g ] T . 
     It should be noted that the average magnitude of color difference between the true color Y and the estimate Ŷ over a set of N real-world objects can be used as a metric to quantify the camera system&#39;s color reproduction performance. The color reproduction errors corresponding to the primary and secondary filters can then be described by the following equations: 
     
       
         
           
             
               
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     In some embodiments, the error introduced by the reconstruction of the spectral distribution can be determined. For example, the spectral distribution can be reconstructed using a linear model. Since the model is linear, the reconstruction is efficient and stable. The linear model for the reconstruction can be expressed as the set of orthogonal spectral basis functions b k (λ): 
         r (λ)=Σ k=1   K σ k   b   k (λ),
 
     where σ k  are scalar coefficients and K is the number of basis functions. By substituting the above-described equation into the cost function, the cost or error function can be described by the following equation: 
     
       
         
           
             
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     These equations can be written as X=F·σ, where F is a M×K matrix: F=∫ λ     min     λ     max   b k (λ)i(λ)c m (λ)dλ, M is the number of color filter channels (for example, color filter array  400  of  FIG. 4  has seven channels, so, M=7), and σ=[σ k ]. The spectral distribution is reconstructed by minimizing ∥F·σ−X∥ 2 . It should be noted that the spectral reflectance of most materials is known to be smooth and is generally positive. Accordingly, the reconstruction approach can be expressed as a constrained minimization as follows: {circumflex over (σ)}=arg min∥{tilde over (F)}·σ−{tilde over (X)}∥ 2 , subject to B·σ≧0, where {tilde over (F)}=[F T αP T ] T , P lk =∂ 2 b k (λ l )/∂λ 2  is a smoothness constraint, α is a smoothness parameter, 1≧L, 1≧k≧K, {tilde over (X)}=[X T 0] T , and B=[b k (λ l )]. This regularized minimization can be solved using quadratic programming. The multispectral image&#39;s mean squared reconstruction error R(C) can then be expressed as: 
     
       
         
           
             
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     where σ n  represents the actual coefficients of the n th  object and {circumflex over (σ)} n  are the reconstructed coefficients. It should be noted that, in some embodiments, the number of basis functions K is 8 and the smoothness parameter α is set to 64.0. 
     In some embodiments, the cost function can include an approach for balancing the extension of dynamic range with signal-to-noise (SNR) ratio. As described previously, to achieve HDR imaging, secondary filters (e.g., filters d, e, f, and g of color filter array  400 ) have lower transmittances than the primary filters (e.g., filters a, b, and c of color filter array  400 ). This can cause deterioration of signal-to-noise ratio (SNR) for the secondary filters. Such a trade-off can be controlled based on the ratio of the exposures of the primary and secondary filters: β=e max /e min , where e max  is the average exposure of the primary filters and e min  is the average exposure of the secondary filters. Accordingly, β can be determined by C from the previously-mentioned equation X=C T IR, where the determined value of β can be used to valance the extension of dynamic range versus the reduction of the signal-to-noise ratio. 
     In some embodiments, dynamic range can be defined as: DR=20 log 10 B full /N r , where V full  represents the full-well capacity of the detector (e.g., V full =3500e − ) and N r  is the root mean square (RMS) of the read-noise of the image sensor. The RMS of the read-noise of the detector can be defined as N r =√{square root over (N shot   2 +N dark   2 )}. For example, N dark  can be set to 33e − . In some embodiments that use the color filter array  400  of  FIG. 4 , it should be noted that N r  does not change, but the maximum detectable gray level becomes βV full . Accordingly, the dynamic range of a camera system using color filter array  400  can be expressed as follows: 
     
       
         
           
             
               DR 
               GAP 
             
             = 
             
               20 
                
               
                   
               
                
               
                 log 
                 10 
               
                
               
                 
                   β 
                    
                   
                       
                   
                    
                   
                     V 
                     full 
                   
                 
                 
                   N 
                   r 
                 
               
             
           
         
       
     
     In some embodiments, the signal-to-noise ration can be defined as: SNR=20 log 10 V/N, where V is the signal and N is the noise. The signal corresponding to a secondary filter can be express using the exposure β as V″ max =V′ max /β, where V′ max  is a signal due to a primary filter. When the signal due to the primary filter is not saturated, the signal due to the secondary filter can be determined from the primary signal. The signal-to-noise ratio for a secondary filter when the primary signal is saturated is the worst-case signal-to-noise ratio for a camera system using mosaic  400 : 
     
       
         
           
             
               S 
                
               
                   
               
                
               N 
                
               
                   
               
                
               
                 R 
                 GAP 
               
             
             = 
             
               20 
                
               
                   
               
                
               
                 log 
                 10 
               
                
               
                 
                   
                     V 
                     full 
                   
                    
                   
                     / 
                   
                    
                   β 
                 
                 
                   N 
                   max 
                 
               
             
           
         
       
     
     where N max =√{square root over (N″ shot   2 +N dark   2 )} and N″ shot =√{square root over (V full /β)}. 
     Because the camera system has a high performance in signal-to-noise ratio and dynamic range when SNR GAP  and DR GAP  are large, the following cost function can be used: 
     
