Patent Publication Number: US-8126323-B2

Title: Method and apparatus for radiance capture by multiplexing in the frequency domain

Description:
PRIORITY INFORMATION 
     This application is a divisional of U.S. application Ser. No. 12/186,396, filed Aug. 5, 2008 now U.S. Pat. No. 8,019,215, which claims benefit of priority of U.S. Provisional Application Ser. No. 60/954,238, filed Aug. 6, 2007, the content of which is incorporated by reference herein in their entirety. 
    
    
     BACKGROUND 
     Description of the Related Art 
     Conventional cameras fail to capture a large amount of optical information. In particular, a conventional camera does not capture information about the location on the aperture where different light rays enter the camera. During operation, a conventional digital camera captures a two-dimensional (2-D) image representing a total amount of light that strikes each point on a photosensor within the camera. However, this 2-D image contains no information about the directional distribution of the light that strikes the photosensor. Directional information at the pixels corresponds to locational information at the aperture. 
     Light-Field or Radiance Photography 
     In contrast to conventional cameras, light-field, or radiance, cameras sample the four-dimensional (4-D) optical phase space, or radiance, and in doing so capture information about the directional distribution of the light rays. This information captured by radiance cameras may be referred to as the light-field, the plenoptic function, or radiance. In computational photography, radiance is a four-dimensional (4-D) record of all light rays in 3-D. Radiance describes both spatial and angular information, and is defined as density of energy per unit of area per unit of stereo angle (in radians). A radiance camera captures radiance; therefore, radiance images originally taken out-of-focus may be refocused, noise may be reduced, viewpoints may be changed, and other radiance effects may be achieved. 
     Conventional cameras, based on 2-D image sensors, are simply integration devices. In a typical setting, conventional cameras integrate over a 2-D aperture to produce a 2-D projection of the four-dimensional (4-D) radiance. Integral, or light-field, photography was proposed more than a century ago to “undo” the integration and measure the complete 4-D radiance arriving at all points on a film plane or photosensor. Thus, integral photography captures radiance as opposed to capturing a flat 2-D picture. The light itself, or radiance, may be mathematically described by the radiance density function, which is a complete representation of light energy flowing along “all rays” in 3-D space. This density is a field defined in the 4-D domain of the optical phase space, the space of all lines in 3-D with symplectic structure. Capturing the additional two dimensions of radiance data allows the rays of light to be re-sorted in order to synthesize new photographs, which may be referred to as novel views. Advantages of radiance photography include gaining information about the 3-D structure of the scene as well as the ability of optical manipulation or editing of the images, such as refocusing and novel view synthesis. 
     Radiance may be captured with a conventional camera. In one conventional method, M×N images of a scene are captured from different positions with a conventional camera. If, for example, 8×8 images are captured from 64 different positions, 64 images are produced. The pixel from each position (i, j) in each image are taken and placed into blocks, to generate 64 blocks. 
       FIG. 1   a  illustrates an exemplary prior art light-field camera, or camera array, which employs an array of two or more objective lenses  110 . Each objective lens focuses on a particular region of photosensor  108 , or alternatively on a separate photosensor  108 . This light-field camera  100  may be viewed as a combination of two or more conventional cameras that each simultaneously records an image of a subject on a particular region of photosensor  108  or alternatively on a particular photosensor  108 . The captured images may then be combined to form one image. 
       FIG. 1   b  illustrates an exemplary prior art integral camera, or plenoptic camera, another type of light-field camera, which employs a single objective lens and a microlens or lenslet array  106  that includes, for example, about 100,000 lenslets. Lenslet array  106  is typically placed a small distance (˜0.5 mm) from a photosensor  108 , e.g. a charge-coupled device (CCD). The raw image captured with a plenoptic camera  102  is made up of an array of small images, typically circular, of the main camera lens  108 . These small images may be referred to as microimages. The lenslet array  106  enables the plenoptic camera  102  to capture the radiance, i.e. to record not only image intensity, but also the distribution of intensity in different directions at each point. Each lenslet splits a beam coming to it from the main lens  104  into rays coming from different “pinhole” locations on the aperture of the main lens  108 . Each of these rays is recorded as a pixel on photosensor  108 , and the pixels under each lenslet collectively form an n-pixel image. This n-pixel area under each lenslet may be referred to as a macropixel, and the camera  102  generates a microimage at each macropixel. The plenoptic photograph captured by a camera  102  with, for example, 100,000 lenslets will contain 100,000 macropixels, and thus generate 100,000 microimages of a subject. Each macropixel contains different angular samples of the light rays coming to a given microlens. Each macropixel contributes to only one pixel in the different angular views of the scene. As a result, each angular view contains 100,000 pixels. 
     Another type of light-field camera is somewhat similar to the plenoptic camera of  FIG. 1   b , except that an array of pinholes is used between the main lens and the photosensor instead of an array of lenslets. 
     Yet another type of light-field camera is similar to the plenoptic camera of  FIG. 1   b , except that a non-refractive cosine mask is used between the main lens and the photosensor instead of an array of lenslets. The cosine mask is a non-refractive element, and modulates the incoming light rays but does not refract the light. The captured image is the convolution of the incoming light field with the mask light field. This camera design captures the 4-D light field directly in the Fourier domain. Thus, a 2-D sensor pixel represents a coded linear combination of several rays. The linear combination can be decoded by software to obtain the 4-D light field. 
     Frequency Domain Analysis of Radiance 
     Techniques for analyzing radiance in the frequency domain have been developed, among which are application of Poisson summation formula to depth representation of scenes, light fields and displays, light transport and optical transforms, Fourier slice theorem applied to refocusing, and others. However, frequency domain analysis has not been applied directly to the understanding and design of light-field, or radiance, cameras in general. Moreover, frequency domain processing has been limited to mask-based radiance cameras that employ sinusoidal (e.g., cosine) masks. 
     SUMMARY 
     Various embodiments of a mask-based radiance camera are described that multiplex radiance in the frequency domain by optically mixing different spatial and angular frequency components of the light received from a scene, and capture the radiance information at a photosensor. Embodiments of a mask-based radiance camera based on an external, non-refractive mask located in front of the main or objective camera lens, rather than between the main lens and the photosensor or film, are described. In addition, an internal mask-based camera based on a medium- or large-format conventional camera with a film back, and a non-refractive mask that may be placed in the film back adjacent to the film, with optional spacers that may be placed between the mask and the film, is described. While both types of mask-based cameras employ periodic masks, neither is limited to sinusoidal (i.e., cosine) masks. 
     Various exemplary embodiments of non-refractive masks are described. The masks are non-refractive; that is, while the masks may act to modulate and/or attenuate the light, the masks do not act to bend the light rays. An exemplary embodiment is a mesh mask, which also may be referred to as a net or screen. The mesh may include horizontally and vertically arranged opaque linear elements or grid lines, which collectively form a grid that modulates, but does not refract, light received from a scene located in front of the camera as the received light passes through the grid. Generally, the opaque grid lines may be equally spaced in the two dimensions. Thus, the opaque grid lines act to form or define rows and columns of periodically spaced transparent (i.e., through which light may pass), non-refractive openings. 
     Another exemplary embodiment of a mask includes transparent circular openings, through which light may pass, in an opaque medium or surface. The circular openings in may be periodically spaced, and arranged in horizontal rows and vertical columns. Other geometric shapes than circles may be used in other embodiments, e.g. squares, hexagons, rectangles, etc. Another exemplary mask is composed of a grid or array of pinholes, and may be referred to as a pinhole mask. The pinholes, which also may be referred to as openings, may typically be, but are not necessarily, circular. The pinholes may be periodically spaced, and arranged in horizontal rows and vertical columns. 
     While various examples of masks are described, in general, any of various types of periodic masks may be used as a non-refractive mask in embodiments. In addition, while the masks are described as periodic, the periodicity may be arbitrary. In other words, the masks that may be used in embodiments of an external mask-based radiance camera are not limited to sinusoidal masks such as sine masks and/or cosine masks. The various embodiments of masks may be used with either the external mask-based radiance camera embodiments or the internal mask-based radiance camera embodiments with appropriate physical configuration to match the particular camera application. 
     Various types of cameras may be used in embodiments of the external mask-based radiance camera, including both film-based and digital cameras, and standard, medium or large-format cameras. A non-refractive mask may be integrated with the camera, or alternatively attachable to the camera, with the mask positioned in front of the main lens so that light from a scene to be photographed arrives at the main lens after passing through and being modulated by the mask. The mask is a non-refractive element, and modulates and/or attenuates the incoming light rays but does not bend the rays. In one embodiment, the main lens may be focused on a plane just behind the mask, between the mask and the main lens. Light is refracted by the main lens onto a photosensor, which may in turn operate to capture a radiance image of the scene. 
     In one embodiment of a method of capturing a radiance image with an external mask-based radiance camera, light from a scene may be received at a mask. The mask is a non-refractive element, and modulates and/or attenuates the incoming light rays but does not bend them. Light that passes through the mask is received at the main lens of a camera. The main lens may be focused on a plane between the mask and the main lens, and proximate to the mask. The received light is refracted by the main lens onto a photosensor of the camera. The photosensor may capture the received light to generate a radiance image of the scene. In some embodiments of the camera, the captured radiance image may be stored to a memory medium or memory device. 
     Embodiments of an internal mask-based radiance camera based on a medium- or large-format film camera with a film back. In one embodiment, a mechanism inside the film back of the film camera holds the mask so that a flat side of the mask is pressed against the film and the surface of mask on which the opaque surface or medium is painted, attached, etc., with openings that are the transparent portion of the mask is away from the film. In one embodiment, the thickness of the mask is such that, when placed against the film, the opaque surface of the mask, and the openings therein, is at a distance f (equivalent to the focal length of the mask) from the film. In one embodiment, spacers may be used between the mask and the film in film holder to increase the distance from the mask and the film to allow f (equivalent to the focal length of the mask) to be changed, for example to match a changed F/number for the main lens. Additional spacers may be added to provide additional spacing. 
     The angular information of radiance images captured with embodiments of an external mask-based radiance camera or with embodiments of an internal mask-based radiance camera may be demultiplexed using an embodiment of a frequency domain demultiplexing method described herein to generate multiple views of a scene. If the radiance was captured to film, the radiance image may be digitized from the film, for example using a film negative or photograph scanner, to generate a digital version of the radiance image that may be stored to a memory medium and/or processed by the frequency domain demultiplexing method. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1   a  illustrates an exemplary prior art light-field camera, or camera array. 
         FIG. 1   b  illustrates an exemplary prior art plenoptic camera. 
         FIG. 2   a  illustrates the geometric representation of a ray as position and angle in an optical system. 
         FIG. 2   b  illustrates the same ray as in  FIG. 2   b , but described as a point, or a vector, in a 2-D space. 
         FIG. 3  illustrates a light-field camera employing an array of pinholes. 
         FIG. 4   a  illustrates, in the frequency domain, a band-limited signal after the array of pinholes. 
         FIG. 4   b  illustrates the shear of the signal after traveling a distance f. 
         FIGS. 4   c  and  4   d  illustrate reconstructing the original signal before the array of pinholes by combining samples at different intersections with the ω q  axis. 
         FIG. 5   a  illustrates an exemplary image obtained from a lens-based radiance camera. 
         FIG. 5   b  illustrates a zoom-in of a region of the image illustrated in  FIG. 5   a.    
         FIG. 5   c  illustrates the magnitude of the 2-D Fourier transforms of the image illustrated in  FIG. 5   a.    
         FIG. 6   a  illustrates an exemplary image obtained from a mask-based radiance camera. 
         FIG. 6   b  illustrates a zoom-in of a region of the image illustrated in  FIG. 6   a.    
         FIG. 6   c  illustrates the magnitude of the 2-D Fourier transforms of the image illustrated in  FIG. 6   a.    
         FIG. 7   a  illustrates an exemplary image obtained from an external mask-based radiance camera. 
         FIG. 7   b  illustrates a zoom-in of a region of the image illustrated in  FIG. 7   a.    
         FIG. 7   c  illustrates the magnitude of the 2-D Fourier transforms of the image illustrated in  FIG. 7   a.    
         FIG. 8  illustrates a method of demultiplexing the angular information of an image captured using a radiance camera, according to one embodiment. 
         FIGS. 9   a  and  9   b  illustrate a method of correcting the effect of waves due to small shifts or misalignments in the FFT, according to one embodiment. 
         FIG. 10  illustrates a frequency domain demultiplexing module, according to one embodiment. 
         FIG. 11   a  shows an exemplary radiance image captured with a lens-based radiance camera. 
         FIG. 11   b  shows the absolute value of the Fourier transform of the radiance image of  FIG. 11   a.    
         FIGS. 12   a  and  12   b  show two stereo views from the frequency domain reconstructed light field of  FIGS. 11   a  and  11   b.    
         FIG. 13  shows a conventional medium-format film camera and a film back, with a computer screen filter, used as a mask, attached to the window just in front of the film, according to one embodiment. 
         FIG. 14  shows two stereo views generated from a radiance image taken using an exemplary mask-based radiance camera. 
         FIG. 15  shows a picture taken through a net, or mesh, located in front of a conventional camera. 
         FIG. 16  shows two stereo views from the radiance generated from the picture shown in  FIG. 15 . 
         FIG. 17  illustrates exemplary net- or mesh-like, non-refractive masks, according to embodiments. 
         FIG. 18  illustrates other exemplary non-refractive masks, according to embodiments. 
         FIG. 19  illustrates an exemplary radiance camera with an external mask attachment, according to one embodiment. 
         FIG. 20  illustrates a method of capturing a radiance image with an external mask-based radiance camera, according to one embodiment. 
         FIG. 21  illustrates an exemplary embodiment of an internal mask-based radiance camera based on a medium- or large-format film camera with a film back. 
         FIG. 22  illustrates an exemplary computer system that may be used in embodiments. 
     
