Patent Publication Number: US-7901095-B2

Title: Resolution scalable view projection

Description:
BACKGROUND 
     1. Field of Invention 
     The present invention is related to a method of generation a light transport matrix T in a projector-camera system where the resolution of the camera is not greater than that of the projector. 
     2. Description of the Related Art 
     When projectors and cameras are combined, hybrid devices and systems that are capable of both projecting and capturing light are born. This emerging class of imaging devices and systems are known in the research community as projector-camera systems. Typically, images captured by one or more cameras, are used to estimate attributes about display environments, such as the geometric shapes of projection surfaces. The projectors in these projector-camera systems then adapt their projected images so as to compensate for shape irregularities in the projection surfaces to improve the resultant imagery. In other words, by using a camera, a projector can “see” distortions in a projected image, and then adjust its projected image so as to reduce the observed distortions. 
     In order to achieve this, the camera and projector need to be calibrated to each other&#39;s imaging parameters so as to assure that any observed image distortion is due to irregularities in the projection environment (i.e. surface irregularities), and not due to distortions inherent to the projector or camera, or due to their relative orientation to each other. 
     Thus, a key problem that builders of projector-camera systems and devices need to solve is the determination of the internal imaging parameters of each device (i.e. intrinsic parameters) and the determination of the geometric relationship between all projectors and cameras in the system (i.e. extrinsic parameters). This problem is commonly referred to as that of calibrating the system. 
     Even after a system has been substantially calibrated, however, the issue of adjusting a projection to compensate for distortions in a projected image is not straightforward. Identifying and compensating for projection distortion can be a very computationally intensive operation, which has traditionally greatly limited its application to non-specialized fields. 
     In an effort to better understand the calibration of projector-camera systems, Applicants studied multi-camera imaging systems found in the field of computer vision. Although such multi-camera imaging systems consist of only image photographing devices, and do not include any image projecting devices, a large body of work concerning the calibration of such multi-camera imaging systems exists in the field of computer vision, and it was thought that one might glean some insight from their approach toward calibrating multiple devices, albeit multiple image photographing devices. 
     A commonly used method in computer vision techniques for calibrating a camera in an imaging system is described in article, “A flexible new technique for camera calibration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330-1334, 2000, by Zhengyou Zhang, which is herein incorporated in its entirety by reference. In this method, multiple images of a flat object marked with a number of known feature points (typically forming a grid) are captured by the camera, with the flat object posed at a variety of known angles relative to the camera. The image location of each feature point is extracted, and since the relative location of each feature point is known, the collection of feature point locations can then be used to calibrate the camera. When two or more cameras are present in an imaging system, the intrinsic parameters as well as the geometric relationship between the present cameras can be estimated by having all cameras capture an image of the flat object at each pose angle. 
     Since projectors and cameras are very similar in terms of imaging geometry, it might seem reasonable to postulate that techniques suitable for calibrating cameras in multi-camera imaging systems might be suitable for calibrating projectors in projector-camera systems. However, since all camera calibration techniques require that the camera requiring calibration (i.e. the imaging device being calibrated) capture a number of images, it would appear that camera calibration techniques cannot readily be applied to projectors since projectors cannot capture images. 
     Therefore, in traditional projector-camera systems, at least two cameras have been needed, in addition to a projector. The two cameras are calibrated first, typically using multi-camera imaging system calibration techniques, to establish a stereo camera pair. More specifically, these systems use a “bootstrapping” procedure to calibrate the two cameras and form the stereo camera pair. As it is known in the art, a stereo camera pair can be used to estimate depth (i.e. achieve a pseudo perspective view) to establish a quasi-depth perception of feature points visible to the stereo camera pair. The calibrated stereo camera pair is then used to calibrate the projector. Basically, the establishment of this quasi-depth perception is used to identify surface depth irregularities of a projection surface and thereby of an image projected onto the projection surface. The projector can then be calibrated to compensate for the surface depth irregularities in the projected image. In essence, to calibrate the projector using this quasi-depth perception, the projector is first made to project feature points onto a display environment (i.e. the projection surface), which may have an irregular surface. The pre-calibrated, stereo camera pair is used to resolve the perspective depth location of the projected points. The projector can then be calibrated to compensate for surface/depth irregularities in the projection surface, as determined by the depth location of the projected points. While this bootstrapping technique is a tested-and-proven calibration method for projector-camera systems, it is not applicable to the calibration of self-contained projector-camera devices, since it requires the use of pre-calibrated; strategically located, external stereo camera pairs, and thus requires much operator intervention. 
     Of related interest is a technique called dual photography proposed by Sen et al. in article, “Dual Photography”, Proceedings ACM SIGGRRAPH, 2005, which is herein incorporated by reference in its entirety. Dual photography makes use of Helmholtz reciprocity to use images captured with real cameras to synthesize pseudo images (i.e. dual images) that simulate images “as seen” (or effectively “captured”) by projectors. That is, the pseudo image simulates a captured image as “viewed” by a projector, and thus represents what a projector-captured image would be if a projector could capture images. This approach might permit a projector to be treated as a pseudo camera, and thus might eliminate some of the difficulties associated with the calibration of projectors. 
     Helmholtz reciprocity is based on the idea that the flow of light can be effectively reversed without altering its light transport properties. Helmholtz reciprocity has been used in many computer graphics applications to reduce computational complexity. In computer graphics literature, this reciprocity is typically summarized by an equation describing the symmetry of the radiance transfer between incoming (ωi) and outgoing (ωo) directions as fr(ωi&gt;ωo)=fr(ω&gt;ωi), where fr represents the bidirectional reflectance distribution function (BRDF) of a surface. 
     Thus, dual photography ideally takes advantage of this dual nature (i.e. duality relationship) of a projected image and a captured image to simulate one from the other. As is described in more detail below, dual photography (and more precisely Helmholtz reciprocity) requires the capturing of the light transport property between a camera and a projector. More specifically, dual photography requires determination of the light transport property (i.e. light transport coefficient) relating an emitted light ray to a captured light ray. 
     When dealing with a digital camera and a digital projector, however, dual photography requires capturing a separate light transport coefficient relating each projector pixel (i.e. every emitted light ray) to each, and every, camera pixel (i.e. every light sensor that captures part of the emitted light ray), at the resolution of both devices. Since a digital projector and a digital camera can both have millions of pixels each, the acquisition, storage, and manipulation of multitudes of light transport coefficients can place real practical limitations on its use. Thus, although in theory dual photography would appear to offer great benefits, in practice, dual photography is severely limited by its physical and impractical requirements of needing extremely large amounts of computer memory (both archival, disk-type memory and active, solid-state memory), needing extensive computational processing power, and requiring much time and user intervention to setup equipment and emit and capture multitudes of light rays for every projection environment in which the projector-camera system is to be used. 
     A clearer understanding of dual photography may be obtained with reference to  FIGS. 1A and 1B . In  FIG. 1A , a “primal configuration” (i.e. a configuration of real, physical devices prior to any duality transformations) includes a real digital projector  11 , a real projected image  13 , and a real digital camera  15 . Light is emitted from real projector  11  and captured by real camera  15 . A coefficient relating each projected light ray (from each projector pixel e within real projector  11 ) to a correspondingly captured light ray (captured at each camera sensor pixel g within real camera  15 ) is called a light transport coefficient. Using the light transport coefficient, it is possible to determine the characteristics of the projected light ray from the captured light ray. 
     In the present example, real projector  11  is preferably a digital projector having a projector pixel array  17  symbolically shown as a dotted box and comprised of s rows and r columns of individual projector pixels e. Each projector pixel e may be the source of a separately emitted light ray. The size of projector pixel array  17  depends on the resolution of real projector  11 . For example, a VGA resolution may consist of an array of 640 by 480 pixels (i.e. 307,200 projector pixels e), an SVGA resolution may have an array of 800 by 600 pixels (i.e. 480,000 projector pixels e), an XVG resolution may have an array of 1024 by 768 pixels (i.e. 786,732 projector pixels e), an SXVG resolution may have an array of 1280 by 1024 pixels (i.e. 1,310,720 projector pixels e), and so on, with greater resolution projectors requiring a greater number of individual projector pixels e. 
     Similarly, real camera  15  is a digital camera having a camera sensor pixel array  19  symbolically shown as a dotted box and comprised of u rows and u columns of individual camera pixels g. Each camera pixel g may receive, i.e. capture, part of an emitted light ray. The size of camera sensor pixel array  19  again depends on the resolution of real camera  15 . However, it is common for real camera  15  to have a resolution of 4 MegaPixels (i.e. 4,194,304 camera pixels g), or greater. 
     Since each camera pixel g within camera sensor pixel array  19  may capture part of an individually emitted light ray from a distinct projector pixel e, and since each discrete projector pixel e may emit a separate light ray, a multitude of light transport coefficients are needed to relate each discrete projector pixel e to each, and every, camera pixel g. In other words, a light ray emitted from a single projector pixel e may cover the entirety of camera sensor pixel array  19 , and each camera pixel g will therefore capture a different amount of the emitted light ray. Subsequently, each discrete camera pixel g will have a different light transport coefficient indicating how much of the individually emitted light ray it received. If camera sensor pixel array  19  has 4,194,304 individual camera pixels g (i.e. has a 4 MegaPixel resolution), then each individual projector pixel e will require a separate set of 4,194,304 individual light transport coefficients to relate it to camera sensor pixel array  19 . Therefore, millions of separately determined sets of light transport coefficients (one set per projector pixel e) will be needed to relate the entirety of projector pixel array  17  to camera sensor pixel array  19  and establish a duality relationship between real projector  11  and real camera  15 . 
     Since in the present example, each discrete projector pixel e requires a separate set of 4,194,304 individually determined light transport coefficients to relate it to real camera  15 , and since real projector  11  may have millions of discrete projector pixels e, it is beneficial to view each set of light transport coefficients as a separate array of light transport coefficients and to collect these separate arrays into a single light transport matrix (T). Each array of light transport coefficients constitutes a separate column within light transport matrix T. Thus, each column in T constitutes a set of light transport coefficients corresponding to a separate projector pixel e. 
     Since in the present example, real projector  11  is a digital projector having an array of individual light projector pixels e and real camera  15  is a digital camera having an array of individual camera pixels g, a light transport matrix T will be used to define the duality relationship between real projector  11  and real camera  15 . In the following discussion, matrix element T ge  identifies an individual light transport coefficient (within light transport matrix T) relating an individual, real projector pixel e to an individual, real camera pixel g. 
     A real image, as captured by real camera  15 , is comprised of all the light rays individually captured by each camera pixel g within camera sensor pixel array  19 . It is therefore helpful to organize a real captured image, as determined by camera sensor pixel array  19 , into a real-image capture matrix, C. Similarly, it is beneficial to organize a real projected image, as constructed by activation of the individual projector pixels e within projector pixel array  17 , into a real-image projection matrix, P. Using this notation, a real captured image (as defined by real-image capture matrix C) may be related to a real projected image (as defined by real-image projection matrix P) by the light transport matrix T according to the relationship, C=TP. 
     The duality transformation, i.e. dual configuration, of the system of  FIG. 1A  is shown in  FIG. 1B . In this dual configuration, real projector  11  of  FIG. 1A  is transformed into a virtual camera  11  and real camera  15  of  FIG. 1A  is transformed into a virtual projector  15 ″. It is to be understood that virtual camera  11 ″ and virtual projector  15 ″ represent the dual counterparts of real projector  11  and real camera  15 , respectively, and are not real devices themselves. That is, virtual camera  11 ″ is a mathematical representation of how a hypothetical camera (i.e. virtual camera  11 ′) would behave to capture a hypothetically projected dual image  13 ″, which is similar to real image  13  projected by real projector  11  of  FIG. 1A . Similarly, virtual projector  15 ″ is a mathematical representation of how a hypothetical projector (i.e. virtual projector  15 ″) would behave to project hypothetical dual image  13 ″ that substantially matches real image  13 , as captured by real camera  15  (of  FIG. 1A ). Thus, the positions of the real projector  11  and real camera  15  of  FIG. 1A  are interchanged in  FIG. 1B  as virtual camera  11 ″ and virtual projector  15 ″. 
     It should be noted that the pixel resolution of the real devices carries forward to their counterpart virtual devices (i.e. dual devices). Therefore, virtual camera  11 ″ has a virtual camera sensor pixel array  17 ″ consisting of s rows and r columns to match the resolution of projector pixel array  17  of real projector  11 . Similarly, virtual projector  15 ″ has a virtual projector pixel array  19 ″ consisting of u rows and u columns to match the resolution of camera sensor pixel array  19  of real camera  15 . 
     If one assumes that dual light transport matrix T″ is the light transport matrix in this dual configuration such that a dual-image capture matrix C″ (which defines dual image  13 ″ as captured by virtual camera  11 ″) relates to a dual-image projection matrix P″ (which defines dual image  13 ″ as projected by virtual projector  15 ″) as C″=T″P″, then T″ eg  would be an individual dual light transport coefficient relating an individual virtual projector pixel g″ to an individual virtual camera pixel e″. 
     Helmholtz reciprocity specifies that the pixel-to-pixel transport coefficient is equal in both directions (i.e. from real projector  11  to real camera  15 , and from virtual projector  15 ″ to virtual camera  11 ″). That is, T″ eg =T ge , which means T″=T T , (i.e. dual light transport matrix T″ is equivalent to the result of the mathematical, matrix transpose operation on real light transport matrix T). Therefore, given light transport matrix T, one can use T T  to synthesize the dual, or virtual, images that would be acquired in the dual configuration. 
     Thus, the light transport matrix T permits one to create images that appear to be captured by a projector, with a camera acting as a second projector. However, as is explained above, the high complexity involved in generating and manipulating light transport matrix T has heretofore greatly limited its application, particularly in the field of calibrating projector-camera systems. 
     Other problems associated with projector-camera systems are how to compensate for light diffusing objects that may obstruct a projector&#39;s line of sight. Of related interest are issues of whether projector-camera systems can be used to achieve more complex images than typical. For example, can such systems combine multiple images from multiple projectors to create a single composite image? Alternatively, can one generate “3-D” images, or other visual effects that previously required more complex equipment and more complex projection setups? Also, can one make better use of the camera in a projector-camera system so that the camera can be an active part of an image creation process. Furthermore, what are the implications of using a low resolution, inexpensive camera in such projector-camera systems? 
     Previous works [Raskar et al. 1998; Underkoffler and Ishii 1998] put forth the concept of intelligent illumination and showed how projectors could be used to enhance workplace interaction and serve as novel tools for problem solving. The projector-camera community has since solved many of the technical challenges in intelligent projectors. In particular, significant advances have been made in automatic mosaicing of multiple projectors [Chen et al. 2002; Yang et al. 2001; Raij et al. 2003; Sukthankar et al. 2001; Raskar et al. 2003]. 
     [Raskar et al. 2001] demonstrated projection onto complex objects. Using previously created 3D models of the objects, multiple projectors could add virtual texture and animation to real physical objects with non-trivial complicated shapes. 
     [Fujii et al. 2005] proposed a method that modified the appearance of objects in real time using a co-axial projector-camera system. [Grossberg et al. 2004] incorporated a piecewise polynomial 3D model to allow a non-co-axial projector-camera system to perform view projection. 
     Projector camera systems have also been used to extract depth maps [Zhang and Nayar 2006], and space-time-multiplexed illumination has been proposed as a means for recovering depth edges [Raskar et al. 2004]. 
     As will be explained more fully below, the present invention addresses the problem of how to determine what a projector needs to project in order to create a desired image by using the inverse of the light transport matrix, and its application will further simplify the calibration of projector-camera systems. 
     SUMMARY OF THE INVENTION 
     The above objects are met in a system that simplifies the generation of transport matrix T, and expands it use to situation where the camera resolution is similar to, or lower than the resolution of the projector. 
     Application of dual photography is simplified by reducing the number of captured images needed to generate a light transport matrix T of (p×q) projector pixel array from (p×q) images to (p+q) images. Manipulation of the light transport matrix T is simplified by replacing the use of a fully populated light transport matrix T with an index associating each projector pixel to only non-zero light transport coefficient values. By eliminating the use of zero-valued light transport coefficients, the memory and processing requirements for implementing dual photography are greatly reduced. This dual photography technique is applied to the calibration of projector-camera systems. 
     In a first embodiment of the present invention, a method of generating light transport coefficients relating a digital projector to a digital camera is shown. In the case where the digital projector has an array of projection pixels and the digital camera has an array of sensor pixels, the method includes: simultaneously activating a first group of projection pixels within the projector to project a first test pattern on a projection scene, any projection pixels not in said first test pattern being maintained dark; capturing a first image of the first test pattern on the projection scene; simultaneously activating a second group of projection pixels within the projector to project a second test pattern on the projection scene, any remaining projection pixels not in the second test pattern being maintained dark, wherein the first and second groups of projection pixels have only one projection pixel in common defining a target projection pixel; capturing a second image of said second test pattern on said projection scene; comparing image pixels of the first image to corresponding image pixels of the second image and retaining the darker of two compared image pixels, the retained image pixels constituting a composite image; and identifying all non-dark image pixels in the composite image, the non-dark image pixels defining non-zero light transport coefficients associated with the target projection pixel. 
     To further expand its use, the light transport matrix is scanned along a row, and only the brightest non-zero values are retained. All other value entries in the row are preferably set to zero. This method facilitates the creation of the inverse of the light transport matrix, but introduces problems when the resolution of the camera is close to, or smaller than, the resolution of the projector. Since the number of camera pixels is lower than the number of projector pixels, it is very likely that a single projector pixel may be captured by multiple camera pixels. Since only the brightest captured value is kept along a row of the light transport matrix, it is likely that captured images pertaining to some of the projector pixels will be lost, i.e. erased or set to zero values. It is therefore necessarily to create new images for the missing projector pixels. 