       
         
           
             
               D 
                
               
                 ( 
                 C 
                 ) 
               
             
             = 
             
               
                 1 
                 
                   DR 
                   GAP 
                 
               
               · 
               
                 1 
                 
                   S 
                    
                   
                       
                   
                    
                   N 
                    
                   
                       
                   
                    
                   
                     R 
                     GAP 
                   
                 
               
             
           
         
       
     
     In some embodiments, each of the above-mentioned cost functions can be combined to provide a total cost function. For example, since each of the above-mentioned cost functions represent a particular dimension of image quality, the total cost function can be expressed as a weighted sum of the individual costs: 
     
       
      
       G=w 
       1 
       {E′+E″}+w 
       2 
       R+w 
       3 
       D  
      
     
     It should be noted that the weights (e.g., w 1 , w 2 , and w 3 ) can be determined according to the image quality requirements of the application for which the camera system is used or manufactured. For example, in some embodiments, w 1 =1.0, w 2 =1.0, and w 3 =1.0 can be used for determining the total cost function. It should also be noted that, since the filters have positive spectral responses (C is to be positive), the enhancement or optimization of C can be expressed as: 
     
       
         
           
             
               C 
               = 
               
                 arg 
                  
                 
                     
                 
                  
                 
                   
                     min 
                     C 
                   
                    
                   
                       
                   
                    
                   G 
                 
               
             
             , 
             
               
                 subject 
                  
                 
                     
                 
                  
                 to 
                  
                 
                     
                 
                  
                 C 
               
               ≥ 
               0 
             
           
         
       
     
     In some embodiments, initial guesses can be assigned to the filter spectral responses. That is, to find the seven spectral response functions in C, initial guesses can be used along with an optimization approach. In one example, the initial guesses for the filter responses can be selected from a set of commercially available optical band pass filters and on-chip filters. In another example, commercial filters can be assigned to each of the seven channels based on one or more of the above-mentioned cost functions (e.g., assigning from a set of 177 commercial filters based on color reproduction error). Accordingly, the primary filters C′ 0  and secondary filters C″ 0  are determined such that: 
     
       
         
           
             
               
                 
                   
                     min 
                     
                       C 
                       0 
                       ′ 
                     
                   
                    
                   
                       
                   
                    
                   
                     E 
                      
                     
                       ( 
                       
                         C 
                         0 
                         ′ 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     
                       C 
                       0 
                       ′ 
                     
                     ∈ 
                     
                       C 
                       0 
                     
                   
                   ) 
                 
               
             
             
               
                 
                   
                     min 
                     
                       C 
                       0 
                       ″ 
                     
                   
                    
                   
                       
                   
                    
                   
                     E 
                      
                     
                       ( 
                       
                         C 
                         0 
                         ″ 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     
                       C 
                       0 
                       ″ 
                     
                     ∈ 
                     
                       C 
                       0 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     where C 0  is the set of commercial filters. 
     In response to assigning seven initial guesses to each of the seven filters, an iterative application can be used to perform a constrained non-linear minimization of 
     
       
         
           
             C 
             = 
             
               arg 
                
               
                   
               
                
               
                 
                   min 
                   C 
                 
                  
                 
                     
                 
                  
                 
                   G 
                   . 
                 
               
             
           
         
       
     
     For example, Mathworks® Matlab® or any other suitable computing program can be used to determine the spectral responses. Using Matlab®, the FMINCON routine can be used to find a minimum of a constrained non-linear multivariable function as described above. However, any other suitable computer program can be used to find the minimum of a constrained non-linear multivariate function. 
       FIG. 6  illustrates the spectral responses of the seven enhanced filters in color filter array  400  of  FIG. 4 . By using the cost function to determine the spectral responses and as a result of the color reproduction term in the cost function, it should be noted that the primary filters a, b, and c represented by curves  605 ,  610 , and  615 , respectively, have spectral responses substantially similar to red, green, and blue filters. Accordingly, the primary filters can be used to obtain RGB images, which essentially cover the entire visible light spectrum. 
     In addition, the spectra captured by the secondary filters d, e, f, and g (represented by curves  620 ,  625 ,  630 , and  635 , respectively), irrespective of their spectral responses, are to be highly correlated with the images obtained using the primary filters. Consequently, anti-aliasing of images produced by secondary filters can be performed. Furthermore, due to the characteristics of the cost function, the secondary filters have lower exposures or transmittances than primary filters. Accordingly, using the primary and secondary filters, high dynamic range information can be obtained and, since the seven filters have different spectra and sample the visible spectrum, their reconstructed images can be used to obtain smooth estimates of the complete spectral distribution of each scene point i.e., a multispectral image. 
     As shown in Table 1 below, the errors in the color reproduction and spectral reconstruction components of the total cost function, the estimated dynamic range, and the signal-to-noise ratio of the initial and enhanced set of seven filters of color filter array  400 . In addition, Table 1 also illustrates the errors in the color reproduction and spectral reconstruction components of the total cost function, the estimated dynamic range, and the signal-to-noise ratio for the red, green, and blue filters in a Bayer mosaic. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Optimization accuracy 
               