    
    
     While the invention is described herein by way of example for several embodiments and illustrative drawings, those skilled in the art will recognize that the invention is not limited to the embodiments or drawings described. It should be understood, that the drawings and detailed description thereto are not intended to limit the invention to the particular form disclosed, but on the contrary, the intention is to cover all modifications, equivalents and alternatives falling within the spirit and scope of the present invention as defined by the appended claims. The headings used herein are for organizational purposes only and are not meant to be used to limit the scope of the description or the claims. As used throughout this application, the word “may” is used in a permissive sense (i.e., meaning having the potential to), rather than the mandatory sense (i.e., meaning must). Similarly, the words “include”, “including”, and “includes” mean including, but not limited to. 
     DETAILED DESCRIPTION OF EMBODIMENTS 
     Various embodiments of a method and apparatus for capturing radiance in the frequency domain, and demultiplexing the captured radiance in the frequency domain, are described. Various embodiments of light-field, or radiance, cameras, including both mask-based and lens-based radiance cameras, are described that multiplex the radiance in the frequency domain by optically mixing different spatial and angular frequency components and capturing the signal via a photosensor (e.g., conventional film or an electronic sensor such as a charge-coupled device (CCD)). Some embodiments of the radiance camera may be based on arrays of “active” optical elements, such as lenses and prisms. Other embodiments of the radiance camera may be based on “passive” optical elements, or masks, such as meshes or grids of circles or pinholes. Both types of radiance cameras may be understood and described according to a mathematical formalism in the frequency domain. 
     In the following sections, a mathematical analysis of radiance cameras in the frequency domain is provided. A method of multiplexing the 3-D radiance onto the 2-D sensor is demonstrated that works in the frequency domain for various radiance cameras, including both lens-based and mask-based radiance cameras. It is also demonstrated that the F/number matching condition known to exist for lens-based radiance cameras is a requirement for all radiance cameras. This helps in constructing and adjusting various mask- and lens-based radiance cameras so that the cameras produce higher quality radiance images. 
     A mathematical method for recovering (demultiplexing) the multiplexed spatial and angular information from the frequency representation is also described, and is shown to be applicable to radiance images captured by both lens-based and mask-based radiance cameras, including radiance images captured with mask-based cameras that employ any periodic mask. This method may be referred to as a frequency domain demultiplexing method. The frequency domain demultiplexing method may, for example, be implemented in a computer software program or module, referred to herein as a frequency domain demultiplexing module. 
     Conventionally, frequency domain demultiplexing methods similar to the frequency domain demultiplexing method described herein have been limited to radiance images captured specifically with mask-based radiance cameras that use sinusoidal (i.e., cosine or sine) masks. Embodiments of the frequency domain demultiplexing method are described for which it is shown that the method may be used to demultiplex radiance information from images captured with mask-based radiance cameras that use any periodic mask, not just sinusoidal masks, and for which it is also shown that the method may to demultiplex radiance information captured with lens-based radiance cameras in addition to mask-based cameras. 
     In addition, embodiments of a radiance camera based on an external mask, e.g. a periodic screen, mesh or grid of openings, such as pinholes or circles, in an opaque surface or element located in front of the main camera lens, rather than between the main lens and the photosensor or film, are described. Furthermore, embodiments of a radiance camera based on an internal periodic but non-sinusoidal mask located in between the main cameral lens and the photosensor or film, are described. 
     Frequency Domain Representation 
     Let r(x) be the radiance in conventional x-space. This can be represented in frequency domain as follows:
 
 R (ω)=∫ r ( x ) e   iω·x   dx   (1)
 
     The following notations are used. The spatio-angular coordinates of a ray at a given plane orthogonal to the optical axis are represented as a vector: 
                   x   =     (         q           p         )             (   2   )               
where q is the location of ray-plane intersection, and p is a vector defining the two angles of that ray at location q. Paraxial approximation is used, assuming the angle is small. A 2-dimensional vector representation of a ray is shown in  FIGS. 2   a  and  2   b .  FIG. 2   a  illustrates the geometric representation of a ray as position and angle in an optical system.  FIG. 2B  illustrates the same ray as in  FIG. 2   b , but described as a point, or a vector x=(q, p), in a 2-D space.
 
     The spatial frequency ω g  and the angular frequency ω p  may be represented in a similar way as a 4-D vector: 
     
       
         
           
             
               
                 
                   ω 
                   = 
                   
                     ( 
                     
                       
                         
                           
                             ω 
                             q 
                           
                         
                       
                       
                         
                           
                             ω 
                             p 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     To simplify the description and the Figures, 2-D radiance with 1-dimensional position q and angle p for each ray may be used. The dot product may be defined as:
 
ω· x=ω   q   q+ω   p   p  
 
Transformations of the Radiance
 
     The following summarizes and extends transformations of radiance in optical systems. A ray x may be transformed as described below. 
     Both lens L and translation T may be described by linear transforms x′=A x  of the ray as a position-angle vector (see equation (2)) by the following matrices: 
                   L   =     (         1       0             -     1   f           1         )             (   4   )               T   =     (         1       t           0       1         )             (   5   )               
where f is the focal length of the lens, and t is the translation (distance of flight). A prism deviates the ray by a fixed angle p prism , so that p′=p+p prism .
 