     To achieve this, present invention provides a system and method for calibrating a projector-camera system having at least one digital projector and one digital camera, the digital projector having an array of projector pixels arranged into of rows and columns, and the digital camera having an array of sensor pixel, the method comprising at least one processing unit to execute the following steps: generating a matrix of light transport values arranged into distinct images, each of the distinct images having a one-to-one correlation with a single corresponding projector pixel within the array of projector pixels, wherein each light transport value has a zero value or has a non-zero value; applying a row scanning process on each row of the array of projector pixels, the row scanning process including: (a) consecutively checking each projector pixel along a row to determine if the distinct image within the matrix corresponding to the projector pixel currently being checked has at least one non-zero value, if it does have at least one non-zero value then labeling the projector pixel currently being check as a non-zero (NZ) projector pixel, and if does not then labeling the projector pixel currently being check as missing (M) projector pixel; (b) grouping consecutive M projector pixels into missing (MS) segments; (c) grouping consecutive NZ projector pixels into non-zero (NZS) segments; (d) identify each MS segment adjoined by a first NZS segment at one end of the MS segment and adjoined by a second NZS segment at the opposing end of the MS segment, conditionally labeling the MS segment as a candidate row-pixel segment; (e) for each candidate row-pixel segment, using the non-zero light transport values of a first distinct image corresponding to a first NZ projector pixel that adjoins a first end of the candidate row-pixel segment and the non-zero light transport values of a second distinct image corresponding to a second NZ projector pixel that adjoins a second end of the candidate row-pixel segment opposite the first end, to generate a new distinct image having at least one non-zero light transport value for each projector pixel within the candidate row-pixel segment, and replacing within the matrix the existing distinct image corresponding to each projector pixel within the candidate row-pixel segment with its generated new distinct image. 
     Preferably, each of the distinct images has a pixel resolution matching the resolution of the camera. Alternatively, the resolution of the camera is not greater than 1.1 the resolution of the projector. 
     In this approach, the distinct images are preferably arranged into rows of the matrix or arranged into columns of the matrix. Additionally, the matrix is a light transport matrix; and each column of the matrix is one of the distinct images; and each distinct image represents an image as created by activation of only one projector pixel within the array of projector pixels and as captured by the camera. 
     It is presently preferred that the matrix is the transpose of a light transport matrix; each row of the matrix is one of the distinct images; and each distinct image represents an image as created by activation of only one projector pixel within the array of projector pixels and as captured by the camera. 
     In an alternate embodiment, step (d) further includes, determining if the number of pixels within the identified MS segment is less than a threshold number of projector pixels, and if it is less then the threshold number then labeling the MS segment as a candidate row-pixel segment, else if it is not less than the threshold number then not labeling the MS segment as a candidate row-pixel segment. The threshold number is preferably at least 6. 
     Also, step (e) includes relabeling each projection pixel within the candidate row-pixel segment as an NZ projector pixel. Step (e) may further includes: dividing the candidate row-pixel segment into two consecutive sub-segments, a first sub-segment and a second sub-segment; for each projector pixel within the first sub-segment, generating a new distinct image that is a copy of the distinct image corresponding to the NZ projector pixel that adjoins the first sub-segment; and for each projector pixel within the second sub-segment, generating a new distinct image that is a copy of the distinct image corresponding to the NZ projector pixel that adjoins the second sub-segment. In this case, it is preferred that the first sub-segment includes at most half of the projector pixels within the candidate row-pixel segment, and the second sub-segment includes the remaining projector pixels within the candidate row-pixel segment. 
     In an alternate embodiment, step (e) includes: (i) superimposing the non-zero light transport values of the first and second distinct images onto a third distinct image having all zero light transport values; (ii) within the third distinct image, grouping the non-zero light transport values superimposed from the first distinct image into a first group of non-zero values, and grouping the non-zero light transport values superimposed from the second distinct image into a second group of non-zero values; (iii) identifying the center of the first group as a first center o 1 ; (iv) identifying the center of the second group as a first center o 2 ; (v) defining a principle x axis bisecting the first center o 1  and second center o 2 ; (vi) projecting each non-zero value pixel within the third distinct image onto the principle x axis; (vii) identifying a range defined as the difference between the maximum and minimum projection values determined in step (vi); (viii) evenly dividing the range into S+2 intervals, where S is the number of projector pixels in the candidate row-pixel segment; (ix) assigning the S projector pixels within the candidate row-pixel segment and the 2 NZ projector pixels adjoining the candidate row-pixel segment according to the S+2 intervals, in a one-to-one fashion, according to their sequence along a row of projector pixels in the projector pixel array; and (x) distributing the non-zero value pixels of the third distinct image to the projector pixels within the candidate row-pixel segment and to the 2 NZ projector pixels adjoining the candidate row-pixel segment according to their assigned segment, based on which segment which segments the projections of the non-zero value pixels fall onto. 
     Preferably, step (v) includes: creating a vector V 12  connecting the first o 1  and second o 2  centers, V 12  being defined as V 12 =o 2 −o 1 ; normalizing vector V 12  and defining normalized vector V 12  as the principle x axis. In this case, it is also preferred that the first center o 1  defines the origin of the principle axis 
     In still an alternate embodiment, step (vi) includes projecting each non-zero value pixel within the third image to the principle x axis. Preferably, each non-zero value pixel is projected to the principle x axis by subjecting each non-zero value pixel to the following relationship: v i =(x i −o 1 ) T *V 12 , where v i  is a scalar representing the location on the principle x axis onto which the non-zero pixel is projected, and x i  is the coordinate of the i th  non-zero pixel within the third distinct image to which the relationship is being applied. 
     In a preferred embodiment, step (x) includes: defining a new image for each projector pixel within the candidate row-pixel segment and to the 2 NZ projector pixels adjoining the candidate row-pixel segment, each new image defined by the non-zero value pixels distributed it. 
     Following the step of applying a row scanning process on each row of the array of projector pixels, it is preferred that this step be followed by applying a column scanning process on each column of the array of projector pixels, the column scanning process including: (a) consecutively checking each projector pixel along a column to determine if the distinct image within the matrix corresponding to the projector pixel currently being checked has at least one non-zero value, if it does have at least one non-zero value then labeling the projector pixel currently being check as a non-zero (NZ) projector pixel, and if does not then labeling the projector pixel currently being check as missing (M) projector pixel; (b) grouping consecutive M projector pixels into missing (MS) segments; (c) grouping consecutive NZ projector pixels into non-zero (NZS) segments; (d) identify each MS segment adjoined by a first NZS segment at one end of the MS segment and adjoined by a second NZS segment at the opposing end of the MS segment, conditionally labeling the MS segment as a candidate column-pixel segment; (e) for each candidate column-pixel segment, using the non-zero light transport values of a first distinct image corresponding to a first NZ projector pixel that adjoins a first end of the candidate column-pixel segment and the non-zero light transport values of a second distinct image corresponding to a second NZ projector pixel that adjoins a second end of the candidate column-pixel segment opposite the first end, to generate a new distinct image having at least one non-zero light transport value for each projector pixel within the candidate column-pixel segment, and replacing within the matrix the existing distinct image corresponding to each projector pixel within the candidate column-pixel segment with its generated new distinct image. 
     Lastly, it is preferred that the matrix have only one non-zero value per row. 
     Other objects and attainments together with a fuller understanding of the invention will become apparent and appreciated by referring to the following description and claims taken in conjunction with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In the drawings wherein like reference symbols refer to like parts. 
         FIGS. 1A and 1B  show a prior art setup for implementation of dual photography. 
         FIGS. 2A and 2B  show a setup for dual photography in accord with the present invention. 
         FIG. 3A  is an example of a light ray projected from a single projector pixel, bouncing off a scene, and forming a small light footprint on a camera sensor pixel array. 
         FIG. 3B  is an example of how a projected image may be represented as an array of information. 
         FIG. 3C  is an example of how a captured light footprint on a camera sensor pixel array may be represented as an array of information. 
         FIG. 4A  is an illustrative example of a projected footprint on a light sensor array within a digital camera resulting from activation of a single projector pixel in a projector. 
         FIG. 4B  is an illustrative example of a column of light transfer coefficients within a matrix T reflecting the example of  FIG. 3A . 
         FIGS. 5A and 5B  show two examples of two columns of projector pixels simultaneously projected onto a scene having a checkerboard pattern. 
         FIGS. 6A and 6B  show two examples of two rows of projector pixels simultaneously projected onto a scene having a checkerboard pattern. 
         FIG. 7A  shows a generated light ray footprint resulting from a single projector pixel, as created by combining the images of  FIGS. 5A and 6A . 
         FIG. 7B  shows a generated light ray footprint resultant from a single projector pixel, as created by combining the images of  FIGS. 5B and 6B . 
         FIG. 8  is a first example of an index associating projector pixels to non-zero valued light transport coefficients, as determined by a light ray footprint as generated in  FIG. 7A  or  7 B. 
         FIG. 9  is second example of an index associating projector pixels to non-zero valued light transport coefficients, as determined by a light ray footprint as generated in  FIG. 7A  or  7 B. 
         FIG. 10A  shows a real captured image taken by a real camera. 
         FIG. 10B  shows a dual captured image, as seen by a real projector, as generated using the method of the present invention. 
         FIG. 11  is an example of a result obtained by use of nomography to calibrate a projector. 
         FIG. 12  shows a projected image distorted by two wine glasses placed between a projector and a scene. 
         FIG. 13  is an example of two overlapping adjacent light footprints produced by two distinct projector pixels. 
         FIG. 14  is an example of a process in accord with the present invention for imposing the display constraint on an arbitrary scene. 
         FIG. 15  is a further step in the imposing of the display constraint as introduced in  FIG. 14 . 
         FIG. 16  is an example of the present invention being applied to color compensation in poster images. 
         FIG. 17  shows the projection scene of  FIG. 12 , but with the present invention&#39;s method of compensating for light distortion applied. 
         FIG. 18  is an exemplary projection setup for using dual photography to create an immersive display system. 
         FIG. 19  shows an immersive projector P 2  being used to simulate a front projector P 1 . 
         FIG. 20  is an example of an image generated using the virtual projector implementation of  FIG. 19 . 
         FIG. 21A  shows the right side of an image projected by a real front projector. 
         FIG. 21B  shows the corresponding left side of the image shown in  FIG. 21A , but in  FIG. 21B , the left side of the image is projected by an immersive projector. 
         FIG. 21C  shows the right side image of  FIG. 21A  joined to the left side image of  FIG. 21B . 
         FIGS. 22A and 22B  show two additional examples of a left side image generated by an immersion projector joined to the right side image generated by a front projector. 
         FIGS. 23A to 23C  show an alternate application of the present invention to recreate in a real room a virtual image created in a virtual model room. 
         FIG. 24  is an example of the application of the technique of  FIGS. 23A-23C  to project an image bigger than the projection space in a real room without image distortion. 
         FIG. 25  is an exemplary projection system in accord with the present invention in a minimal form. 
         FIG. 26  shows a prototype based on the design of  FIG. 25 . 
         FIG. 27  is an alternate view of the setup of  FIG. 26 . 
         FIG. 28A  shows, under ambient lighting, a room with the projection system of  FIGS. 26 and 27  installed. 
         FIG. 28B  shows the room of  FIG. 28A  under immersive projection lighting. 
         FIG. 29A  shows the room of  FIG. 28A  with an uncalibrated projected image. 
         FIG. 29B  shows the room of  FIG. 28A  with a calibrated projected image. 
         FIG. 30A  shows the room of  FIG. 28A  with an uncalibrated blank projection. 
         FIG. 30B  shows the room of  FIG. 28A  with a calibrated blank projection. 
         FIG. 31  is a first step in the application of the present invention to a dome mirror projector. 
         FIG. 32  is a second step in the application of the present invention to a dome mirror projector. 
         FIG. 33  is a third step in the application of the present invention to a dome mirror projector. 
         FIG. 34  demonstrates the application of the present invention to a desired image, to produce a precisely distorted image for projection on a dome mirror projector. 
         FIG. 35  demonstrates the result of projecting the distorted image of  FIG. 34 . 
         FIG. 36  is an alternative design for ceiling-mounted operation. 
         FIG. 37  is an alternate configuration of the present invention. 
         FIG. 38  is still another configuration of the present invention. 
         FIG. 39  shows a set up for mosaicing two, or more, projector-camera systems to create composite image. 
         FIG. 40  shows a composite of images from multiple projector-camera systems. 
         FIG. 41  shows an image as projected by a first of two projector-camera systems in a multi-projector-camera system. 
         FIG. 42  shows an image as projected by a second of two projector-camera systems in a multi-projector-camera system. 
         FIG. 43  is an example of the present invention applied to the projection of a checkerboard pattern onto a complex surface. 
         FIG. 44A  shows a design that uses a single curved mirror  125  and multiple projector-camera pairs  145 . 
         FIG. 44B  shows a design that uses a single mirror pyramid  151  and multiple projector-camera pairs  145 . 
         FIG. 45  shows how multiple large FOV projectors  153   a  and  153   b  can be used to achieve an even larger overall projection FOV. 
         FIG. 46  shows an algorithm consisting of fourth parts for capturing light transport matrix T directly into an efficient representation. 
         FIG. 47  is a pictorial representation of the first part of the algorithm of  FIG. 46 . 
         FIG. 48  is a pictorial representation of the second part of the algorithm of  FIG. 46 . 
         FIG. 49  is a pictorial representation of the third part of the algorithm of  FIG. 46 . 
         FIG. 50  is a pictorial representation of the fourth part of the algorithm of  FIG. 46 . 
         FIG. 51  is a partial view of an exemplary light transport matrix T. 
         FIG. 52  is a pictorial representation of the relationship between each projector pixel within a projector pixel array and each column of light transport matrix T. 
         FIG. 53  is an exemplary row of a projector pixel array illustrating data holes caused by lost light transport information. 
         FIG. 54  is a pictorial representation of a method for filling in the data holes caused by lost light transport information. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     In order to apply camera calibration techniques to projectors, one would require projectors to be able to capture images. That is, if projectors could capture images, then projector-camera systems could be treated like multi-camera systems, and standard camera calibration techniques (described above) might be used to calibrate projector-camera systems. In other words, if a projector could be treated as a pseudo-camera, then it could be calibrated along with a real camera in a manner similar to the camera calibration stage of the multi-camera system described above, and the “bootstrapping” projector calibration stage previously used for calibrating projector-camera systems might be eliminated. 
     With reference to  FIG. 2A , an imaging setup in accord with the present invention may include a real projector  21  and a real camera  25 . Real projector  21  is preferably a digital projector and has an imaging element including an imaging projection array (i.e. projector pixel array  27 ), consisting of p rows and q columns of individual imaging projection elements (i.e. projector pixels j) Projector pixel array  27  is internal to real projector  21 , and is shown for discussion purposes as crossed lines within a dotted square in  FIG. 2A . Real projector  21  is preferably of the liquid crystal display (LCD) type, digital light processing (DLP) type, liquid crystal on silicon (LCOS) type, or other digital projection technology type. 
     Preferably, real camera  25  is a digital camera having an image sensor including a camera sensor pixel array  29  (i.e. light receptor array or image sensor array), consisting of m rows by n columns of individual camera pixels i (i.e. image sensor elements or light receptor pixels). For simplicity, camera sensor pixel array  29  is shown on real camera  25 , but it is to be understood that camera sensor pixel array  29  is internal to real camera  25 . 
     This physical setup using real projector  21  and real camera  25  is preferably called the ‘primal’ setup. Light rays emitted from individual projector pixels j within real projector  21  form real image  23  by bouncing off a projection surface (i.e. display environment or scene), which may have an irregular or flat shape, and some of the light rays eventually reach the image sensor within real camera  25 . In general, each light ray is dispersed, reflected, and refracted in the scene and hits the camera&#39;s image sensor at a number of different locations throughout camera sensor pixel array  29 . Thus, when a single light ray emitted from an individual imaging projector pixel j from real projector  21  reaches real camera  25 , the individually projected light ray forms an m-by-n image on camera sensor pixel array  29 , with each individual camera pixel i receiving a certain amount of light intensity contribution from the projected light ray. 
     Consequently, although it might appear that ideally a single light transport coefficient relating one projector pixel j to one camera pixel i might be determinable by individually turning ON a single projector pixel j to emit a single light ray to hit a single camera pixel i, this is not the case. In reality, the entire camera sensor pixel array  29  will receive a different contribution of light intensity from the singly emitted light ray. Therefore, each light ray emitted from each individual projector pixel j generates a different set, or array, of individual light transport coefficients, one for each camera pixel i within camera sensor pixel array  29 . Consequently, each set (i.e. array) will consist of (m×n) [i.e. m-multiplied-by-n] individual light transport coefficients, one for each camera pixel i. 
     If each set of light transport coefficients is arranged as a column of coefficients to form a composite light transport matrix, T, then the composite light transport matrix T will have a different column of light transport coefficients for each individual projector pixel j. Furthermore, since there is a one-to-one correspondence between each light transport coefficient entry (i.e. matrix element) within each column and each camera pixel i, each column represents the entire image captured at camera  25  resulting from a single projector pixel j being turned ON. Accordingly, the entire (i.e. composite) light transport matrix T will consist of (p×q) [i.e. p multiplied-by q] columns (one column [i.e. captured image] for each individually turned ON projector pixel j) and (m×n) rows (one row for each individual camera pixel i). 
     In the following discussion, it is beneficial to view a real projected image as a first vector, Rprjct, (i.e. real projection vector) having (p×q) elements (one for each projector pixel j), and a resultant real captured image as a second vector, Rcptr, (i.e. real captured image vector) having (m×n) elements (one for each camera pixel i). 
     Using the notation that a real projected image (i.e. an image projected using the entire projector pixel array  27 ) is represented as a (p×q) real projection vector “Rprjct” (i.e. a “p-by-q vector”), and the notation that a correspondingly real captured image (i.e. an image captured by the entire camera sensor pixel array  29 ) is represented as an (m×n) real captured image vector “Rcptr” (i.e. an “m-by-n vector”), then the light transport relationship between real projector  21  and real camera  25  can be written as
 