            
           
           
               
               
               
               
            
               
                   
                 Initial 
                 Enhanced 
                 Bayer 
               
               
                   
                 filters 
                 filters 
                 filters 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 ΔE′(C) 
                 0.0497 
                 0.0429 
                 0.0490 
               
               
                   
                 ΔE″(C) 
                 0.0100 
                 0.0055 
                 N/A 
               
               
                   
                 ΔR(C) 
                 0.0624 
                 0.0610 
                 0.0709 
               
               
                   
                 DR GAP   
                 58.2970 
                 62.9213 
                 56.9020 
               
               
                   
                 SNR GAP   
                 34.7069 
                 32.3694 
                 35.4098 
               
               
                   
                   
               
            
           
         
       
     
     It should be noted that, in response to enhancing the spectral responses of the filters in the generalized assorted pixel color filter array using a cost function, each of the errors in Table 1 have been reduced. It should also be noted that the deterioration of the signal-to-noise ratio is kept low at about 2.3 dB, while the dynamic range is improved by about 4.6 dB. It should further be noted that the errors in color reproduction and spectral reconstruction components of the total cost function are higher with the Bayer mosaic. 
       FIG. 7  shows a schematic diagram of a system  700  for creating multiple image types from a single captured image using a camera system with a color filter array in accordance with some embodiments of the disclosed subject matter. 
     As shown in  FIG. 7 , camera system  700  includes a color filter array  710  that includes primary filters  712  and secondary filters  714 . As described previously, color filter array  710  can be similar to color filter array  400  of  FIG. 4 , where primary filters  712  include three color filters a, b, and c and secondary filters  714  include four color filters d, e, f, and g. The primary filters capture three different spectral images on a rectangular grid with sampling pitch Δs a,b,c =2p, while the secondary filters each sample the incident image on rectangular grids with sampling pitch Δs d,e,f,g =4p through different spectral filters. 
     As also shown in  FIG. 7 , information obtained from the primary filters  712  and secondary filters  714  can be used to generate multiple types of images, such as a monochrome image  720 , a high dynamic range (HDR) monochrome image  730 , a tri-chromatic (RGB) image  740 , a HDR RGB image  760 , and a multispectral image  770 . In some embodiments, a multimodal demosaicing approach with anti-aliasing is applied to generate high resolution images. 
     Referring back to the color filter array  400  of  FIG. 4 , note that there is one color measurement at each pixel. The other colors are estimated from information obtained by neighboring pixels in order to, for example, reproduce high resolution output images irrespective of the type of image (e.g., monochrome image, HDR monochrome image, RGB image, HDR RGB image, multispectral image, etc.). This approach is generally referred to as “demosaicing.” 
     Denoting Λ m  as the set of pixel locations, (i, j), for channel mε{a, b, c, d, e, f, g, a mask function for each filter can be defined as: 
     
       
         
           
             
               
                 W 
                 m 
               
                
               
                 ( 
                 
                   i 
                   , 
                   j 
                 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     1 
                   
                   
                     
                       
                         ( 
                         
                           i 
                           , 
                           j 
                         
                         ) 
                       
                       ∈ 
                       
                         Λ 
                         m 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     otherwise 
                   
                 
               
             
           
         
       
     
     In the color filter array  400  of  FIG. 4  or color filter array  710  of  FIG. 7 , there are seven types of color channels—i.e., a, b, c, d, e, f, and g. Accordingly, the observed data, y(i,j), can be expressed as: 
     
       
         
           
             
               y 
                
               
                 ( 
                 
                   i 
                   , 
                   j 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   
                     m 
                     = 
                     a 
                   
                   , 
                   b 
                   , 
                   c 
                   , 
                   d 
                   , 
                   e 
                   , 
                   f 
                   , 
                   g 
                 
                 
                     
                 
               
                
               
                 
                   
                     W 
                     m 
                   
                    
                   
                     ( 
                     
                       i 
                       , 
                       j 
                     
                     ) 
                   
                 
                  
                 
                   
                     x 
                     m 
                   
                    
                   
                     ( 
                     
                       i 
                       , 
                       j 
                     
                     ) 
                   
                 
               
             
           