     The combined action of several such elements may be described by the composition of all those elements. This provides the ability to build the model of essentially any optical system, such as a multi-element camera lens or radiance camera, as a linear or affine transform. 
     In a non-absorbing optical system, the radiance is conserved. In other words, the radiance does not change along a ray during travel or transformation by optical elements. The mathematical representation of this fact is that any optical matrix is symplectic. The following property of the transforms, that the determinant of any optical matrix is 1, may be used herein:
 
 detA= 1  (6)
 
     The above may also be seen directly from equations (4) and (5). 
     Based on the above-mentioned conservation law that, n a non-absorbing optical system, the radiance is conserved, the radiance r′ after a transform is related to the radiance r before the transform by the following equation:
 
 r ′( x )= r ( x   0 )= r ( A   −1   x )  (7)
 
where x 0  is the ray, which has been mapped into x by the optical transformation A, i.e. x=Ax 0 .
 
     Equation (7) may be expressed in frequency representation as follows: 
                             R   ′     ⁡     (   ω   )       =       ⁢     ∫         r   ′     ⁡     (   x   )       ⁢     ⅇ     ′ω   ·   x       ⁢     ⅆ   x                     =       ⁢     ∫       r   ⁡     (       A     -   1       ⁢   x     )       ⁢     ⅇ     ′ω   ·   x       ⁢     ⅆ   x                     =       ⁢     ∫       r   ⁡     (       A     -   1       ⁢   x     )       ⁢     ⅇ     ′ω   ⁢           ⁢     AA     -   1       ⁢   x       ⁢     ⅆ   x                     =       ⁢     ∫       r   ⁡     (     x   0     )       ⁢     ⅇ     ′ω   ⁢           ⁢     A   ·     x   0           ⁢     ⅆ     x   0                       =       ⁢     R   ⁡     (       A   T     ⁢   ω     )                     (   8   )               
where A T  is the transposed matrix, and equation (6) is used for the change of variables from x to x 0 . Note that this expression is derived for any optical transform A, while conventional works have only considered the special cases.
 
     The above results may be summarized as follows:
 
 x′=Ax   (9)
 
 r ′( x )= r ( A   −1   x )  (10)
 
 R ′(ω)= R ( A   T ω)  (11)
 
Radiance Cameras in the Frequency Domain
 
The Pinhole Light-Field Camera
 
     One type of radiance camera, which may be referred to as a pinhole light-field camera, may be described as an array of pinhole cameras with the same focal distance f, as illustrated in  FIG. 3 . This array of “cameras” may be placed at the focal plane of a conventional camera, typically but not necessarily a large format camera. Note that, in  FIG. 3 , only the focal plane with the array of pinholes is represented. 
     The mathematical representation for the radiance transformations inside a pinhole light-field camera in the frequency domain is described below. This representation may be used throughout the description. 
     Consider a 1-dimensional pinhole light-field camera and the corresponding 2-D radiance. Just before the array of pinholes, the radiance is:
 
 r ( x )= r ( q, p )
 
     Just after the array of pinholes, the radiance is: 
                       r   ′     ⁡     (     q   ,   p     )       =       r   ⁡     (     q   ,   p     )       ⁢       ∑     m   =     -   ∞       ∞     ⁢           ⁢     δ   ⁡     (     q   -   mb     )                   (   1   )               
where b is the pitch (distance between pinholes). In frequency representation this radiance may be written based on the Poisson summation formula as:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             R 
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                               r 
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                                     ( 
                                     
                                       q 
                                       - 
                                       mb 
                                     
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                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     ⅈω 
                                     · 
                                     x 
                                   
                                 
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                   13 
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     Assuming a band-limited signal, this result shows that the radiance after the pinholes consists of multiple copies of the original radiance, shifted in their frequencies by: 
             n   ⁢       2   ⁢   π     b           
for all integers n, as shown in  FIG. 4   a , which illustrates a band-limited signal after the array of pinholes.  FIG. 4   b  illustrates the shear of the signal after traveling a distance f.
 
     After traveling a distance f from the pinholes to the image plane, the radiance is transformed by the translation matrix (5) transposed, according to equation (11). The resultant radiance R f  that reaches the film plane is: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       f 
                     
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                           ∞ 
                         
                       
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                   14 
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     It can be seen that the signal is sheared in the direction of angular frequency. 
     This is represented in  FIG. 4   b , which illustrates the shear of the signal after traveling a distance f. An observation is that a different angular part of each copy intersects with the ω q  axis. Since the film (or sensor) responds only to the zero angular frequency, it records only the thin slice where the spectrum intersects with the ω q  axis. 
     By picking up slices in the image at different angular frequencies and stacking the slices along the ω q  axis, the original signal R(ω q , ω p ) may be reconstructed, as shown in  FIGS. 4   c  and  4   d  which illustrate reconstructing the original signal before the pinhole array by combining samples at different intersections with the ω q  axis. Finally, an inverse Fourier transform may be applied to convert the radiance into the familiar spatio-angular representation r(x). 
     From the above analysis of a pinhole light-field camera in the frequency domain, radiance capture by multiplexing in the frequency domain may be applied to pinhole light-field cameras. A frequency domain demultiplexing method may be applied to the captured radiance to demultiplex the radiance information. The angular information of radiance images captured with a pinhole light-field camera may, for example, be demultiplexed using a frequency domain demultiplexing method as illustrated in  FIG. 8  to generate multiple parallax views of a scene. In addition, while a pinhole array is periodic, the periodicity of the pinholes in the array may be arbitrary. In other words, the pinhole arrays that may be used as a mask in a pinhole light-field or radiance camera are not limited to sinusoidal masks such as sine masks and/or cosine masks. 
     Replacing the Pinhole Array with a Lens Array—the Integral Camera 
     The pinholes in the pinhole light-field camera design may be replaced with lenses. Just as with a single pinhole camera, lenses gather much more light and produce better image quality than small pinholes. Such a radiance camera may be referred to as an integral camera. Different versions of the integral camera have been proposed, including the plenoptic camera illustrated in  FIG. 1   b.    
     An analysis of the integral camera in frequency space may be performed according to the following:
         An array of pinholes, as in the pinhole camera, may be considered, only shifted by a constant (for all pinholes) vector a. Each pinhole is covered by a prism with angle of deviation depending on the shift, defined as       

     
       
         
           
             
               p 
               prism 
             
             = 
             
               
                 a 
                 f 
               
               . 
             
           
         
       
         
         
           
             The superposition of multiple arrays of such pinhole-prisms may be considered, and it may be shown that they all contribute to the final image in the same way. A conventional integral camera may be based on this coherent action of different arrays. Such a camera may be viewed as the limiting case where the plane is made completely of pinhole-prisms and all the light goes through. Each microlens is formed by the corresponding prisms, as a Fresnel lens. 
           
         
       
    
     Following the above derivation for the pinhole light-field camera in equation (13), the radiance after the pinhole-prism array may be expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             R 
                             ′ 
                           
                           ⁡ 
                           
                             ( 
                             ω 
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           ∫ 
                           
                             
                               r 
                               ⁡ 
                               
                                 ( 
                                 
                                   q 
                                   , 
                                   
                                     p 
                                     + 
                                     
                                       a 
                                       f 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 m 
                                 
                                     
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   δ 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       q 
                                       - 
                                       mb 
                                       - 
                                       a 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     ⅈω 
                                     · 
                                     x 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   x 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             b 
                           
                           ⁢ 
                           
                             ∫ 
                             
                               
                                 r 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     q 
                                     , 
                                     
                                       p 
                                       + 
                                       
                                         a 
                                         f 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   ∑ 
                                   n 
                                   
                                       
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     ⅇ 
                                     
                                       in 
                                       ⁢ 
                                       
                                         
                                           2 
                                           ⁢ 
                                           
                                             π 
                                             ( 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               q 
                                               - 
                                               a 
                                             
                                             ) 
                                           
                                         
                                         b 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     ⅇ 
                                     
                                       ⅈ 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             
                                               ω 
                                               q 
                                             
                                             ⁢ 
                                             q 
                                           
                                           + 
                                           
                                             
                                               ω 
                                               p 
                                             
                                             ⁢ 
                                             p 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     q 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     p 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             b 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               n 
                               
                                   
                               
                             
                             ⁢ 
                             
                               
                                 ⅇ 
                                 
                                   - 
                                   
                                     ⅈ 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             ω 
                                             p 
                                           
                                           ⁢ 
                                           
                                             a 
                                             f 
                                           
                                         
                                         + 
                                         
                                           n 
                                           ⁢ 
                                           
                                             
                                               2 
                                               ⁢ 
                                               π 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               a 
                                             
                                             b 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         ω 
                                         q 
                                       
                                       + 
                                       
                                         n 
                                         ⁢ 
                                         
                                           
                                             2 
                                             ⁢ 
                                             π 
                                           
                                           b 
                                         
                                       
                                     
                                     , 
                                     
                                       w 
                                       p 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Note that additional phase multipliers are now present in each term of the sum. After the pinhole-prism array, the light travels a distance f to the film plane. Using equations (5) and (9), the following expression for the radiance at the film (sensor) may be obtained: 
     
       
         
           
             
               
                 R 
                 f 
               
               ⁡ 
               
                 ( 
                 ω 
                 ) 
               
             
             = 
             
               
                 1 
                 b 
               
               ⁢ 
               
                 
                   ∑ 
                   n 
                   
                       
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ⅇ 
                     
                       - 
                       
                         ⅈ 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     f 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ω 
                                       q 
                                     
                                   
                                   + 
                                   
                                     ω 
                                     p 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 a 
                                 f 
                               
                             
                             + 
                             
                               n 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   a 
                                 
                                 b 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     R 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             ω 
                             q 
                           
                           + 
                           
                             n 
                             ⁢ 
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                               b 
                             
                           
                         