Rcptr=T Rprjct
 
where T is a composite light transport matrix that relates real projector  21  to real camera  25 . It is to be understood that light transport matrix T would have been previously generated by, for example, individually and separately turning ON each projector pixel j and determining the corresponding light transport coefficient for each individual camera pixel i.
 
     To recapitulate, since each projector pixel j results in a light ray that is scattered across the entire camera sensor pixel array  29 , each individual camera pixel i will have a differently valued light transport coefficient indicative of an intensity value of a predefined light characteristic it received from each individual projector pixel j. In the present embodiments, this light characteristic is preferably a measure of light intensity. Therefore, each projector pixel j will result in a column of (m×n) individual light transport coefficients, each coefficient indicating an amount of light intensity received by each camera pixel i. Since real projector  21  has (p×q) projector pixels j, light transport matrix T will have (p×q) columns [one for each projector pixel j] and (m×n) rows [one for each camera pixel i] of individual light transport coefficients. Thus, light transport matrix T has traditionally been necessarily huge, consisting of (p×q×m×n) individual light transport coefficients. 
     With reference to  FIG. 2B , a “dual” setup is one where real projector  21  is replaced by a virtual camera  21 ″ having a virtual camera sensor pixel array  27 ″ of equal size as real projector pixel array  27 . Thus, virtual camera  21 ″ has a virtual camera sensor pixel array  27 ″ comprised of p-rows by q-columns of virtual camera pixels j″. Similarly, in this “dual” setup, real camera  25  is replaced by a virtual projector  25 ″ having a virtual projector pixel array  29 ″ of equal size as real camera sensor pixel array  29 . Therefore, virtual projector  25 ″ has a virtual projector pixel array  29 ″ comprised of m-rows by n-columns of virtual projector pixels i″. 
     In this case, a virtual image  23 ″ (as projected by virtual projector  25 ″) would be represented by an (m×n) virtual projection vector, Vprjct″. Similarly, a virtually captured image captured by virtual camera  21 ″ would be represented by a (p×q) virtual captured image vector, Vcptr″. Therefore in this dual setup, virtual captured image vector Vcptr″ relates to virtual projection vector Vprjct″ by a “dual” light transport matrix T″. By the principle of Helmholtz reciprocity, the light transport is equal in both directions. Therefore, the dual light transport matrix T″ for the dual setup (i.e. the duality transformation setup) relates virtual captured image vector Vcptr″ to virtual projection vector Vprjct″ by following relation:
 
Vcptr″=T″Vprjct″
 
     This relationship, however, brings up the problem of how can one determine the multitudes of “virtual” light transport coefficients that make up dual light transport matrix T″, and that relate a virtually projected image to a virtually captured image? Remarkably, it has been found that a duality transformation from a real light transport matrix, T, to its dual light transport matrix, T″, can be achieved by submitting the real light transport matrix T to a transpose matrix operation. Thus, T″ is the transpose of T, such that T″=T T  and Vcptr″=T T Vprjct″. 
     As it is known in the art of matrix computation, the transpose matrix operation of a general [x×y] matrix A is denoted by the addition of a superscript letter “T” (such as A T , for example) and is defined by an [y×x] matrix whose first column is the first row of matrix A, and whose second column is the second row of matrix A, whose third column is the third row of matrix A, and so on. As is readily evident, this matrix operation simply flips the original matrix A about its first element, such that its first element (i.e. at position (1,1)) remains unchanged and the bottom of the first column becomes the end of the first row. Consequently, if one can capture, or otherwise determine, the real light transport matrix T for a primal setup, then the dual light transport matrix T″ for the dual setup is readily computable by flipping the real light transport matrix T to obtain its transpose, T T , as described. In further discussions below, dual light transport matrix T″ and transposed light transport matrix T T  may be used interchangeably. 
     It is noted, however, that if one can provide a simplified method of determining T, one can directly determine T T  from T for the duality transformation. Before proposing a method for simplifying the determination of T, it is beneficial to first discuss some of the difficulties of traditional methods of determining T. 
     As is explained above, real light transport matrix T holds the individual light transport coefficients corresponding between each individual projector pixel j of real projector  21  and all the individual camera pixels i of real camera  25 . Therefore, a determination of each individual light transport coefficient corresponding between an individual projector pixel j and all camera pixels i should avoid light ray contributions from other projector pixels j in projector pixel array  27 . 
     To accomplish this, one may first consider an initial, real projection image_j created by setting a j th  projector pixel to a value of 1 (i.e. is turned ON) and setting to a zero value (i.e. turning OFF) all other elements in projector pixel array  27 . In this case, the j th  projector pixel is the projector pixel under-test for which the light transport coefficients is to be determined. These test conditions are annotated as a real projection vector Rprjct_j that has a value of one at an entry location associated with the j th  projector pixel, and has all other entry locations corresponding to all other projector pixels set to a value of zero. One may then capture real projection image_j to determine its corresponding (m×n) real captured image vector Rcptr_j, which defines the j th  column of matrix T. 
     This method of acquiring a column of light transport coefficients for matrix T for a given j th  projector pixel, suggests that a systematic method for capturing the entire matrix T is to sequentially turn ON each projector pixel j of real projector  21  (one projector pixel at a time), and to capture its corresponding real image, Rcptr_j, with real camera  25 . When all p×q projector pixels j have been sequentially turned ON, and their corresponding real images Rcptr_j have been captured, all the captured image vectors Rcptr_ 1  to Rcptr_(p×q) are grouped to form matrix T. Each captured image vector Rcptr_j constitutes a column of light transport coefficient entries in matrix T. This results in a matrix T having (p×q) columns and (m×n) rows of individual light transport coefficients. 
     This straight forward, and systematic process for determining matrix T, however, is obviously a time-consuming process requiring (p×q) image projection-and-capture steps. Furthermore, the resultant light transport matrix T is very large, consisting of (p×q×m×n) elements. Because of the size of matrix T, computing a dual image is an extremely computation intensive operation requiring matrix multiplication between dual light transport matrix, T T , (which has the same number of elements as matrix T), and a virtual projection vector Vprjct″ (which is a long vector having (m×n) elements). 
     In the past, a scheme for determining matrix T by adaptively turning on multiple pixels of a real projector at a time (such as projector  11  of  FIG. 1A ), has been suggested to speed up the process of determining matrix T. However, this scheme is complex, and requires that an image to be projected be divided into multiple zones, and that one projector pixel from each zone be selected. All selected projector pixels from the multiple zones are turned on simultaneously while assuring that the simultaneously lit projector pixels are selected distant enough from each other to eliminate light interference between each other. Consequently, this scheme is scene-dependent (i.e. dependent upon the display environment) since it is necessary to assure that light interference between individually selected projector pixels is minimized. Since creation of the multiple zones is dependent on the display environment, this scheme requires physical inspection of a projection area and manual setup, and therefore does not easily lend itself to general use or to automation. 
     A feature of the present invention proposes a method of reducing the number of procedural steps in the determination of real light transport matrix T. That is, instead of requiring (p×q) image projection-and-capture steps (and the storing of p×q captured images), the presently proposed method captures at most “p plus q” [i.e. (p+q)] images, and under specific circumstances, described below, the number can be further reduced dramatically. 
     The present method is based on the following assumption: for most projector-camera display applications, any two distinct light rays b and c emitted from real projector  21  will typically hit camera sensor pixel array  29  at distinct parts. That is, any light ray emitted from a projector pixels j will not be dispersed across the entirety of camera sensor pixel array  29 , but will be primarily concentrated within a small group of camera pixels i forming a light footprint within a distinct region of camera sensor pixel array  29 . Furthermore, there will be no, or negligible, overlap in adjacent light footprints (i.e. negligible overlap in the camera pixels i hit by distinct projected light rays from distinct projector pixels j). This ideal light ray distribution condition may be assumed when the resolution of the real camera  25  is much higher (i.e. at least two times higher) than the resolution real projector  21 . 
     This assumption regarding the non-overlap of distinct light rays, off course, is not true in general. For example, the ideal light ray distribution condition may not be achievable when the resolution of real camera is  25  is comparable to, or smaller than, the resolution of real projector  21 , or when a light diffusing material is located between the light path of a projected light ray from real projector  21  to camera sensor pixel array  29 . For example, if the projection scene (i.e. display environment or projection surface/area) includes light diffusing material, such as a glass of milk, the projected light rays will be diffused and the likelihood of significant light overlap between the different light rays at camera sensor pixel array  29  is greatly increased. However, a display setup designed to ensure high resolution projections is likely to be devoid of such light diffusing material, and it is virtually guaranteed that each projected light footprint will be substantially distinct from the next. That is, in venues, or settings, where high resolution projections are desired, it is likely that the venue will be clear of light diffusing articles along the light path of a projected image. 
     In the present first embodiment, it is assumed that the pixel resolution of real camera  25  is at least four times greater than that of real projector  21  and that the projection scene is devoid of any light diffusing material, such that the requirements for the ideal light ray distribution condition are met. Nonetheless, alternative embodiments for more general situations where the resolution of real camera  25  is not restricted to be greater than real projector  21 , or where the light path between real projector  21  and real camera  25  is not limited to be devoid of light diffusing material are provided below. 
     A first pictorial representation of the present assumptions and vector generation method is shown in  FIGS. 3A to 3C . With reference to  FIG. 3A , an illustrative example of the present invention shows real projector pixel array  27  having a pixel resolution smaller than that of real camera sensor pixel array  29 . For ease of discussion, projector pixel array  27  is illustratively shown to consist of only 6 rows and 7 columns (i.e. p=6 and q=7) for a total resolution of 42 projector pixels j (i.e. (6×7) elements). Similarly for ease of discussion, camera sensor pixel array  29  is illustratively shown to consist of only 12 rows and 14 columns (i.e. m=12 and n=14) for a total resolution of 168 camera pixels i (i.e. (12×14) elements). It is noted that the resolution of camera sensor pixel array  29  is four times greater than the resolution of projector pixel array  27 , and no light diffusing material is located within the light path from projector pixel array  27  to camera sensor pixel array  29 , and so the ideal light ray distribution conditions for the present first embodiment are met. 
     In  FIG. 3A , each square within projector pixel array  27  represents a different projector pixel j. In the present example, it is desired to obtain the light transport coefficients for the projector pixel j located at the intersection of the third row and fifth column (i.e. projector pixel j=(5,3) is the projector pixel under test). As explained above, light diffusion, or light scattering, is ignored. Thus, projector pixel j=(5,3) is shown to be turned ON, i.e. lit, as is indicated by a white square, while all other projector pixels are shown to remain turned OFF, as indicated by dark gray squares. Projector pixel j=(5,3) emits light ray  1 , which passes through a projector lens  3  before hitting a projection surface  5  (which is basically a generic projection scene determined by the environment in which real projector  21  and real camera  25  are located). Since light diffusion is ignored in the present example, light ray  1  bounces off projection surface  5  and passes through a camera lens  7  before hitting a group of camera pixels i forming a light ray footprint indicated by circle Ft 1 . Since the resolution of each projector pixel j is much coarser than that of the camera pixels i, the created light ray footprint Ft 1  on camera sensor pixel array  29  covers several camera pixels i, with the camera pixels at the center of light ray footprint Ft 1  (indicated as white squares) receiving the most light intensity, and the camera pixels i along the periphery of light ray footprint Ft 1  (indicated as light gray squares) receiving less light intensity, and those camera pixels i not within light ray footprint Ft 1  (indicated as dark gray squares) receiving no light. The light transport coefficients for each camera pixel i corresponding to the projector pixel under test j=(5,3) may then be determined by determining the light intensity reading of all camera pixels i within camera sensor pixel array  29 . 
     As is explained above, it is beneficial to view projector pixel array  27  as a first vector (real projection vector Rprjct), and to view the resultant captured image across the entirety of camera sensor pixel array  29  as a second vector (real captured image vector Rcptr). A pictorial representation of how real projector vector Rprjct may be constructed is shown in  FIG. 3B . Projector pixel array  27  is first separated into six rows, row_ 1  to row_ 6 , and each row is then turned 90 degrees to form six column segments col_seg_r 1  to col_seg_r 6 . The six column segments col_seg_r 1  to col_seg_r 6  are then joined end-to-end to form one composite column col_ 1  consisting of (p×q) elements (i.e. (6×7) or 42 elements). In the present example, element col_ 1 _row_ 19  (i.e. the 19 th  element down from the top of column col_ 1 ) corresponds to a projector pixel j at array location (5,3) of projector pixel array  27 . Element col_ 1 _row_ 19  is shown as a white square, while all other elements are shown as darkened squares, to indicate that element col_ 1 _row_ 19  is the only projector pixel turned ON. 
     A similar process is applied to camera sensor pixel array  29  in  FIG. 3C  to construct real captured image vector Rcptr. In the present exemplary case, camera sensor pixel array  29  would be broken up into m rows (i.e. 12 rows) that are each turned 90 degrees to form 12 column segments, not shown. In a manner similar to how projector pixel array  27  was rearranged in  FIG. 3B , the 12 column segments are joined end-to-end to form one composite column cmra_col_ 1  (i.e. vector Rcptr) of (m×n) elements (i.e. (12×14) or 168 elements). 
     In the present example, the light ray emitted from projector pixelj at array location (5,3) of projector pixel array  27  is assumed to form light ray footprint Ft 1  on camera sensor pixel array  29 . The span of light ray footprint Ft 1  is indicated by a circle, and encompasses twelve camera pixels i on four different rows of camera sensor pixel array  29 . The camera pixels i within light ray footprint Ft 1  receive at least a part of the light intensity from a light ray emitted from projector pixel j=(5,3) (i.e. matrix element col_ 1 _row_ 19 ) from  FIG. 3A  or  3 B. The camera pixels at the center of light footprint circle Ft 1  receive the most light intensity, and are identified as white squares, and the camera pixels along the perimeter of light footprint circle Ft 1  receive less light intensity and are identified as light gray squares. Those camera pixels not within light ray footprint Ft 1  are shown as dark gray squares. As is illustrated in  FIG. 3C , these white and light gray squares constitute nonzero elements within the captured image, and are shown as white and light gray squares (i.e. nonzero) valued NZ elements interspersed between many zero valued elements (i.e. dark gray squares) within vector Rcptr (i.e. column cmra_col_ 1 ). 
     A second example providing a close-up view of how the light footprint of a single light ray from a projector pixel j may cover several camera pixels i is shown in  FIGS. 4A and 4B . In  FIG. 4A , a partial view of another exemplary camera sensor pixel array  29  shows individual camera pixels i numbered horizontally from 1 to n on the first row, continuing with (n+1) to (2n) on the second row, and (2n+1) to (3n) on the third row, and so on. Following this sequence, it is to be understood that camera pixels i along the bottom-most row would be numbered from (m−1)n+1 to (mn). 
     A second light ray footprint Ft 2  of a single light ray from another exemplary, single, projector pixel j impacting camera sensor pixel array  29  is denoted as a circle. For illustration purposes, those camera pixels i not within light ray footprint Ft 2  [i.e. those camera pixels i not hit by the single light ray emitted from the j th  projector pixel] are shown as deeply darkened, those camera pixels i partly covered by footprint Ft 2  are shown as lightly darkened, and those camera pixels i completely within footprint Ft 2  are shown as having no darkening. As it is known in the art, each camera pixel i that is at least partially covered by light ray footprint Ft 2  will register a light intensity value proportional to the amount of light it receives. This light intensity value may be assigned as the light transfer coefficient for that individual camera pixel i. Alternatively, the light transport coefficient of each camera pixel i may be made proportional to the light intensity value registered by the individual camera pixel i. Nonetheless, those camera pixels i that are not directly hit by the projection light ray from the j th  projector pixel should have a light intensity value of zero (or close to zero, or below a predefined threshold light intensity value), and their corresponding light transport coefficient should likewise have a value of zero (or close to zero). 
     With reference to  FIG. 4B , an example of a captured image vector Rcptr_j [or j th  column of matrix T], as would correspond to the footprint Ft 2  example of  FIG. 4A  is shown. This j th  column of matrix T is illustratively shown as a numbered vertical sequence of light transport coefficients, each corresponding to the numbered camera pixel i of camera sensor pixel array  29  of  FIG. 4A . The numerical sequence of capture vector Rcptr_j preferably follows the horizontally numbered sequence of individual camera pixels i in camera sensor pixel array  29  shown in  FIG. 4A . As shown, only those elements within captured imaged vector Rcptr_j that correspond to camera pixels i covered by light ray footprint Ft 2  have non-zero values, i.e. “NZ”, for light transport coefficients. All other camera pixels have “ZERO” values for light transport coefficients. It is to be understood that “NZ” represents any non-zero light transport coefficient value, and that this value would be related to the amount of light intensity received by the corresponding camera pixel i. Since light ray footprint Ft 2  spans several rows of camera sensor pixel array  29 , each row is sequentially listed in captured image vector Rcptr_j, with several long sequences of zero valued light transport coefficients interspersed between a few non-zero, NZ, valued light transport coefficients. 
     Returning now to the present novel method for determining matrix T, it is first noted that individual light transport coefficients for projector pixel j map to column j of the light transport matrix T. Assuming minimal overlap between projector pixels, it follows that a first set of projector pixels S 1  within imaging projector pixel array  27  maps to a corresponding set of columns (one per projector pixel) in light transport matrix T.
 
[i.e. S1 ⊂{1, . . . , (p×q)}]
 
Furthermore, it is assumed that the first set of projector pixels S 1  includes target projector pixel j, i.e. the target projector pixel under test.
 
     Let Rcptr_S 1  be a first real image captured by real camera  25  of a projected image created by the simultaneous activation of the first set of projector pixels S 1 . 
     Consider now a second set of projector pixels S 2  who share only projector pixel under testj in common with the first set of projector pixels S 1 ,
 
[i.e. S1 ∩S2={j}]
 
Let Rcptr_S 2  be a second real image captured by real camera  25  of a projected image created by the simultaneous activation of the second set of projector pixels S 2 . The light transport coefficients of the j th  column of light transport matrix T (which corresponds to the target projector pixel under test, i.e. corresponds to j) may be directly obtain from real captured images Rcptr_S 1  and Rcptr_S 2  by identifying the one light ray footprint they share in common (i.e. similar to light ray footprints Ft 1  or Ft 2  in  FIG. 3A ,  3 C or  4 A). This common light ray footprint would correspond to a light ray emitted from target projector pixel j, which is the only lit projector pixel j shared in common among first set S 1  and second set S 2 .
 
     The next step, therefore, is to determine how to identify the one light ray footprint commonly shared by both real captured images Rcptr_S 1  and Rcptr_S 2 . A method of identifying this common light ray footprint is to conduct a pixel-by-pixel comparison of both captured images Rcptr_S 1  and Rcptr_S 2 , and to identify the dimmer of two compared pixels. For example, in first captured image Rcptr_S 1  only sensor pixels within individual light ray footprints, each corresponding to the simultaneous lighting of the first set of projector pixels S 1 , will have non-zero (NZ) light intensity values, and all other pixels in captured image Rcptr_S 1  will have zero values, i.e. will be comparatively dark. Similarly in second captured image Rcptr_S 2 , only sensor pixels within light ray footprints corresponding to the second set of simultaneously turned on projector pixels S 2  have non-zero (NZ) light intensity values, and all other pixels will have zero (or dark) values (i.e. below a predefined threshold value). Since the two sets S 1  and S 2  share only the target projector pixel j in common, a direct comparison of both captured images will quickly identify the camera pixels i within the light ray footprint corresponding to projector pixel j by identifying the only non-zero region (i.e. non-dark region) common to both Rcptr_S 1  and Rcptr_S 2 . Stated differently, the intersection of the lit regions (i.e. light ray footprints) of Rcptr_S 1  and Rcptr_S 2  is identified, and this identified light ray footprint will correspond to the target projector pixel under-test, j. 
     A method of accomplishing this is to conduct a pixel-by-pixel comparison of both captured images Rcptr_S 1  and Rcptr_S 2 , and to retain only the darker (i.e. dimmer) of the two compared pixels. This process may be expressed as:
 
Tj≈MIN(Rcptr_S1, Rcptr_S2)
 
where Tj is the j th  column of matrix T, and “MIN” indicates that the lower valued camera pixel (i.e. the darker camera pixel having a lower captured light intensity value) in Rcptr_S 1  and Rcptr_S 2  is retained, and the higher valued (i.e. brighter) camera pixel is discarded. In this way, the only high intensity values that are retained correspond to a light ray footprint common to both S 1  and S 2 .
 