         
       
     
     where x m  is the mth channel&#39;s full resolution image. 
     Referring back to  FIG. 7 , monochrome image  720  can be generated from a single captured image by using information obtained from primary filters  712 . As described previously, information captured by primary filters  712  do not suffer from aliasing. Accordingly, at  722 , missing information for one of the primary filters  712  can be estimated using linear interpolation from other primary filters  712  from color filter array  710 . 
     Monochrome image  720  of a high resolution can be reconstructed using information measured by primary filters. This can be expressed as: 
         I   M ( i,j )={ {circumflex over (x)}   a ( i,j )+ {circumflex over (x)}   b ( i,j )+ {circumflex over (x)}   c ( i,j )}/3 
     where {circumflex over (x)} a (i,j), {circumflex over (x)} b (i,j), {circumflex over (x)} c (i,j) are the full resolution images obtained by interpolating pixels with the primary filters (e.g., primary filters a, b, and c of  FIG. 4 ). For interpolation, a Finite Impulse Response (FIR) Filters F(i,j) can be used and can be expressed as follows: 
         {circumflex over (x)}   v ( i,j )= W   v ( i,j ) y ( i,j )+   W     v ( i,j )[ F ( i,j )* y ( i,j )] 
     where v=a, b, or c, * denotes convolution, and  W   v (i,j)=1−W(i,j). For example, in some embodiments, the fir1 function in Mathworks® Matlab® can be used to find FIR filters of size 30×30 that pass all frequencies, thereby minimizing the loss of high frequencies due to interpolation. 
     In some embodiments, high dynamic range monochrome image  730  can be generated from a single captured image by using information obtained from primary filters  712  and secondary filters  714 . To create a high dynamic range monochrome image (e.g., image  730 ), a low exposure monochrome image  732  can be constructed. At  734 , low exposure monochrome image  732  is constructed using information from secondary filters  714  (e.g., the four secondary filters d, e, f, and g of  FIG. 4 ). These secondary filters  714  have lower exposure and collectively cover the whole visible spectrum. 
     For example, the monochrome values at pixels with filter a (e.g., filter a of color filter array  400  shown in  FIG. 4 ) can be calculated. As shown in  FIG. 4 , color filter array  400  includes four different secondary pixels (e.g., pixels d, e, f, and g) arranged diagonally about each pixel a. Accordingly, the monochrome value at each pixel a can be calculated as the average of the measurements at the four neighboring secondary pixels and can be expressed as: 
     
       
         
           
             
               
                 
                   W 
                   a 
                 
                  
                 
                   ( 
                   
                     i 
                     , 
                     j 
                   
                   ) 
                 
               
                
               
                 { 
                 
                   
                     Q 
                     D 
                   
                   * 
                   
                     y 
                      
                     
                       ( 
                       
                         i 
                         , 
                         j 
                       
                       ) 
                     
                   
                 
                 } 
               
             
             , 
             
               
                 where 
                  
                 
                     
                 
                  
                 
                   Q 
                   D 
                 
               
               = 
               
                 ( 
                 
                   
                     
                       
                         1 
                         4 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         4 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       
                         1 
                         4 
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         4 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
     It should be noted that aliasing caused by half-pixel phase shifts cancel out when adding four pixels in a diagonal neighborhood. The values at pixel a are then interpolated to the other pixels to yield the low exposure monochrome image  732  (ILEM), which can be expressed as: 
         I   LEM ( i,j )= L ( i,j )+ W   s   {Q   D   *L ( i,j )}+ W   b   {Q   H   *L ( i,j )}+ W   c   {Q   V   *L ( i,j )} 
     where: 
     
       
         
           
             
               
                 W 
                 s 
               
                
               
                 ( 
                 
                   i 
                   , 
                   j 
                 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         1 
                       
                       
                         
                           
                             ( 
                             
                               i 
                               , 
                               j 
                             
                             ) 
                           
                           ∈ 
                           
                             { 
                             
                               d 
                               , 
                               e 
                               , 
                               f 
                               , 
                               g 
                             
                             } 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         otherwise 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                    
                   
                     
 
                   
                    
                   
                     Q 
                     H 
                   
                 
                 = 
                 
                   
                     Q 
                     V 
                     T 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             1 
                             2 
                           
                         
                         
                           0 
                         
                         
                           