                         , 
                         
                           
                             f 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ω 
                               q 
                             
                           
                           + 
                           
                             ω 
                             p 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     As explained above, the film (or sensor) only records zero angular frequencies. Therefore, by restricting ω to the ω q  axis, the following expression may be obtained: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       f 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           ω 
                           q 
                         
                         , 
                         0 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       b 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         n 
                         
                             
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ⅇ 
                           
                             - 
                             
                               ⅈ 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       ω 
                                       q 
                                     
                                     ⁢ 
                                     a 
                                   
                                   + 
                                   
                                     n 
                                     ⁢ 
                                     
                                       
                                         2 
                                         ⁢ 
                                         π 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         a 
                                       
                                       b 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   ω 
                                   q 
                                 
                                 + 
                                 
                                   n 
                                   ⁢ 
                                   
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                     b 
                                   
                                 
                               
                               , 
                               
                                 f 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ω 
                                   q 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     An effect of coherence may be easily observed for a small a. It takes place due to the term: 
                 ω   q     ⁢   a     +     n   ⁢       2   ⁢   π   ⁢           ⁢   a     b             
where ω q  is within
 
             π   b         
from the corresponding center (peak), which is at frequency
 
             n   ⁢       2   ⁢   π     b           
in each block. For every exponential term with frequency ω q , there is another term with frequency:
 
                 -   n     ⁢       2   ⁢   π     b       -     ω   q           
inside the same block, but on the other side of the center. Those two frequencies produce opposite phases, which results in a real positive term:
 
             cos   ⁡     (       (       ω   q     +     n   ⁢       2   ⁢   π     b         )     ⁢   a     )           
This term for a small a is close to 1 for all rays.
 
Based on this analysis, the integral camera will also work with lenses for which a can be as big as
 
             b   2         
and the area of the plane is completely covered. All the terms are still positive, but the efficiency of rays far from the center is lower, and high frequencies will be attenuated.
 
     The above analysis proves that the frequency method for multiplexing radiance, described in the case of the pinhole light-field camera, is also valid for a microlens-based integral camera. Similarly, the plenoptic camera, e.g. as illustrated in  FIG. 1   b , and other lens-based radiance cameras that may be shown equivalent to it, can be analyzed using this formulation. 
     From the above analysis, radiance capture by multiplexing in the frequency domain may be applied to lens-based radiance cameras in general. It follows that a frequency domain demultiplexing method may be applied to the radiance captured by a lens-based radiance camera to demultiplex the radiance information. The angular information of radiance images captured with a lens-based radiance camera may, for example, be demultiplexed using a frequency domain demultiplexing method as illustrated in  FIG. 8  to generate multiple parallax views of a scene. 
     Replacing the Pinhole Array with a Mask 
     Radiance cameras that use a periodic sinusoidal mask (e.g., a cosine mask) instead of pinholes or microlenses between the photosensor and the main lens of the camera, and proximate to the photosensor, have been proposed. One way to analyze these mask-based radiance cameras would be to start again with the pinhole formula derived for the pinhole light-field camera, and instead of prisms assume appropriate attenuation at each pinhole. On the other hand, it is also possible to directly derive the result for periodic attenuation functions, such as: 
     
       
         
           
             
               1 
               2 
             
             ⁢ 
             
               ( 
               
                 1 
                 + 
                 
                   cos 
                   ⁡ 
                   
                     ( 
                     
                       
                         ω 
                         0 
                       
                       ⁢ 
                       q 
                     
                     ) 
                   
                 
               
               ) 
             
           
         
       
     
     The radiance after the attenuating mask may be represented as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             R 
                             ′ 
                           
                           ⁡ 
                           
                             ( 
                             ω 
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 ω 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               ∫ 
                               
                                 
                                   r 
                                   ⁡ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ω 
                                         0 
                                       
                                       ⁢ 
                                       q 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     ⅈω 
                                     · 
                                     x 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   x 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 ω 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               1 
                               4 
                             
                             ⁢ 
                             
                               ∫ 
                               
                                 
                                   r 
                                   ⁡ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       ⅇ 
                                       
                                         
                                           ⅈω 
                                           0 
                                         
                                         ⁢ 
                                         q 
                                       
                                     
                                     + 
                                     
                                       ⅇ 
                                       
                                         
                                           - 
                                           
                                             ⅈω 
                                             0 
                                           
                                         
                                         ⁢ 
                                         q 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     ⅈω 
                                     · 
                                     x 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   x 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 ω 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               1 
                               4 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   R 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           ω 
                                           q 
                                         
                                         + 
                                         
                                           ω 
                                           0 
                                         
                                       
                                       , 
                                       
                                         ω 
                                         p 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   R 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           ω 
                                           q 
                                         
                                         - 
                                         
                                           ω 
                                           0 
                                         
                                       
                                       , 
                                       
                                         ω 
                                         p 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     After the mask, the signal travels a distance f to the sensor. Again using equations (5) and (11) the following expression for the radiance may be obtained: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       f 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           ω 
                           q 
                         
                         , 
                         
                           ω 
                           p 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             
                               ω 
                               q 
                             
                             , 
                             
                               
                                 f 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ω 
                                   q 
                                 
                               
                               + 
                               
                                 ω 
                                 p 
                               
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             R 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     ω 
                                     q 
                                   
                                   + 
                                   
                                     ω 
                                     0 
                                   
                                 
                                 , 
                                 
                                   
                                     f 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ω 
                                       q 
                                     
                                   
                                   + 
                                   
                                     ω 
                                     p 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             R 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     ω 
                                     q 
                                   
                                   - 
                                   
                                     ω 
                                     0 
                                   
                                 
                                 , 
                                 
                                   
                                     f 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ω 
                                       q 
                                     
                                   
                                   + 
                                   
                                     ω 
                                     p 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     Again, duplication of the band-limited signal into multiple blocks and shearing proportional to the travel distance may be observed. It is important to note that any periodic mask, not just sinusoidal masks such as cosine masks, may be analyzed this way based on Fourier series expansion and considering individual component frequencies. Samples of the signal on the ω q  axis may be used to reconstruct the complete radiance R(ω). 
     Placing the Array in Front of the Camera 
     Another type of radiance camera may be implemented by placing any one of the optical elements or arrays (mask, microlens array, pinhole array) described as internal elements in relation to the various radiance camera designs in front of the main lens of a conventional camera instead of inside the camera between the photosensor and the main lens, and focusing the camera slightly behind the array. This external array radiance camera design is possible based on the fact that the image inside any camera is 3-dimensional, and is a distorted copy of the outside world. It is clear that the structures placed inside the camera have corresponding structures in the outside world. This is based on the mapping defined by the main camera lens. 
     The photosensor plane corresponds to the plane of focus, and any optical elements in front of the photosensor plane may be replaced by their enlarged copies in the real world, in front of the external plane of focus. Because of this correspondence, and based on the lens formula, optical elements may be built or placed in front of the camera and used as if they were microstructures inside the camera. Later in this document, in the section titled External mask-based radiance camera, a discussion is provided that is directed at replacing a fine mask or screen in front of the photosensor or film, in an area not accessible due to the cover glass, with a non-refractive mask, e.g. a net, mesh or screen, in front of the camera, and embodiments of an external mask-based radiance camera based on this notion are described. 
     Matching the F/Numbers 
     For lens-based radiance cameras that employ an array of microlenses inside the camera, such as the plenoptic camera illustrated in  FIG. 1   b , there exists a restriction that the F/numbers of the main camera lens and the microlenses should be matched. This restriction is based on the following characteristics of or observations about such cameras. If densely packed microlenses in a radiance camera had a smaller F/number than the main (objective) lens, then parts of the images of the main lens would extend beyond the area covered by the corresponding microlens, and would interfere with the images refracted by the neighboring microlens, and vice versa. If the F/number of the microlenses were bigger, then part of the area of the photosensor under each microlens would not be used. Thus, to maximize usage of the photosensor and to minimize interference, the F/number of the microlenses should match the F/number of the main or objective lens in a lens-based radiance camera. 
     Thus, a photographer is not free to change the aperture of the main camera lens without considering the current aperture of the microlenses in a lens-based radiance camera. Whether this restriction is relaxed in any way for other radiance cameras that are not based on microlenses, and whether there exists a quantity equivalent to F/number in cases other than microlenses, are questions that may be addressed via frequency domain analysis of radiance cameras. 
     The final expression for the radiance in all radiance cameras has a second (angular frequency) argument in R equal to fω q , where fis the distance from the pinholes, microlenses or mask to the photosensor. This is a measure for the amount of shear, which can be seen as the tilt of the line fω q  in  FIG. 4   b . Assume a radiance camera is sampling the angular frequency N times, i.e., copies of the signal that intersect with the ω q  axis N times. For example, this could be a mask containing N frequencies at interval ω 0 , or N peaks, including the zero frequency peak. The frequency spectrum of this signal covers an interval of Nω 0  in the horizontal axis. Because of the tilt, those peaks are spread in the vertical ω p  axis in an interval of Nω 0 . Therefore, the following expression holds:
 
2ω p0   =fNω   0   (19)
 
where ωp 0  is the maximal angular frequency of the original signal. The width of the cone of rays (maximal angle of rays) coming to a point on the film plane in a camera is
 
               1   F     ,         
where F is the F/number of the main lens. If the maximal resolution (number of lines) in a radiance camera in an angular direction is N, then the maximal angular frequency would be ω 0 =2πNF. By substituting in equation (19), the following equation may be obtained:
 
 fω   0 =4 πF   (20)
 
     Since the wavelength is b, so that 
                 ω   0     =       2   ⁢   π     b       ,         
the following equation may be obtained:
 
     
       
         
           