     Stated differently, since the contribution of each individual projector pixel j is mapped to distinct parts of the camera sensor pixel array  29 , there is a set L of cameral pixels i among pixels i=1 to i=(m×n) common to captured image Rcptr_S 1  and Rcptr_S 2  that corresponds to the target projector pixel, j.
 
[i.e. L ⊂{1, . . . , (m×n)}]
 
It should again be noted that the target projector pixel, j, is the intersection of projector pixel sets S 1  and S 2 , (i.e. j is the only projector pixel common to both sets S 1  and S 2 ), such that
 
S1 ∩S2={j}
 
Therefore, among the captured camera pixels (in both Rcptr_S 1  and Rcptur_S 2 ) that do not correspond to the target projector pixel, j, (i.e. those camera pixels not in set L, i.e. ∉L), at least one of the compared camera pixels in either Rcptr_S 1  or Rcptr_S 2  will not have received light. Since camera pixels receiving light will be brighter than camera pixels not receiving light, the operation MIN(Rcptr_S 1 , Rcptr_S 2 ), provides an image where only pixels in set L [i.e. ∈L] are lit, which is a good approximation of Tj, i.e. the j th  column in matrix T.
 
     This implies that if sets of adjacent projector pixels in projector pixel array  27  are lit in columns and in rows (or any arbitrary pair of patterns that intersect at only one point, i.e. share only one projector pixel j in common), and a first collection of captured images Rcptr_Sy are made for the lit columns of projector pixels and a second collection of captured images Rcptr_Sx are made for the lit rows of projector pixels, then the light transport coefficients for any individual projector pixel j may be obtained by comparing both collections and identifying the region L where a captured image of a lit column intersect a captured image of a lit row, the intersection corresponding to a light ray projected by activation of projector pixel j, alone. 
     Thus, a method of determining transport matrix T is to collect a first set of images Rcptr_Sy_ 1  to Rcptr_Sy_q, corresponding to q captured images of q lit columns of projector pixels, and construct a second set of images Rcptr_Sx_ 1  to Rcptr_Sx_p corresponding to p captured images of p lit rows of projector pixels. Then for all projector pixels j=1 to j=(p×q) within projector pixel array  27 , there exists a pair of row and column captured images, Rcptr_Sy_a and Rcptr_Sx_b, such that the intersection of Rcptr_Sy_a and Rcptr_Sx_b correspond to a light ray footprint created by activation of the target project pixel under-test, j. Therefore, one needs to construct sets of projection images:
 
Rprjct_Sy — 1 to Rprjct_Sy_q and Rprjct_Sx — 1 to Rprjct_Sx_p
 
where each projection image Rprjct_Sy_ 1  to Rprjct_Sy_q is paired with any of projection images Rprjct_Sx_ 1  to Rprjct_Sx_p such that each pair of projection images shares only one light footprint in common. That is,
 
∀j ∈{1, . . . , (p×q)}∃Rprjct_Sy_a, Rprjct_Sx_b|Rprjct_Sy_a ∩Rprjct_Sx_b_={j}
 
The above formula being interpreted to mean that for all projector pixels j in {1 . . . (p×q)} there exist a pair of projection images, each having a differently constructed pattern such that the intersection of the constructed patterns intersect at a single point (i.e. pattern section, or a single light footprint region) corresponding to a common projector pixel, j. A basic example of such pairs of constructed patterns would be projected pairs of vertical light lines and horizontal light lines. In this case, the intersection of the captured image of a vertical light line and the captured image of a horizontal light line would include all the camera pixels i that correspond to a target projector pixel under-test, j, (i.e. all camera pixels i that lie within a light ray footprint created by a light ray emitted from project pixel under-test j).
 
     Therefore, any column Tj [where j=1 to (p×q)] within transport matrix T can be synthesized from images Rcptr_Sy_ 1  to Rcptr_Sy_q and Rcptr_Sx_ 1  to Rcptr_Sx_p. A scheme that satisfies this property is to use pixel coordinates: let Rprjct_Sx_j be a first projected image such that only pixels with an x-coordinate equal to j are turned ON, and let Rprjct_Sy_k be a second projected image such that only pixels with a y-coordinate equal to k are turned ON. Then MIN(Rprjct_Sx_j, Rprjct_Sy_k) gives an image corresponding to projector pixel (j, k) being turned ON. This process can be better understood with reference to  FIGS. 5A ,  5 B,  6 A,  6 B,  7 A, and  7 B. 
     In  FIG. 5A , a scene, or display environment is illustratively shown as a flat surface  41  with a checkerboard pattern. It is to be understood that the checkerboard pattern is shown purely to facilitate the present description by providing a contrast for projected light lines, and the display environment need not have any pattern and may be of irregular shape. The relative location of each vertical light line and horizontal light line are known since it is known which projector pixels were turned in their creation, and their known relative displacement may be used to calibrate real projector  21 , as is more fully explained below. 
     Firstly, a bright vertical line, or vertical light beam,  47   —   k  (i.e. column of light rays emitted simultaneously from a column of projection pixels), is projected onto surface  41  by real projector  21 . In the present case, bright vertical line  47   —   k  is generated by turning ON all projector pixels within projector pixel array  27  that have a y-coordinate equal to k. Real camera  25  then captures this real image, Rcptr_Sy_k, as one example of a lit column of projector pixels. 
     In  FIG. 5B , where all elements similar to those of  FIG. 5A  have similar reference characters, real projector  21  projects a second bright vertical line  47   —   t  of light rays onto surface  41 . In this case, bright vertical line  47   —   t  is generated by turning ON all projector pixels having a y-coordinate equal to t. Real camera  25  then captures this image, Rcptr_Sy_t, as another example of a lit column of projector pixels. It is to be understood that real projector  21  could project a separate bright vertical line of light rays for each of the q columns of projector pixel array  27 , and real camera  25  could capture a separate image of each projected bright vertical line. 
     With reference to  FIG. 5A , all elements similar to those of  FIGS. 5A and 5B  have similar reference characters and are described above. In the present case, real projector  21  preferably projects a bright horizontal line, i.e. horizontal light beam,  49   —   j  made up of light rays emitted simultaneously from a row of projection pixels onto projection surface  41 . Bright horizontal line  49   —   j  may be generated by turning ON all projector pixels having an x-coordinate equal to j. Real camera  25  then captures this real image, Rcptr_Sx_j, as one example of a lit row of projector pixels. 
     In  FIG. 6B , real projector  21  projects a second bright horizontal line  49   —   r  (made up of simultaneously lit, individual light rays) onto surface  41 . As before, bright horizontal line  49   —   r  may be generated by turning ON all projector pixels having an x-coordinate equal to r. Real camera  25  then captures this real image, Rcptr_Sx_r, as another example of a lit row of projector pixels. It is to be understood that real projector  21  could project a separate horizontal line of light rays for each of the p rows in projector pixel array  27 , and real camera  25  could capture a separate image of each projected bright horizontal line of light rays. 
     If one now conducts a pixel-by-pixel comparison of captured image Rcptr_Sy_k from  FIG. 5A  and captured image Rcptr_Sx_j from  FIG. 6A  (or alternatively compares only their respective bright vertical and bright horizontal lines), using operation MIN(Rcptr_Sx_j, Rcptr_Sy_k) to retain only the darker of two compared image pixels and discarding the brighter of the two, one would generate an image  43 ′, as shown in  FIG. 7A . All elements in  FIGS. 7A and 7B  similar to those of  FIGS. 5A ,  5 B,  6 A, and  6 B have similar reference characters with the addition of an apostrophe, and are described above. 
     Since most of image Rcptr_Sx_j ( FIG. 6A ) is the same as image Rcptr_Sy_k (i.e. they mostly consist of the checkerboard pattern on flat surface  41  with projected light rays), retaining the darker of two compared pixels does not change the majority of the resultant image. That is, if two compared pixels are relatively the same, then electing either pixel over the other does not much affect the resultant image  43 ′. However, when a pixel on bright vertical line  47   —   k  in captured image Rcptr_Sy_k is compared with a corresponding pixel in captured image Rcptr_Sx_j that does not lie on a bright horizontal line  49   —   j , then retaining the darker of the two image pixels will discard the pixel on the bright vertical line from image Rcptr_Sy_k, and retain the plain pixel from image Rcptr_Sx_j, which is illuminated by ambient light and not by a projector pixel. Therefore, bright vertical line  47   —   k  is eliminated from generated image  43 ′. Similarly, when a pixel on bright horizontal line  49   —   j  in image Rcptr_Sx_j is compared with a corresponding pixel in image Rcptr_Sy_k that does not lie on bright vertical line  47   —   k , then retaining the darker of the two pixels will discard the pixel on the bright horizontal line from image Rcptr_Sx_j, and retain the plain pixel from image Rcptr_Sy_k, which is illuminated by ambient light and not by a projector pixel. Consequently, horizontal line  49   —   j  is also eliminated from generated image  43 ′. However, within the region where bright vertical line  47   —   k  intersects bright horizontal line  49   —   j , both compared image pixels are brightly lit pixels showing an impact by a light ray. Comparison of these two image pixels within this intersection region will result in either of the two bright beam pixels being selected for image  41 ′. As a result, image  41 ′ will show a brightly lit region  53  corresponding to a projected light ray emitted from coordinates (j,k) of projector pixel array  27 . Thus, the light transport coefficients for the projector pixel having coordinates (j,k) can be extracted from generated image  53  without having to physically capture an image of a light ray individually projected from the projector pixel at (j,k). 
     A second example is shown in  FIG. 7B , where the combination of real captured images corresponding to  FIGS. 5B and 6B  (which would respectively correspond to real captured images Rcptr_Sy_t and Rcptr_Sx_r following the above-described naming convention), results in a second brightly lit region  55  corresponding to a projected light ray emitted from coordinates (r,t) of projector pixel array  27 . 
     A similar process may be followed to identify the light transport coefficients of every projector pixel j in projector pixel array  27  without having to individually turn ON and project each projector pixel j, one-at-a-time. This method of generating an image of a hypothetically, singularly activated projector pixel to obtain the projector pixel&#39;s light transport coefficients requires at most only (p+q) captured images, one for each row and column of projector pixels in projector pixel array  27  of real projector  21 . Furthermore, once all the pixel projection locations have been identified, the (p+q) captured images may be discarded, and all that needs to be saved is an index and corresponding footprint information. 
     An example of this approach is shown in  FIG. 8 , where a partial view of projector pixel array  27  (illustrated as a crosshatch pattern) is compared to a partial view of camera sensor pixel array  29  (also illustrated as a crosshatch pattern). For illustrative purposes, the cross-hatch pattern illustrating the partial camera sensor pixel array  29  is made denser than the cross-hatch pattern representing projector pixel array  27  in order to better illustrate that, in the present example, pixel density (i.e. resolution) of real camera  25  is preferably greater than the resolution of real projector  21 , and thus a light ray emitted from a single projector pixel j may create a light footprint (such as F 1 ) spanning several camera pixels i. 
     In the present example of  FIG. 8 , an index of real projector pixel array  27  is represented as a partial array with circles  1 ,  2 ,  3 , . . . (q+1) . . . (2q+1) . . . etc. representing individual projector pixels j. Similarly, real camera sensor pixel array  29  is shown superimposed by a corresponding partial array of circular light footprints F 1 , F 2 , F 3 , . . . F(q+1), . . . etc. representing the footprint information corresponding to individually activated projector pixels j (assuming light scattering is ignored). Thus, footprints F 1 , F 2 , F 3 , . . . F(q+1), . . . etc. respectively correspond to projector pixels  1 ,  2 ,  3 , . . . (q+1), etc. 
     Following this approach, only two sets of information need to be stored. A first set of information corresponds to an index of projector pixels j and a second set of information corresponds to footprint information associating groups of camera pixels i with each projector pixel j. In other words, zero-valued coefficients need not be stored, which greatly reduces the memory requirements. 
     A second example of organizing this information is shown in  FIG. 9 , where an index  61  of projector pixels j is shown to point to, or correspond to, group  63  of grayscale (i.e. non-zero valued, or “NZ Grayscale”) camera pixel i information (i.e. corresponding to a resultant light ray footprint). 
     Having shown how to reduce the number of images that need to be captured and stored to generate the needed light transport coefficients, and how to reduce the amount of data that needs to be stored and manipulated, for dual photography, the following next section addresses some of the practical difficulties of implementing dual photography. As is explained above, a light transport matrix T can be very large, and its use (or the use of its transpose, the dual light transport matrix T T ) requires large amounts of active memory (for example, DRAM) and excessive computational processing power/time. Therefore, general use of the dual image has heretofore not been practical. 
     To more efficiently construct a dual image, one first notes that
 
Vcptr″=T T Vprjct″
 
Since the virtual camera sensor pixel array  27 ″ in virtual camera  21 ″ corresponds in actuality to real projector pixel array  27  of real projector  21  (see  FIGS. 2A and 2B ), it is convenient to use the same index j to denote any virtual camera pixel within Vcptr″ obtained by virtual camera  21 ″. Therefore, a relationship between each virtual projector pixel j in a virtual captured image versus a corresponding row of elements in T T  may be denoted as
 
Vcptr″( j )= T   T   j Vprjct″
 
where T T   j  refers to the j th  row in T T .
 
     As is explained above, T T  is the matrix transpose of light transport matrix T (i.e. matrix T turned on its diagonal), and the values of row T T   j  (where j is any value from 1 to (p×q)) therefore correspond to the j th  column of matrix T (i.e. T COL     —     j ). Since each column of T has (m×n) elements (i.e. equivalent to the pixel resolution of real camera  25 ), this would appear to be a very large number of elements. However, recalling that in the present implementation, only a limited number of elements in each column of matrix T are non-zero (i.e. only those corresponding to camera sensor pixels i upon which shone the intersection of a vertical and horizontal light beam, i.e. a light footprint), it is self apparent that only a few of the (m×n) elements within in each column j of matrix T (and subsequently in each row T T   j ) are non-zero. Therefore, it is not necessary to examine all elements in T COL     —     j  when computing Vcptr″ (1). Indeed, as is explained above in reference to  FIGS. 8 and 9 , it is preferred that only a single index  61  showing all individually activated projector pixels j, and their corresponding light ray footprint information  63  be stored. 
     As is shown in  FIG. 9 , index  61  associates a listing of grayscale entries  63  for each projected light ray (from an individual projector pixel j of real projector  21 ). It is to be understood that each group of grayscale entries  63  corresponds to the non-zero entries within each column of T, and only these non-zero grayscale values need to be examined during each matrix operation of a column of light transport matrix T. Thus, the number of calculations needed to determine each value of Vcptr″ (j)=T T   j Vprjct″ is greatly reduced. 
     In other words, this subset of elements, SVcptr″(G), within each column G of matrix T that needs to be processed may be defined as T COL     —   G(a), where a is an index for any virtually captured pixel [i.e. a ∈ {1, . . . , (p×q)}] in Vcptr″(G). Therefore, for each SVcptr″(G), one can define the set of elements to be examined as:
 
SVcptr″( G )={ a|∀z ∈{ 1, . . . , ( p×q )} T   COL     —     G ( a )≧ T T   COL     —     z ( a )}
 
Since in general ∥SVcptr″(G)∥&lt;&lt;(p×q), it takes significantly less time to compute:
 
                 Vcptr   ″     ⁡     (   G   )       =       ∑     α   ∈       SVcptr   ″     ⁡     (   G   )           ⁢         T   G   T     ⁡     (   α   )       ⁢       Vprjct   ″     ⁡     (   α   )                 
than to compute:
 
Vcptr″( j )= T   T   j Vprjct″
 
     An example of a dual image generated using this method is shown in  FIGS. 10A and 10B .  FIG. 10A  shows a primal image, as projected by a real projector.  FIG. 10B  shows the resultant dual image computed by an implementation of the present method. The dual image of  FIG. 10B  represents the image virtually captured by the real projector  27  (i.e. virtual camera  21 ″), or stated differently, the image as “seen” by the real projector  21 . 
     The above discussion shows how to compute dual images efficiently from a reduced set of images, which saves image capture time as well as computation time. As is explained above, the real captured images and dual captured images can be used to calibrate both real camera  25  and real projector  21 , respectively. 
     That is, since the images from the projector&#39;s view cannot be directly captured, a straightforward solution is to construct projector-view images (i.e. dual images) from corresponding camera images using dual photography techniques, and then to calibrate the projector using the constructed images. For example, after having taken several real images at different angles of a known object with carefully measured features, the real camera can be calibrated by using the known dimensions of the object to compensate for distortions in the captured images arising from the different angle views. The virtual images, as seen by the real projector, can then be generated from the same captured images using dual photography techniques, as described above, and the real projector may be calibrated in a manner analogous to the real camera. 
     A possible setback associated with this straightforward method, however, is the difficulty in generating and manipulating the light transport matrix T, and operating on the large image vectors resulting from the large number of camera and projector image pixels. Although this labor-intensive and expensive process is mitigated substantially by using the dual photography method described above, for purposes of calibrating real projector  21  in a projector-camera system, such as that shown in  FIG. 2A , Applicants have developed a novel method that avoids the need for generating a full T matrix and for creating a full dual image, while still taking advantage of some of the benefits of using a dual image (i.e. an image as “seen” by real projector  21 ) to facilitate calibration of real projector  21 . 
     Assuming that a projection scene is relatively flat, then one can greatly reduce the number of light footprints points for which a light transport coefficient needs to be determined. In other words, the generation of a full T matrix can be avoided altogether by noting that to calibrate the projector; one does not need to construct an entire dual image, but only needs to determine the location of a strategic set of points, particularly if the projection surface is not very irregular. For example, the strategic set of points may be the corners of the squares within the checkerboard pattern on flat surface  41  of  FIG. 5A , as seen by the projector. The greater the irregularity of the projection scene surface, the greater the number of points in the strategically selected set of points. Conversely, the flatter the projection scene, the fewer the number of points in the strategic set of points. 
     Applicants have adapted a homography-based method to achieve this goal of a reduced number of points in the strategic set, and thus avoid the generation and manipulation of a full T matrix, or a simplified T matrix described above, and further avoid the full dual-image generation process by incorporating some features of dual-image generation into the calibration process, itself. This alternate embodiment of the present invention directly computes the coordinates of the checker corner features (or other known feature) across the projector-view images without requiring the construction of the dual images and the detection of the corners from the constructed dual images. 
     In this novel method, one may use the real camera to capture images of a planar checkerboard and detect the checker block corners across the captured images. It is to be understood that a checkerboard is being used purely for illustrative purposes, and any scene may be captured. In the present case, since the orientation of the projected vertical and horizontal lines are known, variations in the projected lines can be identified and corrected, irrespective of the scene. Thus, the projector-camera system need not be calibrated beforehand. Rather, all elements of the projector-camera system are calibrated at the same time. 
     Secondly, it is observed that projector images follow the so-called perspective projection model, which relates two (or more) views of a single scene as seen by two (or more) separated sources. That is, different viewing sources will “see” a different view (or image) of the same scene since the different sources are located at different angles to the scene. However, since there is only one real scene (irrespective of the number of views of the scene), one can generate a mathematical relationship between the different views that will associate any point on any one view to a corresponding point on the real scene (and thereby to a corresponding point in each of the other views). 
     If one of these separate views is assumed to be a virtual image as “seen” by a real projector, while a second separate view is deemed to be a real image captured by a real camera, then the perspective projection model (which relates the two views to the common, real scene) would permit one to extract from the captured real image some information relating to the virtual image, without generating a full dual image. 
     Using this approach, Applicants have devised a method of extracting sufficient information for calibrating a real projector without requiring a full dual image. Thus, although the dual image is not fully created, one can still apply a camera calibration technique to a projector, albeit in a roundabout way. 
     Under the perspective projection model, the relationship between two image projections of a planar object from different views is a simple linear projective transformation or homography. This transformation relates the coordinates of any point on the planar object (i.e. a homogeneous coordinate) to the coordinates of a corresponding point on a specific view of the planar object. In the present embodiment, the projector-view image of the planar checkerboard is a homography of the corresponding camera image. Specifically, for any point P on the real scene (i.e. checkerboard), its homogeneous coordinate in the projector-view image Up=(up, vp, 1) and the coordinate in the camera image Uc=(uc, vc, 1) satisfy the following equation,
 
Up=λHUc
 
where λ is a scalar and H is a 3×3 homography transformation matrix (as is known in the art) of which the bottom right entry is set to 1. The pair of corresponding coordinates provides 3 linear equations, where one of the equations determines the scalar and the other two are used to determine H, the homography transformation matrix. Since there are 8 unknown entries in 3×3 matrix H, given the correspondence between N coordinate points (where N≧4) on the checker board, the homography between the projector-view image and the camera image can be recovered by solving the 2N linear equations. The greater the number of N, the lower the error relating coordinate points between the projector-view and the camera image.
 