                             1 
                             2 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     After obtaining low exposure monochrome image  732 , at  736 , a high dynamic range monochrome image  730  can be generated by combining the monochrome images of different exposures and their associated information e.g., the monochrome image  720  generated using primary filters  712  and the low exposure monochrome image  732  generated using secondary filters  714 . 
     In some embodiments, tri-chromatic (RGB) image  740  can be generated from a single captured image by using information obtained from primary filters  712 . As described previously in  FIG. 6 , the primary filters  712  used in color filter array  710 , such as primary filters a, b, and c in  FIG. 4 , have spectral responses similar to red, green, and blue filters. At  742  and  744 , tri-chromatic (RGB) image  740  can be constructed using color reproduction matrix T′ and H′ (a linear transformation from CIE XYZ tristimulus values to sRGB tristimulus values) to combine the information in the {circumflex over (x)} a , {circumflex over (x)} b , {circumflex over (x)} c  images computed using the primary filters. The RGB image can be expressed as: 
         I   RGB ( i,j )= HT′[{circumflex over (x)}   a ( i,j ) {circumflex over (x)}   b ( i,j ) {circumflex over (x)}   c ( i,j )] T    
     As described previously, to calculate the difference of color for color reproduction of a RGB image, the CIE 1931 XYZ color space (created by the International Commission on Illumination), which is based on direct measurements of human visual perception and serves as the basis of which many other color spaces are defined, can be used. The calculation of sRGB tristimulus values (which are employed in some digital cameras or color monitors) from the CIE XYZ tristimulus values is a linear transformation. The CIE XYZ tristimulus values can be defined as Y=A T IR, where Y represents the true tristimulus values and A is a matrix of CIE XYZ color matching functions [  x   y   z ]. The estimated CIE tristimulus values corresponding to the primary filters Ŷ′ can be expressed as an optimal linear transformation: Ŷ′=T′X′, where X′=[x a , x b , x c ] T . The transformation T′ is determined so as to minimize the color difference: min ∥Y−T′X′∥ 2 . 
     In some embodiments, a HDR RGB image  760  can be generated from a single captured image by using information obtained from primary filters  712  and secondary filters  714  of color filter array  710 . To create a high dynamic range tri-chromatic image (e.g., image  760 ), a low exposure tri-chromatic image  750  can be constructed. 
     Full resolution secondary filter images—{circumflex over (x)} d , {circumflex over (x)} e , {circumflex over (x)} f , and {circumflex over (x)} g —can be respectively computed using the d, e, f, and g pixels using bilinear interpolation. However, this can result in severe aliasing. In some embodiments, the aliasing of the secondary filter images can be estimated using information from the primary filter images—{circumflex over (x)} a , {circumflex over (x)} b , {circumflex over (x)} c  at  752 . It should be noted that there is a strong correlation between the spectra of primary filters  712  and secondary filters  714 , as shown by the overlap in  FIG. 6 . For example, when anti-aliasing the full resolution image {circumflex over (x)} e  that corresponds to filter e, it should be noted that filter e has a strong correlation with that of filter a. Accordingly, the interpolated full resolution filter a image {circumflex over (x)} a  at each filter e locations can be sampled. These can then be used to calculate a full resolution image for filter e, which can be expressed as: 
       Ω{ W   e ( i,j ) {circumflex over (x)}   a ( i,j )}
 
     where Ω(•) represents bilinear interpolation. Aliasing can be inferred by subtracting the original {circumflex over (x)} a  image from the interpolated one. Then, to obtain the final estimate of aliasing in channel e, the above-mentioned difference can be scaled by Ψ ae , which is the ratio of the filter transmittances of the a and e pixels, to take into account the difference in exposures of a and e pixels. The estimated aliasing Ψ ae  can be expressed as follows: 
       γ e ( i,j )=[Ω{ W   e ( i,j ) {circumflex over (x)}   a ( i,j )}− {circumflex over (x)}   a ( i,j )]ψ ae  
 
     where: 
     
       
         
           
             
               ψ 
               ae 
             
             = 
             
               
                 ( 
                 
                   
                     ∑ 
                     
                       l 
                       = 
                       1 
                     
                     L 
                   
                    
                   
                     C 
                     e 
                   
                 
                 ) 
               
               
                 ( 
                 
                   
                     ∑ 
                     
                       l 
                       = 
                       1 
                     
                     L 
                   
                    
                   
                     C 
                     a 
                   
                 
                 ) 
               
             
           
         
       
     
     Accordingly, the anti-aliased image {circumflex over (x)} e  can be calculated at  754  as: 
         {circumflex over (x)}   e ( i,j )=Ω{ W   e ( i,j ) y ( i,j )}|γ e ( i,j )
 