             
               
                 
                   
                     f 
                     ⁢ 
                     
                       
                         2 
                         ⁢ 
                         π 
                       
                       b 
                     
                   
                   = 
                   
                     4 
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     F 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     The maximal spatial frequency in the initial band-limited spectrum is 
                 ω   0     2     ,         
and the signal has wavelength  2   b . In this way, the following equation may be obtained:
 
     
       
         
           
             
               
                 
                   
                     f 
                     b 
                   
                   = 
                   F 
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     Thus, all radiance cameras multiplexing in the frequency domain should satisfy the F/number matching condition of equation (22), where F is the F/number of the objective lens, b is the pitch of the pinholes or microlenses, or the period of the lowest frequency in the mask, and f is the distance from the outer surface of the mask or array of pinholes or microlenses (the surface closest to the objective lens) to the sensor for internal mask, pinhole array, and microlens radiance cameras. For external equivalents to internal radiance cameras, such as the external mask-based camera  300  illustrated in  FIG. 19 , f is the distance from the external mask to the plane at which the main lens is focused behind the mask. 
     Demultiplexing in the Frequency Domain 
     The method of frequency domain analysis has been applied in the sections above to the images captured by the various radiance camera designs. In this section, methods of demultiplexing in the frequency domain to render images from radiance captured by the different radiance camera designs is described, and examples from each of the various radiance camera designs are shown. 
     Methods of Demultiplexing 
     In all of the aforementioned radiance camera designs, the 4-dimensional radiance is multiplexed onto the 2-dimensional camera sensor or film. This process of radiance multiplexing is given by equations (14), (16) and (18) for the respective camera designs. It is noted that the entire 4-D light field is encoded in a radiance image captured with a radiance camera. 
       FIGS. 5   a ,  6   a , and  7   a  illustrate exemplary images obtained from the three aforementioned radiance camera designs.  FIG. 5   a  illustrates an exemplary image obtained from a lens-based radiance, or integral, camera.  FIG. 6   a  illustrates an exemplary image obtained from a mask-based radiance camera in which the mask is located internal to the camera between the photosensor and the main lens.  FIG. 7   a  illustrates an exemplary image obtained from an external mask-based radiance camera, such as camera  300  of  FIG. 19 , in which a net or mesh is placed in front of the main lens of a conventional camera.  FIGS. 5   b ,  6   b , and  7   b  illustrate a zoom-in of a region of the images illustrated in  FIGS. 5   a ,  6   a , and  7   a , respectively, and show more detail.  FIGS. 5   c ,  6   c , and  7   c  illustrate the magnitudes of the 2-D Fourier transforms of the images illustrated in  FIGS. 5   a ,  6   a , and  7   a , respectively. The shifted slices or tiles of the transform are visible in each of the images shown in  FIGS. 5   c ,  6   c , and  7   c . Notice that these slices or tiles are placed at equal distances both horizontally and vertically in the case of the lens-based radiance camera image of  FIGS. 5   a - 5   c  and external mask-based radiance camera image of  FIGS. 7   a - 7   c , and only horizontally in the case of the mask-based radiance camera images of  FIGS. 6   a - 6   c . This is due to the use of a mask consisting of only vertical lines in the mask-based radiance camera (see  FIG. 6   b ). In all three cases, examples will be shown of extracting horizontal parallax, but it is noted that extending the method to obtain parallax in both directions is straightforward. 
     There exist several conventional techniques that may be used to extract individual parallax views from a radiance image. Frequency domain multiplexing techniques have been described for radiance images captured with conventional mask-based radiance cameras that specifically use cosine masks, and a frequency domain demultiplexing method may be applied to these radiance images. In the case of lens-based radiance cameras, spatial multiplexing techniques as opposed to frequency domain multiplexing techniques are conventionally used. In an exemplary spatial multiplexing technique, pixels belonging to each “little camera” of a radiance camera (e.g., to each microlens in a microlens array) may be extracted from the captured image, rearranged and put into individual images, so that a 2-D array of 2-D images is obtained. However, the frequency domain analysis of various radiance cameras provided above has shown that frequency domain multiplexing can be applied to lens-based radiance cameras and to pinhole light-field cameras in addition to mask-based radiance cameras, to external mask-based radiance cameras, and to mask-based cameras with masks that are not necessarily sinusoidal masks. It follows that a frequency domain demultiplexing method, such as the one described below, may be applied to radiance images captured with other types of radiance cameras than conventional cosine mask-based radiance cameras. 
       FIG. 8  illustrates a method of demultiplexing the angular information of a radiance image captured using a radiance camera, according to one embodiment. The Figure shows an exemplary application of a frequency domain demultiplexing method to the radiance image illustrated in  FIG. 6   a , which was captured using an internal mask-based radiance camera, but it is noted that the same or a similar method may be applied to radiance images captured with the other types of radiance cameras, including but not limited to lens-based radiance cameras and external mask-based radiance cameras. 
     The frequency domain demultiplexing method illustrated in  FIG. 8  may be based on the separability of the Fourier transform  200  of the original captured radiance image. Depending on the configuration of the optical elements in the radiance camera (whether lens-based or mask-based), three or four dimensions may be multiplexed in the radiance, with two spatial and one or two angular dimensions. For example, a radiance image captured using a mask  302 C of  FIG. 17  may include only one angular dimension, while a radiance image captured with a mask  302 A of  FIG. 17  may include two angular dimensions. With reference to  FIGS. 4   a  through  4   d , the slices or tiles  204  of the 2-D Fourier transform  200  may be extracted, as indicated at  202 . As indicated at  206 , a 2-D inverse Fourier transform (IFFT) is individually applied to each of the slices to obtain intermediate images  208 . As indicated at  210 , the intermediate images  208  are stacked together to form a 3-D image or a 4-D image  212 , depending on the number of angular dimensions in the radiance. Final horizontal parallax images  216  may be obtained by applying a 1-D or 2-D inverse Fourier transform (IFFT) along the angular dimension of the 3-D image or along the two angular dimensions of the 4-D image and unstacking the results, as indicated at  214 . This process is effectively performing a 3-D IFFT. In the general case of horizontal and vertical parallax, the process is extended to 4-D IFFT. Again,  FIG. 8  is directed at extracting horizontal parallax, but it is noted that extending the method to obtain parallax in both horizontal and vertical directions is straightforward. In one embodiment, an extension of the method to extract vertical parallax may apply the same or similar elements  202 ,  206 ,  210  and  214  on the vertical axis of the Fourier transform, e.g. the images shown in  FIGS. 5   c  and  7   c , of the radiance image. 
       FIGS. 9   a  and  9   b  illustrate a method of correcting the effect of waves due to small shifts or misalignments in the FFT, according to one embodiment. Good artifact-free results are very sensitive to determining the location of the centers of the slices or tiles in the Fourier transforms. The Fourier transforms of the images may be obtained by Fast Fourier Transform, which makes the location of the centers of the slices ambiguous due to the discretization. There may be a misplacement error within one pixel around each center, which may cause low-frequency waves in the final parallax images. In one embodiment, this problem may be addressed by multiplying the images before the last 1-D IFFT by a linear phase that corresponds to the subpixel shift in the FFT to more correctly determine the centers of the slices.  FIG. 9   a  shows an image from a mask-based radiance camera before the phase correction is applied to eliminate the low-frequency waves, and  FIG. 9   b  shows the image after the phase correction is applied to eliminate the low-frequency waves. 
     Embodiments of the frequency domain demultiplexing method described above may be implemented in software as or in one or more frequency domain demultiplexing modules. The module(s) may, for example, be implemented in a radiance image processing application or library.  FIG. 22  illustrates an exemplary computer system in which embodiments of the frequency domain demultiplexing module may be implemented. 