     To obtain the corresponding coordinates, 10 white points are preferably projected on a scene, such as the checkerboard pattern. An image of the checkerboard with the projected white points is then captured using a real camera, such as real camera  25 , and the coordinates of the 10 points in the camera image are computed. In the present process, it is only necessary that the ten points be distinguished during the computation of their corresponding coordinates in the captured image. This may be achieved by projecting the ten points sequentially, and determining their corresponding coordinates, in turn. Alternatively, differently colored points may be projected simultaneously, and the different points identified by color. 
     Since the projector preferably projected the points in a known relation to each other, the coordinates of the points in the projected image are known. This results in 10 pairs of corresponding coordinates, one set as captured by the real camera and a second set as projected by the real projector. Once the homography is recovered, the coordinates of the checkerboard corners detected in the camera images can be directly transformed to compute the corresponding corner coordinates in the projector-view images. The projector parameters can then be calibrated using a camera calibration method, such as the one described above. 
     An example of this approach is shown in  FIG. 11 . In  FIG. 11 , the feature capture results are as follows. The circles, or dots, not shown in outline (for example dots  81 ) were used for estimating nomography, while the outlined circles, or dots, (for example dots  83 ) are the corner point features. As can be seen, the outlined dots  83  are on the actual corners, indicating that the projector coordinates for each detected corner has been correctly captured. 
     It is to be understood, however, that instead of projecting dots, one may use the method described above of projecting multiple pairs of lines that intersect at one point (i.e. horizontal and vertical lines as described above), and extracting dot information from the one intersection point of each pair of lines. Furthermore, it is not necessary to project the lines onto a known pattern, such as the checkerboard pattern described above, since the spatial relationship between the pairs of projected lines are already known. That is, since one knows which columns and rows of the projector pixel array are turned on simultaneously, one knows their spatial relation. Furthermore, it is to be noted that the camera need not be pre-calibrated to the particular scene. Rather, the projector and camera may be calibrated at the same time. 
     As is explained above, this approach of using patterns to extract dot information includes a series of pattern projection and image capture steps. Thus, when an uncalibrated camera captures each projected pattern (i.e. vertical or horizontal line), any distortion of the projected pattern due to irregularities on the projection surface may be ignored for the moment. When all the patterns have been captured, the spatial relation (or transformation) between the projector and camera can be made since the spatial relation between the vertical and horizontal lines are known. That is, the spatial relation between the vertical and horizontal lines, as projected, are known since one knows which projector pixels were turned on during their generation. Furthermore, since one knows the true orientation of the vertical and horizontal lines, one can compensate for surface irregularities on the projection scene and view angles. 
     When a captured image of a vertical line is combined with a captured image of a horizontal line to extract dot information, one knows what projector pixel relates to the extracted dot, and one further can determine what group of camera pixels correspond to the extracted dot (as determined by the intersection of the vertical and horizontal lines). Thus, the homography relationship between the projector and camera (i.e. the transformation of one view for the other) can be achieved without need of calibrating either device individually to any specific projection scene (or projection surface). 
     This approach borrows from the above-described, simplified method for generating the transport matrix, T, but reduces the number of needed image projection-and-capture steps from (p+q) to a fraction of (p+q) determined by the number of desired dots. Although as little as 4 dots may be used to calibrate a projector-camera system whose scene environment is a flat projection surface, in order to account for some irregularities in a projection surface, for possible errors in the image capture steps, and for errors in identifying projected light lines or patterns due to existence of ambient light noise, it has been found that projecting seven horizontal light lines and seven vertical light lines to generate as many as fourteen to forty-nine dots is sufficient for overcoming most errors. It is to be understood that the greater the amount of surface irregularities in the projection scene, the greater the number of needed intersecting pairs of lines, but this number of seven vertical lines and seven horizontal lines has been found to be sufficient for many real-world situations. A reason why much fewer than (p+q) dots are needed is because one is not trying to generate a full transport matrix for use in dual photography. Rather, one is interested in identifying a few known relational dots for purposes of finding the nomography relationship between two views, i.e. between the real view from the real camera and the virtual view from the real projector. However, it should be emphasized that an estimate of a full light transport matrix T can be generated from the few relational dots by assuming a smooth transition between the dots. That is, once the homography relationship is obtained, the same relationship of Up=λHUc can be used to fill-in gaps in construction of an estimated light transport matrix T. 
     The above described method for generating transport matrix T can be applied toward solving a different problem. Specifically, given a projector and camera pair, if one would like the camera “to see” a desired view on a given scene (i.e. projection environment), what should the projector illuminate onto the scene in order to produce that desired view? This task is termed “View Projection” hereinafter. 
     Clearly, finding the needed image to be projected requires detailed and precise knowledge about the geometry of the scene and the projection characteristics of the projector device, along with the photometric properties of the scene so that the reflection of light from the projector produces the exact desired view. For display surfaces with uniform albedo, i.e. reflectivity, and parametric shapes, these necessary measurements can be made explicitly. 
     Producing high quality imagery on arbitrarily complex nonparametric shapes with non-uniform surface reflectance properties, however, is still a daunting task requiring familiarity with a wide variety of tools and much human intervention. A goal of the present embodiment is to devise a measurement process that is completely automatic and requires no user intervention or parameter tweaking beyond casual hardware setup. In other words, the projector-camera system that achieves View Projection capabilities should completely calibrate itself with a single touch of a button. 
     A projection system that is capable of displaying correctly on complex surfaces under challenging settings has many real world applications. Making it fully automatic further allows it to be deployed in settings where professional display setup is not always available. Another goal of the present embodiment is to reduce the requirement on exotic imaging equipment, so that high quality calibration can be achieved with off-the-shelf components within an average consumer&#39;s reach. This goal may be achieved by combining some of the features and benefits of the previous embodiments in an automated and simplified process. 
     As is explained above in reference to the image setup of  FIG. 2A , which consists of one real projector  21  and one real camera  25 , real projector  21  has a p-row-by-q-column projector pixel array  27  while the real camera  25  has an m-row-by-n-column camera sensor pixel array  29 . Light rays emitting from real projector  21  bounce off a scene (i.e. projection surface) and some of them eventually reach real camera sensor pixel array  29 . In general, each ray of light is dispersed, reflected, and refracted in the scene and hits camera sensor pixel array  29  at a number of different locations. Thus, the light ray emitted from projector pixel j reaches real camera  25  and forms an m-by-n image across the entirety of camera sensor pixel array  29 , where each camera pixel i receives a certain amount of light. If the image projected is represented as a (p×q)-element real image vector Rprjct, and the image captured is represented as an (m×n)-element captured image vector Rcptr, then the light transport between real projector  21  and real camera  25  can be written as
 
Rcptr=T Rprjct
 
where T is the light transport matrix. As is illustrated in  FIG. 2B , it has been shown that by the principle of Helmholtz reciprocity, the equation
 
Vcptr″=T T Vprjct″
 
can be used to model a “dual” setup where projector  21  is viewed as a p-row-by-q-column virtual camera  21 ″, and real camera  25  is viewed as a m-row-by-n-column virtual projector  25 ″. This enables “Dual Photography”, where one can synthesize images that appear as though they were captured by real projector  21 , with the scene appearing as if illuminated by light emanating from real camera  25 .
 
     View Projection, the object of the present embodiment, addresses a different problem. In View Projection, one is interested in finding a projector image that, when used to illuminate a scene, allows the camera to capture a predefined, desired image. In light transport terms, one is provided with T and with a desired Rcptr, and one wants to recover, i.e. generate, Rprjct. Clearly, Rprjct can be found by the following relation:
 
Rprjct= T   −1 Rcptr
 
If one can determine inverse matrix T −1  of light transport matrix T, then one can generate the required projection image, Rprjct, needed to produce the desired viewable image Rcptr.
 
     The inverse of light transport matrix T has been used in the past for other purposes. For example, in article “A Theory of Inverse Light Transport” by Seitz et al. (IEEE International Conference on Communications, ICC V05), hereby incorporated in its entirety by reference, the inverse of the light transport matrix T is used to analyze the way light bounces in arbitrary scenes. In this approach, a scene is decomposed into a sum of η-bounce images, where each image records the contribution of light that bounces η times before reaching a camera. Using a matrix of “impulse images”, each η-bounce image is computed to infer how light propagates through the scene. 
     The problem being addressed by Seitz et al., of course, is different than that posed by View Projection, where one is provided with, or envisions, a desired image (Rcptr, for example) and wants to infer an unknown projection source that can produce the desired image. In this case, it would be beneficial to compute Rprjct=T −1  Rcptr. It should be noted that calculation (T T  Vprjct″) of Dual Photography would render a virtually captured image, Vcptr″, that is a different view of Rcptr, but does not show the real source of projected image Rprjct. 
     The inverse (T −1 ) of light transport matrix T is more difficult to compute than its transpose, T T , requiring much more computational resources. Indeed, the sheer size of T makes computing T −1  an extremely challenging task requiring tremendous computational resources. Worse, it is not always possible to find the inverse of an arbitrary matrix. That is, some matrixes may not have an inverse. 
     As it is known in the art, the identity matrix, or unit matrix, is denoted by I, and has the property that for a given matrix A, the following relationship holds:
 
AI=IA=A
 
If matrix A were a matrix of order m by n, then the pre-multiplicative identity matrix I would be of order m by m, while the post-multiplicative identity matrix I would be of order n by n.
 
     The multiplicative inverse of a matrix is typically defined in terms of identify matrix I. A left multiplicative inverse of a matrix A is a matrix B such that BA=I, and a right multiplicative inverse of a matrix A is a matrix C such that AC=I If the left and right multiplicative inverse of a matrix A are equal, then the left and right multiplicative inverse are simply called the “multiplicative inverse”, or “inverse”, and is denoted by A −1 . 
     Fortunately, Applicants have found that in many display settings, an approximation to inverse matrix T −1  may suffice. Creating a suitable approximation to the inverse of a matrix, however, requires meeting certain criteria not generally present in light transport matrix T. Therefore, before attempting to create an approximation to inverse matrix T −1 , one needs to first create a suitably modified light transport matrix T that meets the required criteria. 
     In general, T −1 ≠T T . However, one constraint used above in the generation of an estimation of T T  is likewise useful in the construction of T −1 . As is discussed above, for most projector-camera display applications, one may generally use the limitation that any two distinct light rays j and k emitted from a projector will hit a camera sensor at distinct parts, i.e. there is usually little overlap in the camera sensor pixels hit by light from each of the light rays j and k. This characteristic is termed the “Display Constraint” hereinafter. 
     In general, the Display Constraint may be violated if there is significant light scatter in the scene, such as the example given above where the scene consists of a glass of milk, and the light rays are diffused by the milk resulting in significant overlap. Another example of a scene that violates the Display Constraint is shown in  FIG. 12 , which includes two wine glasses between a projector and a projection scene, resulting in significant light scattering. In this example, a projected image consisting of lettering is projected onto the scene and shown to experience much optical distortion due to the wine glasses. Below is provided a method for compensating for such light scattering when the Display Constraint is violated, but for the present discussion, it is first assumed that the Display Constraint is met and each pixel projected will be distinct from the next. 
     As is also explained above, each column of the transport matrix T is the projection image resulting from one pixel from the projector. Thus, all of the column entries have zero values except those corresponding to the camera pixels hit by light emitting from the corresponding projector pixel. 
     Applicants noted that if adjacent light footprints (created by adjacently lit projector pixels) have no overlap, then columns of T would be orthogonal. In real-world applications, by the Display Constraint, overlap between adjacent light footprints is minimal, which implies that light from different projector pixels will mostly hit different camera pixels. As a result, if two columns of T are placed next to each other, the non-zero entries will not line up most of the time, and their dot product will be close to zero. This implies that the columns of T are approximately orthogonal to each other. In the present technique, these minimal, overlapping pixels are removed from the creation of the T matrix, which results in an approximate T matrix that is orthogonal, by definition. As is explained more fully below, these techniques permit further simplification of creation and manipulation of the T matrix. 
     By the Display Constraint, light from different projector pixels will mostly hit different camera pixels, i.e. camera pixels corresponding to different projector pixels will have little overlap with each other. An example of this is shown in  FIG. 13 , wherein a first light footprint, Projector_Pixel_ 1 , from a first projector pixel shares only two perimeter camera pixels  2   i  with an adjacent second light footprint, Projector_Pixel_ 2 , from a second projector pixel. This means that if columns of light transport matrix T are placed next to each other, the non-zero entries among the different columns will not line up most of the time (i.e. only the dimmer non-zero entries will have some overlap, as illustrated by sample pixels  2   i  in the present example), and their dot product will be close to zero. 
     An example of this is shown in  FIG. 14 , where four columns C 1  to C 4  of a light transport matrix T are shown adjacent each other. In column C 1 , circles Pixel_ 1   c  and Pixel_ 1   d  identify the brightest parts (i.e. the center parts) of a first pixel&#39;s light footprint that is recorded as an image within column C 1 . As shown, the full light footprint includes other less bright pixel groups Pixel_ 1   b  and Pixel_ 1   e . However, for ease of explanation, only the brightest parts of each light footprint within each column C 1 -C 4  is identified by circles since the circles identify those pixel coefficient entries that will be retained in construction of a modified (i.e. estimated) light transport matrix T. As was shown in  FIG. 13 , the brightest part of any one light footprint does not overlap with the brightest part of any other light footprint. Thus, if the dimmer, perimeter sections of a light footprint (i.e. pixel groups Pixel_ 1   b  and Pixel_ 1   e ) are ignored to form a resized light footprint, then it can be assured that no overlap will exist between the resized light footprints. As is shown in  FIG. 14 , if one compares adjacent circled groups of pixels along adjacent columns, one will note that none of the circled pixels in adjacent columns overlap. For example, partial light footprint Pixel_ 1   c  in column C 1  is offset in the horizontal direction from partial light footprint Pixel_ 2   c  in column C 2 , which in turn is offset from partial light footprint Pixel_ 3   c  in column C 3 , which is likewise offset from partial light footprint Pixel_ 4   c  in column C 4 , and so on. Similarly, none of the other bright partial light footprint circled sections (i.e. Pixel_ 1   d , Pixel_ 2   b , Pixel_ 3   b , and Pixel_ 4   b ) within each column line up with each other in the horizontal direction. 
     Since by the Display Constraint light from different projector pixels will mostly hit different camera pixels, most column entries have zero-value entries, except those corresponding to a light footprint defining camera pixels hit by light emitted from a corresponding projector pixel. This means that if two columns of T are placed next to each other, the non-zero entries will not line up most of the time, and their dot product will be close to zero. This implies that the columns of T are orthogonal to each other. 
     This is particularly true if the dim perimeter parts of a light footprint are ignored, and one works with the resized light footprints consisting of only the brightest sections of a light footprint. In this case, columns of T become truly orthogonal to each other, meaning that that the transpose of a specific column multiplied by any column other than itself will produce a zero result. 
     Thus, the orthogonal nature of T may be assured by the way in which T is constructed in the above-described process for generating a modified T. As is explained above, the amount of memory necessary for storing T is likewise reduced by storing only nonzero values within each column. In the modified T, this process is simplified because the nonzero values to be stored can be quickly identified by storing only the brightest value within each row of light transport matrix T, which automatically creates resized light footprints. For example in  FIG. 14 , one may identify the brightest pixels along rows of T (i.e. among adjacent image information corresponding to separate projector pixels and arranged in adjacent columns of T) as indicated by circles Pixel_ 1   c  to Pixel_ 4   c , Pixel_ 2   b  to Pixel_ 4   b  and Pixel_ 1   d . If only the brightest valued entry within each row is retained, one obtains the structure of  FIG. 15 . The identified bright sections may then be combined into a vector representation  40  of modified matrix T (hereinafter light transport matrix T and modified light transport matrix T are used interchangeably, unless otherwise stated). As shown, vector representation  40  removes any overlap between adjacent columns of T by limiting itself to only the brightest pixel values within each row of T, and effectively imposes the Display Constraint on an arbitrary scene. Consequently, this construct of T is orthogonal, by design. It is to be understood that to fully define modified light transport matrix T, one only needs a second matrix column, or second array, to hold index information indicating which groups of partial light footprint sections belong to the same resized light footprint. In the present case, for example, groups Pixel_ 1   c  and Pixel_ 1   d  are part of a single resized light footprint corresponding to a first pixel. Similarly, light footprint sections Pixel_ 2   b  and Pixel_ 2   c  together form a second resized light footprint for a second pixel, and so on. 
     To determine an approximation to the inverse of T (i.e. to determine an approximation to inverse matrix T −1 ), it is beneficial to first note that AA −1 =I, and that the identity matrix I is comprised of a matrix having entry values set to a value of “one” (i.e. 1) along a diagonal from its top left corner (starting at matrix location (1,1)) to its bottom right corner (ending at matrix location (r,g)), and having entry values set to “zero” (i.e. 0) everywhere else. In order to compute T −1 , one first defines a matrix {hacek over (T)} such that each column in {hacek over (T)} is comprised of normalized values, with each column in {hacek over (T)} corresponding to a column in light transport matrix T (or equivalently, to a column in the modified light transport matrix). That is,
 
 {hacek over (T)}r=Tr /(∥ Tr ∥) 2   , r= 1, 2, 3 , . . . , pq  
 
where {hacek over (T)}r is the r th  column of {hacek over (T)}. Since matrix operation ∥Tr∥ defines the square root of the sum of the squares of all values in column r of matrix T, the square of ∥Tr∥ is simply the sum of the squares of all the values in column r. That is,
 
                 (        Tr        )     2     =     {       ∑     ɛ   =   1       ɛ   =   g       ⁢       (     Tr   ɛ     )     2       }           
By dividing each value entry in column r by the sum of the squares of all the values entries in column r, operation {Tr/(∥Tr∥) 2 } has the effect of normalizing the value entries in column r of matrix T. If one now takes the transpose of {hacek over (T)}r, i.e. flips it on its side such that the first column becomes the top row and the last column becomes the bottom row, the result will be rows of elements that are the normalized values of corresponding columns of elements in T. Therefore, for every column in T, one has the following result:
 
( {hacek over (T)}r   T )×( Tr )=1
 
and
 
( {hacek over (T)}r   T )×( T ω)=0, for  r≠ω 
 
In other words, multiplying a column of T with a corresponding row in {hacek over (T)}r T  always results in a value of 1, and as one multiplies all the columns in T with the corresponding row in {hacek over (T)}r T , one produces a matrix with numeral 1&#39;s along its diagonal, and one may place zeroes everywhere else to fully populate the produced matrix.
 