     In addition, other anti-aliased secondary can be similar calculated. 
       FIG. 8  shows examples of low exposure RGB images calculated from the secondary filters through the anti-aliasing approach. For example, image  810  shows a low exposure RGB image calculated from secondary filters  714  without anti-aliasing. It should be noted that false color artifacts  812  caused by aliasing are present. Image  820  shows the downsampled image Ω{W e (i,j){circumflex over (x)} a (i,j)} calculated using the pixels with primary filter a of primary filters  712 . Image  830  then shows the aliasing Y e (i,j) estimated using the downsampled image  820  and the full resolution image for channel a. It should be noted that the brightness of image  830  is enhanced for visualization. Accordingly, image  840  is a lower exposure RGB image obtained after anti-aliasing using image  830 , which provides the estimation of aliasing. Image  840  shows the efficacy of the anti-aliasing approach, where false color artifacts (e.g., artifacts  812  in image  820 ) can be removed. 
     A low exposure RGB image can be obtained by multiplying the secondary filter images by a color reproduction matrix at  756 , which can be expressed as: 
         I   LERGB ( i,j )= HT″[{circumflex over (x)}   d ( i,j ) {circumflex over (x)}   e ( i,j ) {circumflex over (x)}   f ( i,j ) {circumflex over (x)}   g ( i,j )] T    
     where T″ is the color reproduction matrix and H is the linear transformation from CIE XYZ to sRGB. 
     After obtaining low exposure RGB  750 , at  758 , a high dynamic range RGB image  760  can be generated by combining the tri-chromatic (RGB) images of different exposures and their associated information e.g., the RGB image  740  and the low exposure RGB image  750 . 
     In some embodiments, a multispectral image  770  can be generated from a single captured image using information from primary filters  712  and secondary filters  714  of color filter array  710 . For multispectral image  770 , the spectral reflectance of an object can be reconstructed using images {circumflex over (x)} a , {circumflex over (x)} b , {circumflex over (x)} c  and anti-aliased images {circumflex over (x)} d , {circumflex over (x)} e , {circumflex over (x)} f , and {circumflex over (x)} g  at  772 . In some embodiments, a HDR RGB image  760  can be generated from a single captured image by using information obtained from primary filters  712  and secondary filters  714  of color filter array  710 . 
     As described previously, the spectral distribution is reconstructed by minimizing the expression: ∥F·σ−X∥ 2 . In some embodiments, the reconstruction approach can be expressed as a constrained minimization as follows: {circumflex over (σ)}=arg min∥{tilde over (F)}·σ−{tilde over (X)}∥ 2 , subject to B·σ≧0, where {tilde over (F)}=[F T αP T ] T , P lk =∂ 2 b k (λ l )/∂λ 2  is a smoothness constraint, a is a smoothness parameter, 1≧L, 1≧k≧K, {tilde over (X)}=[X T  0] T , and B=[b k (λ 1 )]. This regularized minimization can be solved using quadratic programming. 
       FIG. 9  shows an original image  910  of a Circular Zone Plate (CZP) and multiple images  920 ,  930 ,  940 , and  950  generated from a single captured image of the original image  910  in accordance with some embodiments. It should be noted that original image  910 , which serves as the ground truth, shows a CZP image calculated using a diffraction-limited model of a lens with a f-number of 5.6 and a 1.0 μm pixel size. Using a camera system with a generalized assorted pixel color filter array or mosaic (e.g., color filter array  400 , color filter array  710 , etc.) having multiple primary filters and multiple secondary filters to capture an image, multiple image types e.g., a demosaiced monochrome image  920 , a demosaiced tri-chromatic (RGB) image  930 , a demosaiced and anti-aliased low exposure monochrome image  940 , and a demosaiced and anti-aliased low exposure tri-chromatic (RGB) image  950  can be generated. 
       FIG. 10  shows Modulation Transfer Function (MTF) calculations for each image  910 ,  920 ,  930 ,  940 , and  950 . As shown, curve  1010  is associated with original image  910 , curve  1020  is associated with monochrome image  920  and tri-chromatic (RGB) image  930 , curve  1030  is associated with low exposure monochrome image  940 , and curve  1040  is associated with low exposure tri-chromatic (RGB) image  950 . Note that curve  1010  for monochrome image  920  and tri-chromatic (RGB) image  930 , which were generated using primary filters of the color filter array, is substantially similar to curve  1020  associated with original image  910 . Note also that the low exposure monochrome image  940  has a MTF of about 0.1 at 0.1754 fs, while the low exposure tri-chromatic (RGB) image  950  has a MTF of about 0.1 at 0.1647 fs. For standard monochrome and RGB, this generally occurs at 0.2125 fs. This demonstrates that the camera mechanisms that use a color filter array with multiple primary filters and multiple secondary filters and a multimodal demosaicing approach allows a user to control the trade-off between spatial resolution and radiometric details of the recovered image. 
     Additional examples of images generated from a single captured image are shown in  FIGS. 11 and 12 . Note that ground truth images  1110  of  FIGS. 11 and 1210  of  FIG. 12  are calculated using a diffraction-limited model of a lens with a f-number of 5.6 and a 1.0 μm pixel size. Image  1120  of  FIG. 11  and image  1220  of  FIG. 12  show examples of raw images captured using a camera system having a generalized assorted pixel color filter array or mosaic with primary filters and secondary filters. Image  1130  of  FIG. 11  and image  1230  of  FIG. 12  show examples of demosaiced monochrome images generated using raw image  1120  and  1220 , respectively, and the information obtained from the primary filters. Image  1140  of  FIG. 11  and image  1240  of  FIG. 12  show examples of high dynamic range monochrome images generated using raw image  1120  and  1220 , respectively, and the information obtained from the primary and secondary filters. Image  1150  of  FIG. 11  and image  1250  of  FIG. 12  show examples of tri-chromatic (RGB) images generated using raw image  1120  and  1220 , respectively, and the information obtained from the primary filters. Image  1160  of  FIG. 11  and image  1260  of  FIG. 12  show examples of high dynamic range tri-chromatic (RGB) images generated using raw image  1120  and  1220 , respectively, and the information obtained from the primary and secondary filters. 