       FIG. 10  illustrates a frequency domain demultiplexing module, according to one embodiment. Radiance image  410  may be captured with any of a variety of radiance cameras, including but not limited to various lens-based radiance cameras, internal mask-based radiance cameras, external mask-based radiance cameras, internal and external mask-based radiance cameras that use periodic masks that are not necessarily sinusoidal (e.g., cosine) masks, radiance cameras that use an internal or external net, screen or mesh as a mask rather than a conventional mask, and pinhole light-field cameras. Frequency domain demultiplexing module  400  obtains or receives an input radiance image  410 . Frequency domain demultiplexing module  400  performs a frequency domain demultiplexing method, for example as described in  FIG. 8 , on the input image  410  to generate multiple output images  440 , for example multiple parallax views of a scene for which the radiance information was captured in radiance image  410 . In one embodiment, during the method, the method of correcting the effect of waves due to small shifts or misalignments in the FFT, as described above in reference to  FIGS. 9   a  and  9   b , may be applied. Output images  440  may, for example, be stored to a storage medium  450 , such as system memory, a disk drive, DVD, CD, etc. 
     In one embodiment, frequency domain demultiplexing module  400  may provide a user interface that provides one or more textual and/or graphical user interface elements, modes or techniques via which a user may view or control various aspects of frequency domain demultiplexing. For example, the user interface may include user interface elements that allow a user to select input and output files, to specify optical characteristics of the radiance camera used to capture the input radiance image, and so on. 
     Radiance Camera Embodiments 
     Embodiments of the frequency domain demultiplexing method of  FIG. 8  may be applied to radiance images captured with various types of radiance cameras, including but not limited to lens-based radiance cameras and mask-based radiance cameras. Several embodiments of different types of radiance cameras are described below. 
     Lens-Based Radiance Cameras 
     Embodiments of the frequency domain demultiplexing method of  FIG. 8  may be applied to radiance images captured with lens-based radiance cameras, for example radiance images captured with a plenoptic camera such as plenoptic camera  102  illustrated in  FIG. 1   b . To illustrate the frequency domain demultiplexing method of  FIG. 8  for a lens-based camera, a simulation of the plenoptic/integral camera may be accomplished by taking multiple images of a scene with one conventional lens camera. For example, as a simulation of the plenoptic/integral camera, 49 images of a scene may be taken from equally spaced locations. The centers of projection may be arranged on a plane as a 7×7 grid, with the cameras pointed perpendicular to the plane. The final image is made up of 7-pixel×7-pixel blocks, each of which consists of 49 pixels taken from the same location in all 49 images.  FIG. 11   a  shows an exemplary radiance image taken using the above apparatus. Zoomed area  250  shows the formed blocks.  FIG. 11   b  shows the absolute value of the Fourier transform of the radiance image of  FIG. 11   a . To obtain horizontal parallax, the frequency domain demultiplexing method of  FIG. 8  may be applied, with 7 views. Two images resulting from this process are shown in  FIGS. 12   a  and  12   b , which show two stereo views from the frequency domain reconstructed light field of  FIGS. 11   a  and  11   b . Small parallax is visible in this stereo pair at close examination. Note that the left and right images are switched. The left and right images are switched so that stereo fusion can be achieved with crossed eyes observation. Note that the embodiments of lens-based radiance cameras described herein are exemplary, and not intended to be limiting. Other embodiments of lens-based radiance cameras, using various types of film-based or digital camera designs, are possible and contemplated. 
     Mask-Based Radiance Camera Implementations 
     Embodiments of the frequency domain demultiplexing method of  FIG. 8  may be applied to radiance images captured with non-refractive mask-based radiance cameras. Various different mask-based radiance camera designs are described that may be used to illustrate the frequency domain demultiplexing method of  FIG. 8  for images captured with a mask-based radiance camera. In order to achieve good resolution, a small value of the largest period b, on the order of 0.1 mm, may be used. With F/number of the main lens equal to 4, the mask may be placed about 0.4 mm from the surface of the sensor, which may not be possible due to the cover glass. Because of the cover glass restriction, embodiments of a mask-based radiance camera based on a film-based, medium format camera may be used. Reasons for using a medium format camera may include the larger image that gives potential for higher resolution and easier access to the film back, where modifications may be made to convert the conventional camera into a mask-based radiance camera. The embodiments of mask-based radiance cameras described herein are exemplary, and not intended to be limiting. Other embodiments using other types of film-based or digital camera designs and/or other types of masks are possible and contemplated. 
     In an exemplary embodiment of a mask-based radiance camera, a Contax™ 645 medium format film camera with a film back may be used (see  FIG. 13 ). Note that various other film cameras with film backs could be substituted for the Contax™ camera. Exemplary embodiments that use different mesh, net or screen masks are described. Refer to  FIG. 17  for exemplary net- or mesh-like non-refractive masks. 
     In a first exemplary embodiment, a picture of a poster displaying a computer-generated grid is taken, and then the negative is used as a mask in front of the film in the film back. In one embodiment, the computer-generated grid is a 2-D cosine mask with 3 harmonics in both spatial dimensions. The spacing of 0.5 mm may be achieved by placing the developed negative between two thin glasses to form a non-refractive mask. The film that is being exposed slides directly on the surface of the glass. 
     In a second exemplary embodiment, a computer screen filter, e.g. a 3M™ computer screen filter, may be used as a non-refractive mask in front of the film in the film back. In one embodiment, the computer screen filter contains about 14 black lines per mm, and the lines are sandwiched between transparent plastic material 0.2 mm thick. As a result, the F/number of the mask is approximately 3.  FIG. 13  shows a Contax™ 645 camera, and the film back with a 3M™ computer screen filter attached to the window just in front of the film, according to one embodiment. Using this embodiment, a high-resolution radiance image of 14 samples/mm may be captured, where each sample contains complete angular information. 
     Results obtained with the second exemplary embodiment of a non-refractive screen mask are shown herein, but note that the results from the first exemplary embodiment of a non-refractive screen mask are similar. 
     A sequence of parallax movies, which are generated from pictures captured by the above exemplary internal mask-based radiance camera at different apertures, may be used to illustrate that the optimal F/number exemplary mask-based radiance camera is approximately 5.6. This value is slightly higher than the expected 3 or 4. Possible reasons are the refractive index of the plastic material, which increases optical path, and possible micro-spacing between the film and the 3M™ filter due to mechanical imperfection/dust. 
       FIG. 14  shows two stereo views generated, using a frequency domain multiplexing method as illustrated in  FIG. 8 , from a radiance image taken using the exemplary mask-based radiance camera with the mask at F/5.6. Selected areas show detail in which it is easy to spot parallax. Note that, as in  FIGS. 12   a  and  12   b , the left and right images are switched. 
     External Mask-Based Radiance Cameras 
     In the section titled Placing the array in front of the camera, it was demonstrated that a non-refractive mask or screen in front of the photosensor or film may be replaced with a non-refractive mask, e.g. a net or screen, array or grid of pinholes, etc., in front of the main camera lens. To demonstrate the method of frequency domain multiplexing for a radiance camera based on such an external mask, pictures may be taken with a conventional camera through a net, mesh, or screen in front of the camera (see mask  302 A of  FIG. 16  to view what such a net, mesh or screen may look like). For the demonstration, a conventional camera with an 80 mm lens and a digital back, without any modifications to the camera, may be used. For the demonstration, the mesh is placed approximately 2 meters (m) from the camera, and the camera is focused on a plane about 10 centimeters (cm) behind the net or mesh. With this apparatus, the cover glass problem of the mask-based radiance cameras described above may be overcome. Note that the above-described apparatus is exemplary, and not intended to be limiting. Embodiments of an external mask-based radiance camera  300  that employs similar principles demonstrated via the above exemplary apparatus are illustrated in and described in reference to  FIG. 19 . 
     By differentiating the lens equation: 
                 1   a     +     1   b       =     1   f           
the following is obtained:
 