     Therefore, in the case of matrix T, where columns are (or are made) orthogonal to each other, and given the specific construction of matrix {hacek over (T)}, it has been shown that the transpose of {hacek over (T)} is equivalent to the inverse of T (i.e. {hacek over (T)} T =T −1 ), by definition, or at least {hacek over (T)} is a left multiplicative inverse of T. Therefore, Rprjct={hacek over (T)} T  Rcptr. 
     Note that only the part of the projector pixels that actually hit the camera sensor can be recovered. For the projector pixels not hitting any of the camera pixels, the corresponding columns in T contain zeros and the above equation of {hacek over (T)}r=Tr/(∥Tr∥) 2  is undefined. In such cases it is preferred that the corresponding columns in {hacek over (T)} be set as zero columns. Thus, {hacek over (T)} T  is the inverse of the part of T that covers the overlapping area of the field-of-views of the projector and the camera. It only recovers the projector pixels in Rprjct that fall in the overlapping area and blacks out the other pixels. 
     In the following discussion, matrix {hacek over (T)} T  is called the View Projection matrix, such that given a desired view c, one can find an image p defined as p=({hacek over (T)} T  c) such that projecting image p produces a scene which, when viewed from the reference camera location, has the same appearance as c. Since {hacek over (T)} T  is effectively an approximation of inverse matrix T −1  (or a modified version of matrix T −1  created by imposing upon it the display constraint) and is used as such herein, for the rest of the following discussion inverse matrix T −1  and View Projection matrix {hacek over (T)} T  may be used interchangeably, unless otherwise indicated. In this case, it is to be understood that the symbol T −1  generally represents the inverse of both the unmodified and the modified (i.e. approximated) forms of the light transport matrix T since its modification is optional, for ease of use. 
     Although an efficient method for computing an approximation of true inverse matrix T −1  has been demonstrated, in its raw form, inverse matrix T −1  is still a staggeringly large matrix, requiring large amounts of storage and of course significant computation power just to perform matrix multiplication. Again, approximation of the true inverse matrix T −1  is generally subject to the Display Constraint. 
     Therefore, it is advisable to apply to View Projection matrix {hacek over (T)} T  a similar compacting technique as is described above for reducing transpose matrix T T . In this manner, View Projection matrix {hacek over (T)} T  may be reduced to a representation consisting of a first column of nonzero entries and a second column of corresponding index values. Recall that the Display Constraint dictates that the non-zero entries in the distinct columns of T do not line up on the same row. This means that on each row of T, there will only be one non-zero entry (or one entry above a predefined threshold value) and one need only store that entry along with its column index in order to represent the entire matrix T (with the understanding that any non-stored T entry location is designated with a zero value, by default). Consequently, the space needed to store T can be reduced down to that of a single column of nonzero T entries plus a second column of index values. 
     In a real world setting, however, many of the unlit entries of T (i.e. those camera pixels i not within a light footprint) will not be exactly zero due to sensor noise (i.e. light picked up by a camera pixel, but which does not originate from a singularly designated, i.e. turned ON, projector pixel). Thus, there may be multiple non-zero values in each row, with one non-zero value entry corresponding to a light footprint and the remaining non-zero entries being due to light noise. In such cases, it is likely that the one non-zero entry corresponding to a light footprint will have the highest value in the row (i.e. have the brightest value). Therefore in order to filter out light noise, the maximum valued entry in each row (i.e. the row-maximum value) may be identified and designated as the one non-zero entry for that row that corresponds to a light footprint. This eliminates any low valued entries due to light noise. By identifying these row-maximum entry values, one obtains the non-zero values of each column. Therefore, for each projector pixel, one identifies the corresponding set of camera pixels, along with the distribution of light intensity among these camera pixels. Typically, this set of corresponding camera pixels (associated to any given projector pixel) is a very small subset of the entire camera image. Since only the entries from these corresponding camera pixels need to be considered during matrix operations, performing view projection image transformation using this sparse matrix representation is very efficient. 
     In practice, it is preferred to use a fixed pattern scanning technique, which exploits the Display Constraint to speed up the process. As is explained above, the contribution from a projector pixel j maps to column j of the light transport matrix T. It follows that a set of projector pixels S 1  ⊂ {1, . . . , p×q} maps to a corresponding set of columns in T. Considering two such sets of pixels S 1  and S 2  where S 1  ∩ S 2 ={j}, let the two images captured when turning on the two sets of pixels be C S1  and C S2 . It can be shown that
 
Tj≈MIN(C S1 , C S2 )
 
where Tj is the j th  column of T. C S1 , C S2  are the sum of the respective columns of T, i.e. C S =Σ j∈S  Tj. Given that the contribution of each individual projector pixel j is mapped to distinct parts of the camera sensor, there is a common set of pixels l ⊂ {1, . . . , m×n} in the captured images C S1 , C S2  that corresponds to projector pixel j. Now, as S 1  ∩ S 2 ={j}, for the rest of the captured image pixels ∉l, at least one of the images would not have received light from one of the projector pixel sets, S 1  or S 2 . Since pixels receiving light will be brighter than pixels not receiving light, MIN(C S1 , C S2 ) renders an image where only pixels ∉l are lit, which is a good approximation of Tj.
 
     This implies that if one can construct sets
 
Y1, . . . , Yp, X1, . . . , Xq,
 
where
 
∀ i  ∈{1, . . . , ( p×q )}, ∃ X   j   , Y   k   |X   j    ∩Y   k   ={i},  
 
one can synthesize Tj where j=1, . . . , p×q from images C X1 , . . . , C Xq  and C Y1 , . . . , C Yp .
 
     In the above discussion, each pair of projected images has only one intersection point in common. One construction that satisfies this property uses the following pixel coordinates as indicators: let Xj be an image creating by turning on only those pixels whose x-coordinate is equal to j, and Yk be an image created by turning only those pixels whose y-coordinate is equal to k. Then MIN(Xj , Yk) gives an image with coordinates (j, k) turned ON. Using this method, capturing p+q images allows one to synthesize all p×q columns of T. It should be noted that while this method for capturing T is faster than capturing each pixel individually, and simpler than the adaptive scheme described in the reference to Sen et al., 2005, discussed above, this method may still be slower than other schemes previously proposed in the literature. In particular, a method for establishing projector-camera pixel correspondence is to project the X and Y coordinates in time sequential binary codes, in theory allowing all correspondence to be captured in log(N) time, where N is the maximum coordinate value. In practice, this method places stringent requirements on the camera sensor resolution, especially when the least significant bits are being projected. One may also propose to project multiple stripes simultaneously to cut down the number of images needed. This however requires some way to identify each of the stripes captured in an image, e.g. specifying a subset of landmark stripes or exploiting color coding or other distinguishing patterns. These schemes may indeed speed up the capture process, however they introduce additional complexity and brittleness, and often one needs to tweak the feature detection parameters so the schemes work correctly. In the presently preferred embodiment, there are no parameters to be tweaked and unless there is a hardware failure, a one-time capture typically yields a good copy of a view projection matrix. 
     As is explained above, the view projection matrix, {hacek over (T)} T , is capable of compensating for geometric and photometric distortions introduced by projection optics, display surfaces, and their relative positioning. The efficacy of the view projection capabilities of {hacek over (T)} T  was tested by means of a number of experiments. 
     To illustrate that one can control the desired appearance of a scene, it was shown that one can dynamically change the appearance of a printed poster. An example is shown in  FIG. 16 . The Poster image on the left is shown under white light. The poster image on the right is shown under view projection illumination. The faces of the cubes are made to appear to have the same color. A sheet of white paper placed over the poster image on the right reveals the actual projected image used to produce this desired view. It was found that viewers quickly lose track of the original appearance of the poster after seeing the animation. While it is difficult to show in a paper, this scene as observed by the camera is quite close to that observed by the human eye. The poster was animated by cycling the colors in the poster and making the colors appear uniform. Movies were also shown over the poster while photometric compensation took place in real time to eliminate all traces of the underlying poster image. 
     As mentioned earlier, the presently preferred construct of transport matrix T depends on the Display Constraint, which stipulates minimal overlap between adjacent light footprints resulting from adjacently lit projector pixels. An example of a situation where the Display Constraint is not upheld is discussed above in reference to  FIG. 12 , where two wine glasses are placed between a projector and a projection surface, or scene. As is further explained above, however, the view projection matrix {hacek over (T)} T  can enforce (i.e. impose) the Display Constraint on an arbitrary scene by selecting only the brightest pixel within each row of T in the creation of an approximation of the inverse of T. In order to test the efficacy of the algorithms of the present invention under various conditions (not just only under the Display Constraint, from which they derive their efficiency and justification) an experiment was conducted to determine how they behave when the Display Constraint is not satisfied. The experiment illustrated in  FIG. 12  was set up with the pair of wine glasses placed between the projector and its display surface. Due to the significant amount of distortions introduced by the glassware, there is much overlap between the sensor light footprints of distinct projector pixels. To this scene was then applied the same acquisition method and same algorithms for enforcing the Display Constraint in the generation of the view projection matrix {hacek over (T)} T . It was found that the computed view projection matrix {hacek over (T)} T  was able to remove most of the unwanted distortion and allow the projector to display a corrected image on the designated display surface, as shown in  FIG. 17 . 
     Stated differently, when the Display Constraint cannot be guaranteed, the method of generating view projection matrix {hacek over (T)} T , which forces a pseudo Display Constraint on a projection environment, can be applied to the generation of the light transport matrix T. Specifically, when generating the simplified light transport matrix T, as described above in reference to  FIGS. 13-15 , one may select the brightest pixel in each row to identify components of a light footprint, whereby light diffusion error and light noise error can be greatly reduced or eliminated. 
     Having defined a method for approximating the inverse of light transport matrix T, it will now be shown how dual photography can be used with an immersive display system to achieve advanced and complex setups. 
     With reference to  FIG. 18 , in a preferred embodiment, a conventional front projector P 1  (similar to real projector  21  of  FIG. 2A ) is used in conjunction with an immersive projector P 2 . As indicated by field-of-view (i.e. FOV)  91 , in the presently preferred embodiment, the portion of a display surface covered by the FOV  91  of front projector P 1  is a subset of the field-of-view of immersive projector P 2 , as indicated by field-of-view lines  93 . Since FOV  91  of front projector P 1  is entirely within the scope of FOV  93  of immersive projector P 2 , it would be desirable to have immersive projector P 2  simulate the projected image produced by front projector P 1 . However, it should be emphasized that in general FOV  91  of front projector P 1  does not necessarily need to overlap any part of FOV  93  of immersive projector P 2 . 
     Although it is not necessary for FOV  93  of immersive projector P 2  to overlap part of FOV  91  of front projector P 1 , it is desirable that two light transport matrices separately associating a camera C to front projector P 1  and to immersive projector P 2  be created. As it would be understood, the two transport matrices may be generated separately since the FOV&#39;s of P 1  and P 2  do not necessarily overlap. 
     However, in the specific example of the presently preferred embodiment, camera C is placed such that the FOV  95  of camera C is a superset of FOV  91  of front projector P 1  and a subset of FOV  93  of immersive projector P 2 . As indicated by FOV lines  95 , the field-of-vision of camera C completely encompasses FOV  91  of front projector P 1 , but is entirely engrossed by FOV  93  of immersive projector P 2 . To simulate a projected image from front projector P 1  using immersive projector P 2 , one first determines a first light transport matrix, T 1 , relating a first projected image p 1  from front projector P 1  to a first captured image c 1  captured by camera C such that c 1 =T 1 p 1 . One then determines a second light transport matrix, T 2 , relating a second projected image P 2  from immersive projector P 2  to a second captured image c 2  captured by camera C such that c 2 =T 2 p 2 . Consequently, one has the following relation
 
c 1 =T 1 p 1  
 
and
 
c 2 =T 2 p 2  
 
In order to simulate projected image p 1  from front projector P 1  using immersive projector P 2 , one needs c 1  (i.e. the captured, projected image from front projector P 1 ) to be the same as c 2  (i.e. the captured, projected image from immersive projector P 2 ), i.e. one needs
 
c 2 =c 1  
 
which lead to the relation:
 
T 2 p 2 =T 1 p 1  
 
solving for p 2  (i.e. the image projected by immersive projector P 2 ), one obtains the following relation:
 
 p   2 =( T   2   −1 )( T   1   p   1 )
 
This means that to create image p 1 , one can project the image directly using front projector P 1 ,or the same effect can be achieved by projecting a transformed image [defined as (T 2   −1 )(T 1 p 1 )] on immersive projector P 2 . Note that the view projection matrices naturally convey the projector-camera correspondence into projector-projector correspondence.
 
     Such a projection is shown in  FIG. 19 , where immersive projector P 2  is used to simulate a front projector, such as projector P 1  of  FIG. 18 . In  FIG. 19 , a virtual projector P 1 ″, as simulated by immersive projector P 2 , is denoted by dotted lines. Therefore, image p 1 , as projected by front projector P 1  of  FIG. 18 , can be recreated by projecting transformed image (T 2   −1 )×(T 1 p 1 ) on immersive projector P 2  of  FIG. 19 . By doing so, viewers  100   a ,  100   b , and  100   c  do not have to concern themselves with occluding any front projector, i.e. P 1  or P 1 ″. Clearly, the image is distortion free and movie playback on the virtual projector can be run in real time. It is to be understood that T 2   −1  can be replaced by an approximation matrix {hacek over (T)}T 2 , as explained above. As stated before, view projection matrix {hacek over (T)} T , which approximates an inverse matrix T −1 , can be freely substituted for T −1  in the following discussions, unless otherwise stated. 
     An example of an image generated using this virtual projector implementation is shown in  FIG. 20 . A front projected image  101  is simulated using a large field-of-view display system. Projector  103 , located along the bottom of  FIG. 20  is part of the large field-of-view display system, and is used to generate image  101  shown in the center of  FIG. 20 . 
       FIGS. 21A to 21C  illustrate the quality of the simulation by showing a real front-projected image ( FIG. 21A ) and a simulated front-projected image ( FIG. 21B ) seamlessly coupled together side-by-side ( FIG. 21C ).  FIG. 21A  shows the right side of a front-projected image projected by a real front projector, such as P 1  of  FIG. 18 .  FIG. 21B  shows the corresponding left side of the front-projected image of  FIG. 21A , but in  FIG. 21B  the left side of the front-projected image is projected by an immersive projector, such as P 2  of  FIG. 19 , to simulate a virtual front projector, such as P 1 ″ of  FIG. 19 . 
       FIG. 21C  illustrates the quality of the simulated, left-side, front-projected image created by the immersive projector. In  FIG. 21C , the right-side, front-projected image of  FIG. 21A  is shown joined to the left-side, front-projected image of  FIG. 21B . This results in a side-by-side seamless registration of the right-side and left-side images created by a real front projector and a simulated, virtual front projector, respectively. 
     Two additional examples showing side-by-side comparisons of real front-projected images created by a real front projector and simulated front-projected images created by an immersive projector are shown in  FIGS. 22A and 22B . In both  FIGS. 22A and 22B , the left half of the shown image is created by an immersive projector to simulate a display from a virtual front projector, and the right side half of the shown image is created by a real front projector. 
     An alternate application of the present technique is better understood with reference to  FIGS. 23A to 23C . In the present example, immersive projector P 2  of  FIG. 23C  is used to create various ambient lighting effects (i.e. virtual environments). If the camera is positioned such that its FOV covers a significant portion of the display room, one can use view projection to create an immersive environment where the walls are lit according to a virtual model. To achieve this, camera C is therefore positioned such that its FOV covers a significant portion of a display room  111 , as shown in  FIG. 23A . In  FIG. 23A , camera C and immersive projector P 2  are positioned such that the FOV of camera C encompasses most, if not all of (and preferably more than) the FOV of immersive projector P 2 . In the present example, P 2  is shown as an immersive projector, but projector P 2  may be any type of a projector, such a front projector. To establish a relationship between camera C and projector P 2 , a light transport matrix T 3  relating camera C to projector P 2  is captured, i.e. determined, using any of the methods described above. Once this is done, an image c 3  as viewed (i.e. captured) by camera C will be related to a projected image p 3 , as projected by projector P 2 , according to the following relationship:
 
c 3 =T 3 p 3  
 
which results in
 
 p   3 =( T   3   −1 )×( c   3 )
 
Consequently, one can build a virtual model of display surfaces of room  111 . This constructed virtual model room (i.e. virtual room)  111 ″, shown in  FIG. 23B , may be a computer simulation, for example. Once virtual room  111 ″ is created, various simulated lighting effects (or projected images or floating images) may be added to virtual room  111 ″. For example,  FIG. 23B  shows virtual room  111 ″ being lit by candle light from a large candle  113 . The computer model further models the position and resolution of camera C (of  FIG. 23A ), shown as dotted box C in  FIG. 23B . The computer model then “captures” (i.e. creates) a synthetic view c 3 ″ of virtual room  111 ″ from the viewpoint camera C to simulate a real image of virtual room  111 ″ as if it had been captured by real camera C of  FIG. 23A . The simulated lighting effects of  FIG. 23B  can then be recreated in real room  111  of  FIG. 23C  using P 2  by projecting transformed image (T 3   −1 )×(c 3 ″).
 