     It should be noted that the texture and color of saturated regions in the monochrome and RGB images become visible in the corresponding high dynamic range images. As also shown in  FIGS. 11 and 12 , more detail is shown in the high dynamic range monochrome image than in the high dynamic range tri-chromatic (RGB) image. 
     In addition,  FIGS. 11 and 12  show examples of multispectral images generated using information obtained and calculated from primary and secondary filters. For example, image  1170  of  FIG. 11  and image  1270  of  FIG. 12  shows 31-band multispectral images (400-700 nm, at 10 nm intervals) of several static scenes capturing by using a tunable filter and a cooled CCD camera. The corresponding reconstructed spectral reflectance curves  1180  and  1280  show that the reconstructed spectral reflectance (identified by the dashed line) is substantially similar to the spectral reflectance of the ground truth image. 
     Alternatively, some camera systems can use a different generalized assorted pixel color filter array to capture a single image of a scene and control the trade-off between image resolution, dynamic range, and spectral detail to generate images of multiple image types. 
     For example,  FIG. 13  shows an example of an 8×8 color filter array  1300  that includes five different color filters—e.g., Green (G), Red (R), Blue (B), Yellow (Y), and Emerald (E). Each color filter has two exposures e.g., a bright exposure and a dark exposure, where color (C) denotes a bright pixel and color (C′) denotes a dark pixel. For example, pixel G denotes a bright green pixel, while pixel G′ denotes a dark green pixel. 
     As shown in  FIG. 13 , the bright and dark green channel samples every two lines in the horizontal and vertical directions and sample at every line in the diagonal direction. Accordingly, the horizontal and vertical sampling frequency of bright and dark green channel is f HV /2 and the diagonal sampling frequency of bright and dark green channel is f D , where f HV  is the horizontal and vertical sampling frequency and f D  is the diagonal sampling frequency of the image sensor. In addition, the horizontal and vertical Nyquist frequency of bright and dark green channel is half of the sampling frequency or f HV /4, while the diagonal Nyquist frequency of bright and dark green channel is f D /2. Referring back to  FIG. 13 , bright and dark red, blue, yellow, and emerald channels sample every four lines in the horizontal and vertical directions and sample every two lines in the diagonal direction. Accordingly, the horizontal and vertical sampling frequency of bright and dark red, blue, yellow, and emerald channels is f HV /4 and the corresponding diagonal sampling frequency is f D /2. Thus, the horizontal and vertical Nyquist frequency of bright and dark red, blue, yellow, and emerald channels is f HV /8 and the corresponding diagonal Nyquist frequency is f D /4. 
     To further illustrate the Nyquist frequencies of color filter array  1300  of  FIG. 13 , the Nyquist or usable frequency region  1400  in the frequency domain is shown in  FIG. 14 . As shown, the Nyquist region of the bright and dark green channel is located in the substantially square area identified by a full line  1410  and the Nyquist region of the bright and dark red, blue, yellow, and emerald channels is located in the substantially square area identified by a dashed line  1420 . 
     In another example of a color filter array in accordance with some embodiments of the disclosed subject matter,  FIG. 15  shows an example of a 9×9 color filter array  1500  that includes five different color filters e.g., Green (G), Red (R), Blue (B), Yellow (Y), and Emerald (E). 
     As shown in  FIG. 15 , the green channel samples every line in the horizontal and vertical directions and samples at every line in the diagonal direction. Accordingly, the horizontal and vertical sampling frequency of dark green channels is f HV  and the diagonal sampling frequency of green channels is f D . In addition, the horizontal and vertical Nyquist frequency of green channels is half of the sampling frequency or f HV /2, while the diagonal Nyquist frequency of green channels is f D /2. Referring back to  FIG. 15 , red, blue, yellow, and emerald channels sample every two lines in the horizontal and vertical directions and sample every two lines in the diagonal direction. Accordingly, the horizontal and vertical sampling frequency of red, blue, yellow, and emerald channels is f HV /2 and the corresponding diagonal sampling frequency is f D /2. Thus, the horizontal and vertical Nyquist frequency of red, blue, yellow, and emerald channels is f HD /4 and the corresponding diagonal Nyquist frequency is f D /4. 
     The Nyquist frequencies of color filter array  1500  are further illustrated in  FIG. 16 . As shown,  FIG. 16  shows that the Nyquist region of the green channel is located in the substantially diamond area identified by a full line  1610  and the Nyquist region of the red, blue, yellow, and emerald channels is located in the substantially diamond area identified by a dashed line  1620 . 
     Similarly, as described above, multiple image types can be generated from a single captured image using a camera system with a color filter array, such as color filter array  1300  of  FIG. 13  or color filter array  1500  of  FIG. 15 , in accordance with some embodiments of the disclosed subject matter. For example, the information obtained from the five color filters with two exposures of color filter array  1300  of  FIG. 13  can be used to generate a monochrome image, a high dynamic range (HDR) monochrome image, a tri-chromatic (RGB) image, a HDR RGB image, and a multispectral image. In another example, the information obtained from the five color filters of color filter array  1500  of  FIG. 15  can be used to generate multiple types of images, such as a monochrome image and a tri-chromatic (RGB) image. As also described above, a demosaicing approach with anti-aliasing can be applied to generate images of multiple types. 
     In some embodiments, using one of color filter arrays  1300  or  1500 , a linear regression model of local color distribution can be used to reduce aliasing effects. For example, it has been determined that there are strong inter-color correlations at small local areas (e.g., on a color-changing edge). These local color distributions in an image can be expressed by the following linear regression model: 
     