     
       
         
           
             
               da 
               
                 a 
                 2 
               
             
             = 
             
               - 
               
                 db 
                 
                   b 
                   2 
                 
               
             
           
         
       
     
     Therefore, moving the focus by da=10 cm away from the net or mesh produces a movement of: 
                 -   da     ⁢       b   2       a   2         =     0.16   ⁢           ⁢   mm           
away from the photosensor surface. At the same time, the image of the 2 mm grid of the net or mesh has been reduced linearly to 0.08 mm, which gives an F/number of about 3, and high resolution.
 
       FIG. 15  shows a picture taken through a net, or mesh, using the apparatus described above. The Figure shows an image taken through a mesh with a conventional camera at F/number 4.  FIG. 16  shows two stereo views from the radiance generated from the picture shown in  FIG. 15 . The left and right images are switched. The two stereo views of the scene may be reconstructed using a method of demultiplexing in the frequency domain as illustrated in  FIG. 8 . 
       FIGS. 17 and 18  illustrate exemplary non-refractive masks, according to embodiments. With appropriate physical adaptations for mounting or otherwise attaching the masks in or to the cameras, the exemplary masks may, be used as internal masks, e.g. placed in front of the photosensor, for example in a film-back camera as previously described, or as external masks placed in front of the main camera lens, for example in an external mask-based radiance camera as illustrated in  FIG. 19 . The masks are non-refractive; that is, while the masks may act to modulate and/or attenuate the light, the masks do not act to bend the light rays. It is important to note that the exemplary non-refractive masks are not limited to sinusoidal (e.g., cosine or sine) masks. In other words, any periodic mask may be used in mask-based radiance cameras as described herein. 
       FIG. 17  shows examples of net- or mesh-like, non-refractive masks. Mask  302 A illustrates a rectangular mesh-like mask. Mask  302 B illustrates a circular mesh-like mask  302 B. A mesh, which also may be referred to as a net or screen, may include horizontally and vertically arranged opaque linear elements or grid lines, which collectively form a grid that modulates, but does not refract, light as the light passes through the grid. Generally, the opaque grid lines may be equally spaced in the two dimensions. Thus, the opaque grid lines act to form or define rows and columns of periodically spaced transparent (i.e., through which light may pass), non-refractive openings. In this example, the openings are square; however, in some embodiments, the openings may be rectangular but not square. In some embodiments, a mesh mask may include only horizontal or vertical lines, as illustrated in mask  302 C. 
       FIG. 18  shows examples of some other types of non-refractive masks  302  that, like the masks illustrated in  FIG. 17 , may be used as internal or external masks. Mask  302 D illustrates an exemplary circular mask that includes transparent circular openings, through which light may pass, in an opaque medium or surface. The circular openings in this example are periodically spaced, and arranged in horizontal rows and vertical columns. Other geometric shapes than circles may be used for the openings in other embodiments, e.g. squares, hexagons, rectangles, etc. Mask  302 E illustrates an exemplary circular mask that is composed of a grid of pinholes in an opaque medium or surface. The pinholes, which may also be referred to as openings, may typically be, but are not necessarily, circular. Like the circular openings of mask  302 D, the pinholes in this example are periodically spaced, and arranged in horizontal rows and vertical columns. Note that the pinhole mask  302 E appears similar to mask  302 D. However, the use of the term “pinhole” indicates that the size of the openings is very small, as in the optical notion of a “pinhole camera”; very small openings, or pinholes, have optical effects that are not produced, or not produced to the same degree, by larger openings. In other words, a mask with very small openings may be referred to as a pinhole mask, while a mask with larger openings, such as mask  302 D, is not technically a pinhole mask. Mask  302 F illustrates another pinhole mask, this one rectangular instead of circular, and in which the pinholes are aligned differently than the pinholes in mask  302 E. 
     In various embodiments, the opaque grid lines of the masks  302  illustrated in  FIG. 17 , or the opaque surface or medium through which there are openings of the masks  302  illustrated in  FIG. 18 , may be composed of a metal, a paint, an alloy, a plastic, a composite material, or any other suitable opaque substance, composition or material capable of being arranged, affixed or applied to form the linear, opaque elements of a grid, or an opaque surface through which openings may be provided, for use in an optical device. In some embodiments, the grid or surface with openings may be affixed to, painted on, or otherwise attached to a surface of a non-refracting, transparent sheet of glass. In other embodiments, the grid or surface with openings may be sandwiched between two non-refracting, transparent sheets of glass. Other non-refracting, transparent materials than glass may be used. One skilled in the art will recognize that other materials and methods of manufacturing such masks are possible. 
     Note that the openings in masks  302 , including but not limited to the exemplary masks illustrated in  FIGS. 17 and 18 , may be variably spaced without restriction on the spacing other than periodicity. That is, while the openings are arranged periodically, the distribution of the openings in the masks  302  is not limited to, for example, the cosine, or integer multiples of the cosine. In other words, while sinusoidal masks such as cosine masks may be used in embodiments, periodic masks other than sinusoidal masks may also be used in embodiments. 
     In some embodiments of an external mask-based camera such as camera  300  of  FIG. 19 , a mask  302  may be integrated with, or alternatively may be coupled to and decoupled from, an attachment, e.g. a tube. The attachment, in turn, may be coupled to and decoupled from a camera body. Alternatively, the mask  302  may be integrated with the attachment. As yet another alternative, the mask  302  may be integrated with the camera body, in front of the main or objective lens of the camera. In one embodiment, the mask  302 , when integrated with or coupled to a camera, is positioned so that the horizontal and vertical grid lines, or rows and columns of openings, are horizontal and vertical with respect to the camera, i.e. the photosensor, so that the horizontal grid lines (or rows of openings) are parallel to the horizontal axis of the photosensor, and the vertical grid lines (or columns of openings) are parallel to the vertical axis of the photosensor. 
     The exemplary masks of  FIGS. 17 and 18  are not intended to be limiting; other geometric shapes may be used for masks  302 , the number, thickness and spacing of the opaque elements or grid lines in mesh-like masks such as masks  302 A,  302 B, and  302 C may vary, and as mentioned above, the size, shape, and spacing of the openings in other types of masks such as masks  302 D,  302 E and  302 F may vary. 
     Embodiments of a radiance camera based on an external, non-refractive mask located in front of the main or objective camera lens, rather than between the main lens and the photosensor or film, are described.  FIG. 19  illustrates an exemplary radiance camera with an external mask, according to one embodiment. External mask radiance camera  300  may include a camera body  310 . Various types of cameras may be used in some embodiments, including both film-based and digital cameras, and standard, medium or large-format cameras. Thus, photosensor  330  may be conventional film or a device for digitally capturing light, for example a CCD, and photosensor  330  may be integrated into camera body  310  or mounted to camera body  310  via a film back such as in the camera shown in  FIG. 13 . Main (objective) lens  320  may be any of a variety of types of lenses, including lenses with different focal lengths or other optical characteristics, and may be integrated into camera body  310  or mounted to camera body  310  via an external lens attachment. 
     A non-refractive mask  302 , such as exemplary mesh-like masks  302 A through  302 C illustrated in  FIG. 17 , or exemplary masks  302 D through  302 F illustrated in  FIG. 18 , may be integrated with camera body  310  in front of the main lens  320 , or alternatively attachable to camera body  310  or to an external attachment  304 , with mask  302  positioned in front of the main lens  320  so that light from a scene to be photographed arrives at the main lens  320  after passing through the mask  302 . The mask  302  is a non-refractive element, and as such modulates and/or attenuates the incoming light rays but does not bend them. The mask  302  is not limited to sinusoidal masks such as cosine masks or sine masks; any arbitrary periodic mask may be used. In one embodiment, the mask  302  may be mounted in or integrated with a mask attachment  304 . Mask attachment  304  may be integrated with camera body  310 , or alternatively may be coupled to and decoupled from camera body  310  in front of main lens  320 . Note that the illustrated shape and size, including the length, of mask attachment  304  is exemplary, and not intended to be limiting. Furthermore, the sizes and spacing of the mask  302 , mask attachment  304 , main lens  320 , camera body  310 , and photosensor  330  are exemplary and not intended to be limiting. In other words, external mask-based radiance camera  300  is not necessarily drawn to scale. 
     In one embodiment, the main lens  320  may be focused on a plane  322  just behind the mask  302 , between the mask  302  and the main lens  320 . Light from plane  322  is refracted by main lens  320  onto photosensor  330 , which may in turn operate to capture a radiance image of the scene, e.g. when a shutter of the camera  300  is triggered. An exemplary radiance image captured with a mask (in this example, a net- or mesh-like mask) located in front of the main camera lens is shown in  FIG. 15 . 
     The angular information of radiance images captured with embodiments of external mask radiance camera  300  may be demultiplexed using an embodiment of the frequency domain demultiplexing method described in  FIG. 8  to generate multiple views of a scene. The frequency domain demultiplexing method may be implemented in an embodiment of a frequency domain demultiplexing module as illustrated in  FIG. 10 . 
     In general, embodiments of an external mask radiance camera  300  as described herein may include, in addition to the above-described elements, any other type of elements and features commonly found in digital cameras or other cameras including but not limited to conventional light-field and plenoptic cameras and medium- or large-format film cameras, and may also include additional elements and features not generally found in conventional cameras. Camera  300  may include a shutter, which may be located in front of or behind objective lens  320 . Camera  300  may include one or more processors, a power supply or power source, such as one or more replaceable or rechargeable batteries. Camera  300  may include a memory storage device or system for storing captured images or other information such as software. In one embodiment, the memory system may be or may include a removable/swappable storage device such as a memory stick. Camera  300  may include a screen (e.g., an LCD screen) for viewing scenes in front of the camera prior to capture and/or for viewing previously captured and/or rendered images. The screen may also be used to display one or more menus or other information to the user. Camera  300  may include one or more I/O interfaces, such as FireWire or Universal Serial Bus (USB) interfaces, for transferring information, e.g. captured images, software updates, and so on, to and from external devices such as computer systems or even other cameras. Camera  300  may include a shutter release that is activated to capture a radiance image of a subject or scene. Camera  300  may include one or more manual and/or automatic controls, for example controls for controlling optical aspects of the camera such as shutter speed, aperture, and the location of focal plane  322  of the main lens  330 , one or more controls for viewing and otherwise managing and manipulating captured images stored in a memory on the camera, etc. 
       FIG. 20  illustrates a method of capturing a radiance image with an external mask-based radiance camera, according to one embodiment. The external mask-based radiance camera multiplexes radiance in the frequency domain by optically mixing different spatial and angular frequency components of the light received from a scene, and captures the radiance information at a photosensor. As illustrated at  400 , light from a scene may be received at a mask  302 . The mask  302  is a non-refractive element, and modulates and/or attenuates the incoming light rays but does not bend them. Light that passes through the mask  302  is received at the main lens  320  of a camera  300 , as indicated at  402 . The main lens  320  may be focused on a plane  322  between the mask  302  and the main lens  320 , with the plane  322  proximate to the mask  302 . The received light may be refracted by the main lens  320  to a photosensor  330  of the camera  300 , as indicated at  404 . The photosensor  330  may capture the received light as a radiance image of the scene, as indicated at  406 . As indicated at  408 , the angular information in the captured radiance image may be demultiplexed using an embodiment of the frequency domain demultiplexing method described in  FIG. 8  to generate multiple views of the scene. The frequency domain demultiplexing method may be implemented in an embodiment of a frequency domain demultiplexing module as illustrated in  FIG. 10 . 
     In some embodiments, the captured radiance image and/or the multiple views generated by the frequency domain demultiplexing method may be stored to a memory medium or memory device. Note that, if the radiance image was originally captured to film, i.e. if the camera is a film camera, the radiance image may be digitized from the film or from a photograph produced from the film, for example using a film negative or photograph scanner, to generate a digital version of the radiance image that may be stored to a memory medium and/or processed by the frequency domain demultiplexing method of  FIG. 8 . 