     Another application of the present technique is when it is desired to project an image  117  that is bigger than the walls of a real room  111 . As was discussed above, various techniques may be used to calibrate a real projector to compensate for the angles of the walls and ceiling to the projection wall of real room  111 , but the present embodiment solves this problem using a different approach. In the present example, virtual room  111 ″ of  FIG. 23B  is assigned dimensions similar to real room  111  of  FIGS. 23C and 24 , and image  117  is superimposed in an undistorted fashion onto virtual room  111 ″. An undistorted image  117  may be superimposed onto virtual room  111 ″ (i.e. a computer generated room) using any number of known image processing software. A view c 3 ″ (i.e. a synthetic captured image) of image  117  without distortion on virtual room  111 ″ from the viewpoint of camera C is then created. Immersive projector P 2  is then made to project transformed image (T 3   −1 )×(c 3 ″) to recreate the undistorted oversized image  117  on a wall of real room  111 . 
     An example of this technique is shown in  FIG. 24 , where an image  115 ′ is projected within a room  119  having wall dimensions smaller than the projection dimensions of image  115 ′. The result is an undistorted projection  115 ′ that does not require calibrating the projector (such as projector P 2 , above) to compensate for curvatures (or other irregularities) on the projection surface. 
     As seen, virtual projectors and environments can be combined to create an immersive movie viewer. Since the virtual environment is also an active visual field, one can animate the larger field of view display to create a more engaging experience. 
     The above-described techniques may be applied to the creation of large field-of-view (i.e. large FOV) displays. A large FOV creates a sense of immersion and provides a more engaging experience for a viewer. The present approach describes an immersive projection system with a very large FOV. The system is also designed with a built-in large FOV camera/light sensor that is able to capture light from the areas covered by projection&#39;s FOV. The sensor allows the system to adapt the projected light to optimize image quality and to more generally allow the system to interact with its environment. Although the present system is primarily motivated by the desire to display surround video content, it is important to note that this projection system can also be used to view conventional video content. 
     With reference to  FIG. 25 , an exemplary projection system in accord with the present invention in its minimal form consists of the following components: a projector  121 ; a camera  123 , which can be a digital still camera or a digital video camera; curved mirror  125 , which can be spherical or otherwise; and mounting mechanisms for the above components. Light from projector  121  is reflected off curved mirror  125  before reaching a display surface  127 , which can be any surface, including building walls, floors, ceilings, and dedicated projection screens. Display surface  127  can also be arbitrarily shaped. Reflecting the projected light off the curved mirror enlarges the projector FOV. Light rays from the environment, which may or may not have originated from the projector, also reflect off the curved mirror  125  before reaching the camera. This similarly enlarges the camera FOV. 
       FIG. 26  shows an implementation based on the design of  FIG. 25 . All elements in  FIG. 26  similar to those of  FIG. 25  have similar reference characters and are described above. The present construction highlights one of the key applications of smart projector-camera systems, which is to build immersive multi-wall virtual environments. Thus, the present example uses a simple panoramic projection setup consisting of a conventional front projector  121 , a high-resolution digital still camera  123 , and a hemispherical curved mirror  125 . In the present setup, curved mirror  125  (which may be termed a ‘dome projector’) is a low-cost hemispherical plastic security mirror dome of the type used community convenience stores. This type of mirror dome costs at least three orders of magnitude less than a professionally designed and fabricated optical-grade mirror. Furthermore, the mounting mechanism was also made from inexpensive parts available from typical hardware and building material stores. As such, there is virtually no guarantee of conformance to elegant mathematical models. In experimenting with this construct, it was further found that mirror dome  125  is quite far from a true hemispherical surface (or any simple parametric form, for that matter). 
       FIG. 27  is an alternate view of the setup of  FIG. 26 , and shows the view of mirror  125  as seen (very roughly) from the viewpoint of camera  123 . As can be seen in the reflection of mirror  125  in  FIG. 27 , camera  123  is able to “see” the floor, at least three vertical walls, and the ceiling by means of the reflection in mirror  125 . 
     In  FIG. 28A , a room under ambient lighting with the present projection system installed.  FIG. 29A , an uncalibrated dome projector displays an image of a checkerboard in the room of  FIG. 28A . Clearly there is a significant amount of nonlinear geometric distortion in the displayed image.  FIG. 29B  shows the same setup, but with geometric compensation using the view projection matrix. As the image was shot from the location of the reference camera, straight lines in the view remain straight across multiple walls. 
     The view projection matrix {hacek over (T)} T  also compensates photometrically for the color and intensity variations as well as non-uniformity in the display surface albedo/reflectance properties.  FIG. 30A  shows an uncalibrated dome projector displaying a uniform intensity image. As can been seen, the resulting image is significantly darker towards the top left and right corners of the front wall. In  FIG. 30B , the same uniform intensity image is projected in a calibrated dome projector, which is shown to produce a more uniform intensity. 
     Having a compact representation for all things necessary for view projection makes it easy to analyze display systems that could be overwhelmingly difficult to calibrate otherwise. It also makes it practical to build these systems and have them precisely calibrated up to the limits of the display and imaging hardware. In this section we illustrate the usefulness of T −1  with a few examples using the dome projector, described above, in combination with a front projector. Even using a consumer-level, commodity, panoramic projector like the one used in the present example, however, it was found that the view projection matrix still enabled one to put together compelling, immersive display setups. 
     It is further to be understood that the present dome projector setup of  FIGS. 25-30  can be used in place of immersive projector P 2  of  FIGS. 18 and 19  to achieve simulation of a front projector, as described above, or in place of immersive projector P 2  in  FIGS. 23A-23C  to create virtual environments, as described above. 
     An example of immersive projection lighting created using the present projector-camera system is shown in  FIG. 28B . As can be seen, the present projection-camera system is able to project images onto the two walls as well as the ceiling. An example of how this effect can be achieved using the view projection matrix, {hacek over (T)} T , is illustrated in  FIGS. 31-33 . 
     The view projection matrix {hacek over (T)} T  is first generated using any of the methods described above. As is explained above, when a projection surface consists primarily of a flat surface (or conjoined flat surfaces), forty-nine (49) or fewer, reference points may be generated using seven vertical light lines and seven intersecting horizontal light lines to approximate a full view projection matrix, {hacek over (T)} T , and still achieve a good level of projector to camera calibration. In this case, the missing matrix entry values may be extrapolated from the 49 reference points since the projection surface is assumed to be flat. 
     However, since in the examples of  FIGS. 25-30 , the projection surface consists of a curved mirror, it is preferred that a full view projection matrix {hacek over (T)} T  at the resolution of the projector be generated. Since in the present example projector  121  has a resolution of p×q projector pixels, calibration between projector  121  and camera  123  should be achieved by generating p×q light transport reference points. 
     With reference to  FIG. 31 , projector  121  of  FIG. 26  individually projects a series of q vertical lines VL_ 1  to VL_q onto the projection surface (i.e. mirror  125  of  FIG. 26  in the present case), which are individually, and automatically, captured by camera  123  of  FIG. 26 . In  FIG. 32 , projector  121  then individually projects p horizontal lines HL_ 1  to HL_p that are in turn individually, and automatically, captured by camera  123 . As is explained above, the captured vertical and horizontal lines are each individually combined to identify their uniquely coincident reference point (i.e. light footprint). This process is continued until all unique intersecting points are identified (shown as white circles in  FIG. 33 ), and their light transport information extracted. It is to be understood that although the vertical and horizontal lines emitted from projector  121  are perfectly vertical and horizontal, the resultant projected lines on dome mirror  125  will follow the curvature of dome mirror  125 . 
     As is explained above, it is preferred that the Display Constraint be enforced in the construction of view projection matrix {hacek over (T)} T . Thus, if one has a desired image, C, and wants to determine how to transform a projection image P in order to display desired image C undistorted onto room  111  via mirror  125 , one needs to project a distorted version of image C defined as P={hacek over (T)} T ×C. 
     This is illustrated in  FIG. 34 , where Desired Image C is written as a vector  200  consisting of m×n image pixel entry values, C 1  to C m×n . Vector  200  is multiplied with the created view projection matrix {hacek over (T)} T , which consists of (p×q) rows and (m×n) columns to produce a transformed image P, written as vector  201  and consisting of (p×q) image pixel entry values to be respectively applied their corresponding one of the (p×q) projector pixels of projector  121 . The resultant transformed image P is shown to consist of p rows and q columns. 
     With reference to  FIG. 35 , transformed image P from  FIG. 34  is sent to projector  121  as Projector LCD Image P, and is projected onto mirror  125 . The resultant image  203  on room  111  is an undistorted representation of Desired Image C of  FIG. 34 . 
     It is to be noted that in the present case, camera  123  and projector  121  were not calibrated prior to creating transformed image P. Rather, the distortion of transformed image P inherently compensates for issues related to a lack of calibration between camera  123  and projector  121  due to it having been constructed using view projection matrix {hacek over (T)} T , which includes calibration compensating information for the camera-proj ector pair. 
     As it would be understood, if the desired image were a video image, then the view projection matrix {hacek over (T)} T  would be applied to the video image. That is, since a video image is comprised of a plurality of still images arranged in sequence, one would apply the view projection matrix {hacek over (T)} T  transformation to each of the sequenced still images to produce a transformed video projection. 
     It should further be emphasized that the FOV of projector  121  and the FOV of camera  123  are in general different, and may or may not overlap. When there is a significant overlap in the two FOV&#39;s, images captured by camera  123  can be used as feedback for improving the quality of a projected image from projector  121  in a manner similar to those described above. For example, feedback from camera  123  to projector  121  can be used to compensate for variations in the shape and reflectance properties of a display surface (as seen by camera  123 ) so that a projected image appears as though it were projected on a flat white surface. 
     The FOV of camera  123  may also include areas not covered by the FOV of projector  121 . For example, while the FOV of projector  121  covers the front and side walls of test room  127  shown in  FIGS. 28A and 28B , the camera may capture areas outside the projector&#39;s FOV, possibly including areas where viewers are located. This allows the system to adapt and interact with viewers by detecting and tracking either the viewers or the viewers&#39; pointing devices. It may be possible for camera  123  to track small lights mounted, for example, on remote controls and facilitate user interaction. 
     With reference to  FIG. 36 , an alternate configuration based on the construct of  FIG. 25  but geared toward ceiling-mounted operation is shown. All elements similar to those of  FIG. 25  have similar reference characters and are described above. 
     In  FIGS. 37 and 38 , two additional alternate configurations are shown. All elements in  FIGS. 37 and 38  similar to those of  FIG. 25  have similar reference characters and are described above. 
     In  FIG. 37 , a planar mirror  141  is used to fold the optical path so that projector  121  and camera  123  can be placed under the curved mirror  125 , thereby achieving a smaller footprint.  FIG. 38  shows a booth design for enclosing projector  121 , camera  123 , curved mirror  125 , and flat mirror  141  within a booth  143  for display booth operation. Using this construct, one can simultaneously produce two projection images; a first front (or rear) projection image on a first Display Surface A and a second rear projection image on a second Display Surface B. 
     In several of the designs described above, the projector and cameras do not have common optical centers. However, it is possible to design projector-camera pairs with collocated optical centers. While a system with collocated optical centers allows the projector and the camera to have identical field-of-visions, a system with non-collocated optical centers has the potential to allow 3D reconstruction of its environment. 
     Up to this point, the provided examples have consisted of one camera and one projector, but as suggested above, multiple camera and projector combinations may be used. This poses the problem of how to seamlessly integrate, or combine, two or more projected images from two or more projectors that have different field-of-visions, or combine multiple projectors to create a large field of vision display. Therefore, before expounding on some of the visual effects possible by using multiple projectors, it may be beneficial to first discuss how multiple projectors may be seamlessly used together. That is, the efficacy of the present invention as applied to a single projector, single camera system, can be expanded to systems having multiple projectors and one (or more) camera(s). 
     The development of a non-parametric method for calibrating projector-camera systems and for solving the above described “view projection problem” is discussed above. That is, the view projection problem of defining how a projector should illuminate a scene (i.e. defining what a projector should project onto the scene) so that a camera sees a desired view is discussed above. In summary, the above-described development first provided multiple constructs for a light transport matrix T that relates an image p projected from a projector to an image c captured by a camera by the relationship c=Tp. Then, an interim working matrix, {hacek over (T)}, was defined as being populated by the following relationship:
 
 {hacek over (T)}r=Tr /(∥ Tr ∥) 2   , r= 1, 2, 3 , . . . , pq   (1)
 
where {hacek over (T)}r is the r th  column of {hacek over (T)} and pq is the number of columns in T. It was then shown that under the Display Constraint, one can define the view projection matrix {hacek over (T)} T  as:
 
 {hacek over (T)}   T   =T   −1  
 
which leads to the following relation:
 
p={hacek over (T)} T c
 
     As is explained above, the Display Constraint comes from the observation that in a typical projector camera setup for information display purposes, any two distinct light rays emitting from distinct projector pixels will typically hit a camera sensor pixel array at distinct parts, i.e., there is usually little overlap in the camera pixels i hit by light from each of the distinct light rays. It implies that the columns of T are orthogonal to each other, which enables the normalization process in equation (1) to lead to the inverse of T. As is further explained above, in situations where the Display Constraint is not observed naturally, one can modify the construct of T to artificially impose the Display Constraint by declaring the brightest pixel in each row of T as being part of a light footprint resulting from a distinct light ray, and declaring all other pixels in the same row of T to be zero-valued entries. This operation forces T to become orthogonal, and thereby permits the application of equation (1). 
     The view projection matrix {hacek over (T)} T  thus solves the view projection problem, defined as: given a desired view c; find an image p={hacek over (T)} T c such that projecting p produces a scene which, when viewed from the location of the reference camera, has the same appearance as c. 
     In practice, large field-of-view displays, virtual living stations with surrounding wall screens, and many other applications require the use of more than one projector due to the limited field-of-view of each individual projector. In general when using more than one projector, the field-of-views of these projectors partially overlap with each other. To achieve view projection (or indeed to project a continuous image across the different field-of-views) using display systems having two or more projectors, one needs to mosaic the individual projectors to generate the desired camera view. 
     An example of mosaicing using the present invention is presented using a two-projector display system. It is to be understood that the present approach to constructing a mosaic display may be extended to display systems having three or more projectors, since extension to systems consisting of more projectors is straightforward. That is, the process described below for combining first projection image from a first projector with second projection image from a second projector can be applied to combining the second projection image of the second projector with a third image of a third projector to create a mosaic image combining the first, second, and third projection images. Similarly, the same process can be applied to combine the third projection image with a fourth projection image from a fourth projector to create a mosaic image that combines the first, second, third, and fourth projection images. 
     Consider a multi-projector display system consisting of two projectors with an overlapping field-of-view (FOV), and a camera with a FOV that is a superset of both projectors. The light transport matrixes T 1  and T 2  (separately determined) respectively relating each projector image p 1  and p 2  to corresponding camera captured image c 1  and c 2 , give the following equations:
 
c 1 =T 1 p 1  
 
and
 
c 2 =T 2 p 2  
 
     To display an image that spans the field-of-vision, FOV, of both projectors, one needs to find c 1  and c 2  such that a composite image, c, combines c 1  and c 2  seamlessly. That is, composite image c, which is defined as c=c 1 +c 2 , is the desired image as observed by the camera. More specifically, one needs to compute the appropriate projector images p 1  and p 2  to display the desired composite image c, by solving the following linear equation,
 
 c=c   1   +c   2  
 
or
 
 c =( T   1   p   1 )+( T   2   p   2 )=[ T   1   T   2   ][p   1   p   2 ] T   (2)
 