       
         
           
             
               
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     It should be noted that a pixel at location (i,j) in color filters arrays  1300  or  1500  can be represented by either (R i,j , g i,j , b i,j , y i,j , e i,j ), (r i,j , G i,j , b i,j , y i,j , e i,j ), (r i,j , g i,j , B i,j , y i,j , e i,j ), (r i,j , g i,j , b i,j , Y i,j , e i,j ), or (r i,j , g i,j , b i,j , y i,j , E i,j ), where R i,j , B i,j , Y i,j , and E i,j  denote the known red, green, blue, yellow, and emerald components of the color filter array and r i,j , g i,j , b i,j , y i,j , e i,j  denote the unknown components of the color filter array. In addition, it should also be noted that the estimates of r i,j , g i,j , b i,j , y i,j , e i,j  are denoted as {circumflex over (R)} i,j , Ĝ i,j , {circumflex over (B)} i,j , Ŷ i,j , and Ê i,j . 
     The resulting Fourier transforms of V GR  and M R  are as follows: 
     
       
         
           
             
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     Using these expressions, the aliasing of R can be estimated. 
     In some embodiments, using one of color filter arrays  1300  or  1500 , directional smoothing can be used to reduce aliasing effects. For example, to reduce the computational cost of anti-aliasing, directional smoothing can be used when the local statistics (e.g., V GG , V GR , M G , and M R ) are calculated. As shown in  FIG. 17 , one-dimensional smoothing along a direction to the local area of the color filter array  1710  is applied at  1720  to obtain one-dimensional signals of colors  1730 . Then, an anti-aliasing approach is applied to the one-dimensional color signals  1730  at  1740 . After anti-aliasing, color data for each phase is obtained at  1750 . Local statistics (e.g., V GG , V GR , M G , and M R ) can then be calculated at  1760  using the anti-aliased one-dimensional color data. 
     It should be noted that the directional smoothing approach can be applied in any suitable direction. For example, the smoothing approach can be applied in the horizontal, vertical, right-ascending diagonal ( ), and right-descending diagonal direction ( ). It should also be noted that the direction of smoothing can be selected based at least in part on the direction of the local texture (e.g., horizontal smoothing for horizontal stripes). 
     In some embodiments, the directional smoothing approach for several directions (e.g., horizontal, vertical, right-ascending diagonal, and right-descending diagonal direction) is performed and anti-aliasing, computing local statistics, and output color interpolations are also performed for each direction. By measuring magnitudes of the gradient and local color variance of the anti-aliased one-dimensional signals, residual aliasing for each direction can be evaluated. In some embodiments, the direction that provides the smallest residual aliasing can be selected as the suitable direction of the interpolation filter. 
       FIG. 18  shows a portion of an original image  1810  and multiple images  1820 ,  1830 ,  1840 , and  1850  generated from a single captured image of the original image  1810  in accordance with some embodiments. Using a camera system with a generalized assorted pixel color filter array or mosaic, such as color filter array  1300  of  FIG. 13 , having five different color filters, each having two exposures (e.g., a bright exposure and a dark exposure) to capture an image and the anti-aliasing approach described above, multiple image types e.g., a monochrome image  1820 , a tri-chromatic (RGB) image  1830 , a high dynamic range (HDR) monochrome image  1840 , and a HDR RGB image  1850  can be generated. It should be noted that, by sacrificing spatial resolution, the quality of the spectrum and the dynamic range can be improved. 
     In some embodiments, hardware used in connection with the camera mechanisms can include an image processor, an image capture device (that includes a generalized assorted pixel color filter array, such as the one in  FIG. 4 ), and image storage. The image processor can be any suitable device that can process images and image-related data as described herein. For example, the image processor can be a general purpose device such as a computer or a special purpose device, such as a client, a server, an image capture device (such as a camera, video recorder, scanner, mobile telephone, personal data assistant, etc.), etc. It should be noted that any of these general or special purpose devices can include any suitable components such as a processor (which can be a microprocessor, digital signal processor, a controller, etc.), memory, communication interfaces, display controllers, input devices, etc. The image capture device can be any suitable device for capturing images and/or video, such as a portable camera, a video camera or recorder, a computer camera, a scanner, a mobile telephone, a personal data assistant, a closed-circuit television camera, a security camera, an Internet Protocol camera, etc. The image capture device can include the generalized assorted pixel color filter array as described herein. The image storage can be any suitable device for storing images such as memory (e.g., non-volatile memory), an interface to an external device (such as a thumb drive, a memory stick, a network server, or other storage or target device), a disk drive, a network drive, a database, a server, etc. 
     Accordingly, generalized assorted pixel camera systems and methods are provided. 
     Although the invention has been described and illustrated in the foregoing illustrative embodiments, it is understood that the present disclosure has been made only by way of example, and that numerous changes in the details of implementation of the invention can be made without departing from the spirit and scope of the invention, which is only limited by the claims which follow. Features of the disclosed embodiments can be combined and rearranged in various ways.