       FIG. 21  illustrates an exemplary embodiment of an internal mask-based radiance camera based on a medium- or large-format film camera with a film back. In conjunction with current high-resolution scanners used to digitize captured images from negatives or prints, large-format film camera embodiments may be capable of up to 1 gigapixel, or even higher, resolution for the flat or light-field representation of the 4D radiance (the raw radiance image).  FIG. 13  shows an exemplary embodiment of a conventional medium-format film camera and a film back, with a computer screen filter, used as a mask, attached to the window just in front of the film. Another exemplary embodiment may, for example, be implemented in large-format film camera using a 135 mm objective lens  530  and 4×5 format film as the “photosensor” (in medium- and large-format cameras, single negatives of film are generally placed in a film holder  502  or cartridge that can be inserted into and removed from the camera body). Other objective lenses and/or other medium or large film formats, for example 8×10 format film, may be used in various embodiments. 
     Radiance camera  500  includes a mask  406 . Mask  406  is a non-refractive optical element, and as such modulates and/or attenuates light rays but does not bend them. Mask  506  may be a mesh-like mask such as exemplary mesh-like masks  302 A through  302 C illustrated in  FIG. 17 , or another type of mask such as exemplary masks  302 D through  302 F illustrated in  FIG. 18 . Note, however, that the mask  506  will generally be rectangular, and sized to match the format of the film camera. The mask  506  is not limited to sinusoidal masks such as cosine masks or sine masks; any arbitrary periodic mask may be used. 
     In one embodiment, a mechanism inside a film holder  502  of the large-format film camera holds the mask  506  so that the flat side of the glass base of the mask  506  is pressed against the film and the opaque surface of the mask  506  (the surface of mask  506  on which the opaque surface or medium is painted, attached, etc., with openings that are the transparent portion of the mask) is away from the film. In one embodiment, the thickness of the mask  506  is such that, when placed against the film, the opaque surface of the mask  506 , and the openings therein, is at a distance f (equivalent to the focal length of the mask  506 ) from the film. Other configurations of masks  506  are possible, and the configuration of the medium- or large-format film camera with a film back  502  makes it possible to easily change configurations of masks by simply using a different mask  506 . In one embodiment, microsheets  504  of glass may be used in the assembly as spacers or shims between the mask  506  and the film in film holder  502  to increase the distance from the mask  506  and the film to allow f (equivalent to the focal length of the mask  506 ) to be changed, for example to match a changed F/number for main lens  530 . An exemplary thickness of a microsheet  504  that may be used is 0.23 mm. Additional microsheets  404  may be added to provide additional spacing. The ability to insert or remove microsheet glass  504 , to insert or remove one or more microsheets  504  of glass, and the availability of microsheet glass  504  in different, precisely known thicknesses may provide spacing in a rigorously controlled manner. In some embodiments, other mechanisms than microsheet glass  504  may be used as spacers between the mask  506  and film holder  502  to adjust the distance between the mask  506  and film holder  502 . 
     As illustrated in  FIG. 21 , in one embodiment, the film holder  502  and mask  406  may be coupled to create assembly  510 . One or more microsheets  504  may optionally be inserted between the film holder  502  and mask  506  to provide additional spacing as necessary or desired. The assembly  510  may then be inserted into the large-format film camera. The combination of the large-format film camera and the assembly  510  effectively forms a radiance camera  500 . Radiance camera  500  may then be used to capture a radiance image of a scene on the film in film holder  502 . The assembly  510  may then be removed from the camera  500 , disassembled, and the film may be appropriately processed. The film negative and/or a print of the radiance image may then be digitized, for example using a high-resolution scanner or a device that generates digital images from negatives. The digitized radiance image may be stored to a storage device, such as a disk drive, DVD, CD, etc. The digitized radiance image may be demultiplexed according to the frequency domain demultiplexing method, implemented in a frequency domain demultiplexing module executing on a computer system. 
     Exemplary System 
     Various components of embodiments of a method for demultiplexing captured radiance in the frequency domain, as described herein, may be executed on one or more computer systems, which may interact with various other devices. One such computer system is illustrated by  FIG. 22 . In the illustrated embodiment, computer system  700  includes one or more processors  710  coupled to a system memory  720  via an input/output (I/O) interface  730 . Computer system  700  further includes a network interface  740  coupled to I/O interface  730 , and one or more input/output devices  750 , such as cursor control device  760 , keyboard  770 , audio device  790 , and display(s)  780 . In some embodiments, it is contemplated that embodiments may be implemented using a single instance of computer system  700 , while in other embodiments multiple such systems, or multiple nodes making up computer system  700 , may be configured to host different portions or instances of embodiments. For example, in one embodiment some elements may be implemented via one or more nodes of computer system  700  that are distinct from those nodes implementing other elements. 
     In various embodiments, computer system  700  may be a uniprocessor system including one processor  710 , or a multiprocessor system including several processors  710  (e.g., two, four, eight, or another suitable number). Processors  710  may be any suitable processor capable of executing instructions. For example, in various embodiments, processors  710  may be general-purpose or embedded processors implementing any of a variety of instruction set architectures (ISAs), such as the x86, PowerPC, SPARC, or MIPS ISAs, or any other suitable ISA. In multiprocessor systems, each of processors  710  may commonly, but not necessarily, implement the same ISA. 
     System memory  720  may be configured to store program instructions and/or data accessible by processor  710 . In various embodiments, system memory  720  may be implemented using any suitable memory technology, such as static random access memory (SRAM), synchronous dynamic RAM (SDRAM), nonvolatile/Flash-type memory, or any other type of memory. In the illustrated embodiment, program instructions and data implementing desired functions, such as those described above for a method for demultiplexing radiance in the frequency domain, are shown stored within system memory  720  as program instructions  725  and data storage  735 , respectively. In other embodiments, program instructions and/or data may be received, sent or stored upon different types of computer-accessible media or on similar media separate from system memory  720  or computer system  700 . Generally speaking, a computer-accessible medium may include storage media or memory media such as magnetic or optical media, e.g., disk or CD/DVD-ROM coupled to computer system  700  via I/O interface  730 . Program instructions and data stored via a computer-accessible medium may be transmitted by transmission media or signals such as electrical, electromagnetic, or digital signals, which may be conveyed via a communication medium such as a network and/or a wireless link, such as may be implemented via network interface  740 . 
     In one embodiment, I/O interface  730  may be configured to coordinate I/O traffic between processor  710 , system memory  720 , and any peripheral devices in the device, including network interface  740  or other peripheral interfaces, such as input/output devices  750 . In some embodiments, I/O interface  730  may perform any necessary protocol, timing or other data transformations to convert data signals from one component (e.g., system memory  720 ) into a format suitable for use by another component (e.g., processor  710 ). In some embodiments, I/O interface  730  may include support for devices attached through various types of peripheral buses, such as a variant of the Peripheral Component Interconnect (PCI) bus standard or the Universal Serial Bus (USB) standard, for example. In some embodiments, the function of I/O interface  730  may be split into two or more separate components, such as a north bridge and a south bridge, for example. In addition, in some embodiments some or all of the functionality of I/O interface  730 , such as an interface to system memory  720 , may be incorporated directly into processor  710 . 
     Network interface  740  may be configured to allow data to be exchanged between computer system  700  and other devices attached to a network, such as other computer systems, or between nodes of computer system  700 . In various embodiments, network interface  740  may support communication via wired or wireless general data networks, such as any suitable type of Ethernet network, for example; via telecommunications/telephony networks such as analog voice networks or digital fiber communications networks; via storage area networks such as Fibre Channel SANs, or via any other suitable type of network and/or protocol. 
     Input/output devices  750  may, in some embodiments, include one or more display terminals, keyboards, keypads, touchpads, scanning devices, voice or optical recognition devices, or any other devices suitable for entering or retrieving data by one or more computer system  700 . Multiple input/output devices  750  may be present in computer system  700  or may be distributed on various nodes of computer system  700 . In some embodiments, similar input/output devices may be separate from computer system  700  and may interact with one or more nodes of computer system  700  through a wired or wireless connection, such as over network interface  740 . 
     As shown in  FIG. 22 , memory  720  may include program instructions  725 , configured to implement embodiments of a method for demultiplexing radiance in the frequency domain as described herein, and data storage  735 , comprising various data accessible by program instructions  725 . In one embodiment, program instructions  725  may include software elements of a method for demultiplexing radiance in the frequency domain as illustrated in the above Figures. Data storage  735  may include data that may be used in embodiments. In other embodiments, other or different software elements and data may be included. 
     Those skilled in the art will appreciate that computer system  700  is merely illustrative and is not intended to limit the scope of a method for demultiplexing radiance in the frequency domain as described herein. In particular, the computer system and devices may include any combination of hardware or software that can perform the indicated functions, including computers, network devices, internet appliances, PDAs, wireless phones, pagers, etc. Computer system  700  may also be connected to other devices that are not illustrated, or instead may operate as a stand-alone system. In addition, the functionality provided by the illustrated components may in some embodiments be combined in fewer components or distributed in additional components. Similarly, in some embodiments, the functionality of some of the illustrated components may not be provided and/or other additional functionality may be available. 
     Those skilled in the art will also appreciate that, while various items are illustrated as being stored in memory or on storage while being used, these items or portions of them may be transferred between memory and other storage devices for purposes of memory management and data integrity. Alternatively, in other embodiments some or all of the software components may execute in memory on another device and communicate with the illustrated computer system via inter-computer communication. Some or all of the system components or data structures may also be stored (e.g., as instructions or structured data) on a computer-accessible medium or a portable article to be read by an appropriate drive, various examples of which are described above. In some embodiments, instructions stored on a computer-accessible medium separate from computer system  700  may be transmitted to computer system  700  via transmission media or signals such as electrical, electromagnetic, or digital signals, conveyed via a communication medium such as a network and/or a wireless link. Various embodiments may further include receiving, sending or storing instructions and/or data implemented in accordance with the foregoing description upon a computer-accessible medium. Accordingly, the present invention may be practiced with other computer system configurations. 
     CONCLUSION 
     Various embodiments may further include receiving, sending or storing instructions and/or data implemented in accordance with the foregoing description upon a computer-accessible medium. Generally speaking, a computer-accessible medium may include storage media or memory media such as magnetic or optical media, e.g., disk or DVD/CD-ROM, volatile or non-volatile media such as RAM (e.g. SDRAM, DDR, RDRAM, SRAM, etc.), ROM, etc. A computer-accessible medium may also include transmission media or signals such as electrical, electromagnetic, or digital signals, conveyed via a communication medium such as network and/or a wireless link. 
     The various methods as illustrated in the Figures and described herein represent exemplary embodiments of methods. The methods may be implemented in software, hardware, or a combination thereof. The order of method may be changed, and various elements may be added, reordered, combined, omitted, modified, etc. 
     Various modifications and changes may be made as would be obvious to a person skilled in the art having the benefit of this disclosure. It is intended that the invention embrace all such modifications and changes and, accordingly, the above description to be regarded in an illustrative rather than a restrictive sense.