     In such a setting, a camera pixel is lit by either one projector pixel from one of the projectors or two projector pixels simultaneously from respective projectors. In the former case, the camera pixel gives a linear equation on the corresponding projector pixel in p 1  or p 2 . In the latter case where the camera pixel falls in an overlapping part of the FOVs of the two projectors, it gives a linear equation on the two corresponding projector pixels in p 1  or p 2 , respectively. Since each projector pixel covers a number of camera pixels, it is constrained by a number of linear equations. Thus such equations from all the camera pixels form an overconstrained linear system on the projector images p 1  or p 2 . 
     However, one can no longer directly compute the view projection matrix for the two-projector system (i.e. one cannot compute the inverse of [T 1 T 2 ] directly), as could be done in the single-projector cases, describe above. This is because the projection of pixels from different projectors can overlap with each other, and therefore the Display Constraint no longer holds between the multiple projectors, i.e. columns in T 1  are not necessarily orthogonal to columns in T 2 , and consequently Eq. (1), above, can not be used to compute the view projection matrix for [T 1 T 2 ]. Therefore, instead of directly solving the linear system in Eq. (2), an alternating linear solution has been developed by iterating two steps until convergence, as follows: 
     1. Given the current estimate of p 2 , compute p 1 =({hacek over (T)} 1 ) T (c−T 2 p 2 ) 
     2. Given the current estimate of p 1 , compute p 2 =({hacek over (T)} 2 ) T (c−T 1 p 1 ) 
     In the two formulas immediately above, ({hacek over (T)} 1 ) T  and ({hacek over (T)} 2 ) T  are the respective view projection matrices for the two projectors, and P 2  is set equal to zero at the initial iteration step. Since the view projection matrices naturally convey the correspondence and mapping between the images of the projectors and the camera, it does not take many iterations for p 1  and p 2  in formulas 1 and 2 (immediately above) to converge to their respective complementary image. In practice, the iteration process takes only a few iterations cycles (typically 5 or less) for p 1  and p 2  to converge to respective complementing images that, when combined form a mosaic image. That is, p 1  will converge to a first image and p 2  will converge to a second image, and when images of p 1  and p 2  are projected and superimposed, they will form a combined mosaic image. 
     With reference to  FIG. 39 , an example of a system that implements this process includes a first projector-camera system pair  221  and a second projector-camera system pair  223 . First projection-camera system pair  221  includes a first projector  21   a and a first camera  25   a  related by a first view projection matrix {hacek over (T)} 1  constructed using any method described above. Similarly, second projection-camera system pair  223  includes a second projector  21   b  and a second camera  25   b  related by a second view projection matrix {hacek over (T)} 2  also constructed using any of the above-described methods. It is further to be understood that {hacek over (T)} 1  and {hacek over (T)} 2  are each generated independently such that second projector-camera system pair  223  is off while {hacek over (T)} 1  is generated and first projector-camera system pair  221  is off while {hacek over (T)} 2  is generated. 
     First projector-camera system pair  221  has a first field-of-vision FOV_ 1  defining a first projection region Reg_ 1 , and second projector-camera system pair  223  has a second field-of-vision FOV_ 2  defining a second projection region Reg_ 2 . As shown, Reg_ 1  and Reg_ 2  overlap each other within an area identified by crosshatch marks. This overlap region is further labeled Reg_ 1 +2. It is to be understood that the size of overlap region Reg_ 1 +2 is made large for purposes of explanation and that a small overlap region is more typical, although the amount of overlap is not critical to the present application. 
     It is to be noted that formulas p 1 =({hacek over (T)} 1 ) T (c−T 2 p 2 ) and p 2 =({hacek over (T)} 2 ) T (c−T 1 p 1 ) account for geometric (i.e. spatial) and photometric (i.e. lighting) compensation, and thus take care of any blending needs between adjoining displayed images. This is because light transport matrix T incorporates geometric information, and consequently so does view projection matrix {hacek over (T)}. Furthermore when identifying the light footprint information for a typical light transport matrix T, full photometric information is also obtained due to each captured pixel in a light footprint having a value-entry for any light intensity variation of the three color (RGB) sub-components of each camera pixel (see for example, the white and shades of gray blocks that make up light footprints Ft 1  in  FIG. 3   a  or Ft 2  in  FIG. 4A ). 
     However, if matrix T were to be constructed in a binary ON/OFF manner (that is, pixels within a light footprint are classified as being fully ON (i.e. classified as having a light intensity value of 255 in a typical luminosity scale of 0 to 255) and pixels outside the light footprint are classified as being fully OFF (i.e. classified as having a light intensity value of 0), then this binary ON/OFF manner of constructing matrix T would not have much photometric information (since it would in effect be a black-and-white image). However, this binary ON/OFF manner of constructing matrix T would still have full geometric information so that the above two formulas (1) for p 1  and (2) for P 2  would still be able to determine the geometric mosaicking of multiple projection images. In such a case, however, an additional light blending step (described below) would be helpful to blend light intensities of multiple projection images when mosaicking the multiple projection images. 
     Another situation where such an additional step for blending the light intensity of multiple projected images may be useful is in situations where light transport matrix T is created from a limited number of light footprint information. As is explained above, this would apply to situations where a projection surface (or scene) is flat, and the light transport matrix T is estimated using a limited number of intersecting patterns to generate a limited number of identified light footprints. Since the projection surface is flat, the missing geometric information within light transport matrix T can be inferred using homography techniques. The estimated light transport matrix T thus provides full geometric information, but it does not generally provide photometric information for locations between the limited number of identified light footprints. 
     If the projection surface is assumed to be uniformly white in color (such as a white projection canvas or a white wall), then a light blending step may be skipped by simply insert white-color information as photometric information for all identified and inferred light footprint information. Alternatively, if one assumes that the projection surface is of a uniform color, but not necessarily white, then one can define an estimated photometric reading by assigning it the photometric information value of one (or of an average of two or more) identified light footprint(s). One can then populate the photometric information of all inferred light footprints with the thus defined estimated photometric reading. 
     Returning now to the topic of the additional light blending step, it has been found that even if one has a light footprint matrix T generated by using all available pixel information (i.e. a full T, not an estimated T), a projected image can still benefit from an additional light blending step when mosaicking multiple projected images. In this case, the information obtained from the additional light blending step may constitute an additional parameter to the blending results obtained from the formulas p 1 =({hacek over (T)} 1 ) T (c−T 2 p 2 ) and p 2 =({hacek over (T)} 2 ) T ( c −T 1 p 1 ). 
     A simplified view of multiple projection regions demonstrating this additional light blending step is shown in  FIG. 40 . As before, Reg_ 1  is the projection region provided by projector-camera system pair  221 , and Reg_ 2  is the projection region provided by projector-camera system pair  223 . For ease of explanation, Reg_ 1  is denoted by vertical hatch lines and Reg_ 2  is denoted by horizontal hatch lines. Overlap region Reg_ 1 +2 is therefore denoted by the intersection of vertical and horizontal hatch lines. Of particular interest is the definition of a desired projection region  225  denoted by a darkened outline spanning across parts of regions Reg_ 1 , Reg_ 2 , and Reg_ 1 +2. Desired projection region  225  defines the region upon which a desired combined (i.e. mosaic) image is to be displayed. Desired projection region  225  has image contributions from both projector-camera systems  221  and  223 . Within desired projection region  225 , Reg_A identifies that part of image region  225  provided solely by projector-camera system  221 , Reg_C identifies that part of image region  225  provided solely by projector-camera system  223 , and Reg_B identifies that part of image region  225  provided by a combination of projector-camera systems  221  and  223 . It is to be understood that Reg_B might also be provided solely by either one of projector-camera systems  221  or  223 , but it has been found that visual artifacts at an image border (where an image provided by one projector-camera system ends and a second image projected by a second projector-camera system begins) can be mitigated, or eliminated, by blending the transition between projector-camera systems. Therefore, in the presently preferred embodiment, both projector-camera systems  221  and  223  contribute to the image created within Reg_B. The question at hand is, how much (i.e. what parts) of an image within Reg_B each of projector-camera systems  221  and  223  provides. 
     For ease of explanation,  FIG. 41  shows Reg_ 1  and that part of desired image  225  within the FOV of projector-camera system  221 , i.e. Reg_B. The vertical hatch lines indicate that part of Reg_ 1  that is made dark due to it not contributing to desired image  225 . Arrows  1 A,  1 B,  1 C, and  1 D indicate how normalized light intensity is varied as one moves away from a border of region Reg_ 1  toward Reg_B, and approaches a border of desired image  225  that is provided by projector-camera system  223 . Area Reg_A is outside the combined section Reg_B, and is provided solely by projector-camera system  221 . As one traverses arrow  1 A from the left border of Reg_ 1  and approaches Reg_B, arrow  1 A is shown dark to indicate that all image components are provided by projector-camera system  221 . Following arrow  1 A, as one enters Reg_B, arrow  1 A is initially dark and is then lightened (i.e. shown as thinning stripes) to indicate that the image intensity is initially strongly provided by projector-camera system  221 , but the intensity drops off as one traverses from left to right along Array  1 A toward the right border of Reg_B and Reg_ 1 . Arrow  1 D indicates that initially at the right end of Reg_B, no intensity is provided by projector-camera system  221 , but the light intensity from projector-camera system  221  is increased as one traverses from right to left within Reg_B toward the end-point of arrow  1 A. Similarly, arrow  1 B indicates that the light intensity falls as one traverses down arrow  1 B away from a border of region  225 . Likewise, arrow  1 C indicates that light intensity falls as one traverses up arrow  1 C away from a border of region  225 . 
     In other words, light intensity variations in an image can be expressed as a factor of a defined maximum light intensity value, i.e. the normalized value. This normalized valued multiplied by a factor of 1, would provide the defined maximum light intensity value, and the same normalized value multiplied by a factor of 0.5 would provide half the defined maximum light intensity value. By altering the normalized value within a given region, one can alter the brightness of the image within the given region. In the present case, undesirable light artifacts are avoided at the borders by providing a gradual blending of image borders, rather than abruptly changing light intensity of pixels at the borders. 
     A similar construct is shown in  FIG. 42  from the point of view of projection-camera system  223 . Again, the horizontal hatch lines indicate that part of Reg_ 2  that is made dark due to it not contributing to desired image  225 . Also like before, Reg_B indicates the blending area where an image is provided by a combination of both projector-camera systems  221  and  223 , and region Reg_C indicates that part of desired image region  225  provided solely by projector-camera system  223 . Arrows  2 A,  2 B,  2 C, and  2 D are indicative of how the normalized light intensity is varied as moves from Reg_ 2  toward the borders of desired region  225 , and within Reg_B toward that part of desired image  225  provided by projection-camera system  221 . Area Reg_C defines that part of desired image  225  that is outside combined region Reg_B, and is provided solely by projector-camera system  223 . As one enters region Reg_B, arrow  2 A indicates that the image intensity is initially strongly provided by projector-camera system  223 , but the intensity drops off as one traverses from right to left along Array  2 A toward the left border of region Reg_B. Arrow  2 D indicates that initially at the left end of region Reg_B, no intensity is provided by projector-camera system  223 , but the light intensity from projector-camera system  223  is increased as one traverses from left to right within Reg_B. Similarly, arrow  2 B indicates that the light intensity falls as one traverses down arrow  2 B away from a border of desired region  225  within Reg_B. Likewise, arrow  2 C indicates that light intensity falls as one traverses up arrow  2 C away from a border of region  225 . 
     For each pixel projected by projector-camera system  221 , a determination is made of the projected pixel&#39;s proximity to the left, right, upper, and lower border of the Reg_ 1  and Reg_B. These parameters affect the normalized light intensity of the projected pixel. The closer a projected pixel from projector-camera system  221  is to any one border of Reg_ 1 , the higher parameter contribution for that border, which makes for a brighter normalized intensity. The same is true for a projected pixel from projector-camera system  223  with determination of the projected pixel&#39;s proximity to the four borders of Reg_ 2  and Reg_B. Additionally, the light intensity of projected pixels close to border is adjusted as one approaches the border so as to avoid abrupt light intensity changes at the borders. 
     For example in  FIG. 40 , as one travels along arrow A 1  in region Reg_A from left to right one will be moving from the left border of desired image region  225 , which is provided wholly by projector-camera system  221 , to the left border of region Reg_ 2 , and region Reg_B which is provided by both projector-camera systems  221  and  223 . As one moves along arrow A 1  from left to right, one is moving further away from the left border of Reg_ 1 , but the image is provided solely by projector-camera system  221 , and so the normalized light intensity of pixels produced by projector-camera system  221  is at its normal, highest value. When one reaches the end of arrow A 1  and reaches the beginning of arrow A 2 , projector-camera system  221  is still at its highest normalized value since light blending is just about to begin. At the start of arrow A 2 , projector-camera system  221  has its highest normalized light intensity and projector camera system  223  has its lowest normalized light intensity. As one moves along arrow A 2  from left to right (i.e. from Reg_A provided exclusively by projector-camera system  221  toward Reg_C provide exclusively by projector-camera system  223 ), the normalized light intensity of projector-camera system  221  is lowered from its highest to it lowest normalized light intensity value. Conversely, as one moves along arrow A 2  from left to right, the normalized light intensity of projector-camera system  223  is raised from its lowest normalized light intensity value to its highest. It is to be understood that this light transition is not necessarily linear. For example, the greater changes in normalized light intensity of projector camera system  221  preferred occur as one approaches the right border or Reg_B along arrow A 2 . 
     Therefore, at the end of arrow A 2 , projector-camera system  223  is at its highest normalized light intensity, and projector-camera system  221  is at its lowest normalized light intensity. As one moves from left to right along arrow A 3 , projector-camera system  223  provides all the projected light and no light contribution is provided by projector-camera system  221 . Thus, within region Rev_C, projector-camera system  223  may be maintained at its highest normalized light intensity. 
     The above formulas p 1 =({hacek over (T)} 1 ) T (c−T 2 p 2 ) and p 2 =({hacek over (T)} 2 ) T (c−T 1 p i ), using both geometric and photometric information, was tested by mosaicing two projectors on a statue, as shown in  FIG. 43 . This is not a contrived example, as the two projectors are required to cover all surfaces visible from the camera. Using this setup, a seamless, distortion-free (from the point of view of the camera) image was successfully projected on the statue. One can also easily perform relighting by view projecting images of the statue under varying illumination conditions. 
     Construction of large FOV projector systems using the above method of combining multiple projectors is shown in  FIGS. 44A and 44B . In  FIG. 44A , a single curved mirror  125  is used in combination with multiple projector-camera pairs  145 . In  FIG. 44B , a single mirror pyramid  151  is used with multiple projector-camera pairs  145  to achieve a large FOV. With the construct of  FIG. 44B , the optical centers of all projectors can be collocated within the mirror pyramid, creating a single virtual large FOV projector. Similarly, the camera optical centers can also be collocated to create a single virtual large FOV camera. 
       FIG. 45  shows that multiple large FOV projectors  153   a  and  153   b  (such as those shown in  FIGS. 44A ,  44 B, or other large FOV projector systems) can be used to achieve an even larger overall projection FOV. One or more conventional projectors  155  can also be used in combination. As seen, the FOV of projector  153   a , as indicated by dotted line  157  overlaps the FOV of projector  154   b , as indicated by dotted line  159 , by an overlap amount  161 . The images from projectors  153   a  and  153   b  may be combined using the method described above for combining multiple projectors. 
     Although the above discussed methods of generating and using the light transport matrix T are more efficient than previous approaches, the above discussed methods still requires substantial memory resources during generation of the light transport matrix T. Once generated, however, the memory and computational resource requirements for using the light transport matrix T can be efficiently reduced, as is described above. It is therefore desirable to now describe a method of reducing the memory requirements for generating light transport matrix T. 
     Above are described methods for calibrating multiple projectors so that they can be automatically aligned. Also described are methods for efficiently storing the light transport matrix T, as well as methods for reducing the number of capture image needed from p×q down to p+q. Further described is how to synthesize each column of light transport matrix T by
 
Tj≈MIN(C S1 , C S2 )
 
where Tj is the j th  column of T. S 1  and S 2  are sets of pixels where S 1  ∩ S 2 ={j}, and C S1 , C S2  are the sum of the respective columns of T, i.e. C S =Σ j∈S  Tj.
 
     Recall that the display constraint dictates that the non-zero entries in the distinct columns of T do not line up on the same row. This means that on each row of T, there will only be one non-zero entry and one just needs to store that entry along with its column index. This means one can reduce the space needed to store T down to that for a single column plus a column of index values. In a real world setting, due to sensor noise many of the unlit entries of T will not be exactly zero. One can simply identify the maximum value in each row and designate that entry as the one non-zero entry for the row. 
     Note that even though the final representation of the light transport matrix needs only a single column of light values and a single column of index values, during the capture process one is still storing p+q images. When the capture device has mass storage capability, such as a built-in hard disk drive, this may not be an issue. However, if the capture device needs to work with a small memory footprint, this may not be acceptable. Additionally, computing the columns of T from the p+q images also takes a significant amount of computational resources. Fortunately, as is shown below, it is possible to capture the light transport matrix using just enough memory to hold two copies of the matrix (in its efficient format), and a single capture image. 
     A key aspect of the present embodiment for capturing light transport matrix T is that one captures matrix T directly into its efficient modified format, instead of first storing the p+q images, then generating the full light transport matrix T from the p+q images, and then compressing the resultant full matrix T into its efficient modified format. In order to do this, one needs a way to compute the column of light values and the column of index values directly from the set of p+q images as each image is captured. 
     In the previous approach described above, the sparse matrix representation is computed by finding the maximum value in each row. The column index of that maximum value entry gives one the index value, and the maximum value itself gives one the light transport value entry for that row. This means that in the end, only the maximum light value is necessary, and so the present embodiment finds the maximum values during image capture. 
     The above reasoning assumes that one is finding the maximum values across the columns of light transport matrix T. However, one is not capturing the columns of matrix T directly, and instead one is approximating each of the p×q values from the p+q images. A pseudo code (or procedural steps) describing the present embodiment is shown in  FIG. 46 . That is,  FIG. 46  shows a Small Memory Footprint Algorithm for capturing light transport matrix T directly into its efficient, sparse representation T=(Tval, Tid). 
     Recall that the present approach requires the construction of sets
 
 Y={Y 1, . . . ,  Yp } and  X={X 1, . . .  Xq} 
 
such that
 
∀ i ∈{ 1, . . . , ( p×q )}, ∃ X   j   , Y   k   |X   j    ∩Y   k   ={i},  
 
     In  FIG. 46 , each P X (P Y ) is a projector image formed by turning the set of pixels in a particular Xi(Yj) ON, and each C X (C Y ) is the corresponding captured image when P X (P Y ) is projected. The capture phase captures C Y  where Y=1 . . . p and C X  where X=1 . . . q. Note that the algorithm does not need to store all the C X ,C Y  images. Instead, a pair MaxX, MaxY images and IdX, IdY index images are computed and stored. The sparse light transport matrix T=(Tval, Tid) is then computed from these four images. The pseudo code function IntersectionIndex maps a 2D (2-dimensional) index (j, k) back to a 1D (1-dimensional) index i (the relationship between i, j, k is defined immediately above). 
     The basic insight that allows this efficient procedure to work is the following: if a captured camera pixel&#39;s value is at its maximum when projector pixel i is turned ON, then it should also be at its maximum when projector images Pj, Pk are projected, because the projector pixel i should be turned ON in both projector images Pj, Pk. Thus finding the maximum across the set of images Px where X=1 . . . q and across P Y  where Y=1 . . . p will identify the Pj, Pk images whose camera pixels were at their maximum values when projector pixel i was turned ON. 
     Without this algorithm, it was previously necessary to store all the p+q images. Using this algorithm, one only needs to store the MaxX, MaxY and IdX, IdY images, and the Tval, Tid images. Similarly without this algorithm, it was previously necessary to synthesize the p×q columns of light transport matrix T before it could be compressed into efficient Tval, Tid format. This requires O(p×q×m×n) time. The new algorithm runs in O((p+q)×m×n) time. 
     It is to be understood that the present embodiment for reducing memory and computational resources in the generation of light transport matrix T may be substituted for any of the above-described applications of light transport matrix T. 
       FIGS. 47 to 50  provide an alternate, pictorial rendition of the present embodiment of  FIG. 46 . In  FIG. 47 , the first part of the pseudo code of  FIG. 46  creates three matrices (or arrays) MaxX[i], MaxY[i], and Tval[i], and populates the three matrices with zero values. Each of the three matrices is of size (m×n), i.e. the size of the camera pixel density. For ease of explanation, each of the three matrices is illustratively shown as a rectangular matrix to resemble the arrangement of a projector pixel array and a camera photo sensor array, as discussed above. It is to be understood, however, that the actual arrangement of the data set is irrelevant, as long each datum&#39;s relationship to the data set, as whole, is known. 
     In  FIG. 48 , the next part of the pseudo code of  FIG. 46  compares each pixel location Cx[i] of each newly captured image Cx of a projected vertical line (created by simultaneously turning ON all the projector pixels within one column of a projector pixel array) to the current contents of MaxX. Whenever a current pixel value within captured image Cx is brighter than the value currently stored in the correlating location within MaxX, the correlating location within matrix MaxX is overwritten with the currently captured image&#39;s brighter pixel value. When comparing the first captured image, C 1 , to MaxX, this process results in MaxX storing a copy of the first captured picture C 1  since matrix MaxX was initially populated with all zero values. When comparing the second and subsequent captured images (C 2  to C q ) to the current values of MaxX, any previously stored bright vertical line is retained, and any new bright vertical line from the currently captured image is written to MaxX. Thus, by the time this process is applied to the last captured image (i.e. C q ) of the last projected vertical line, MaxX has stored all q vertical lines from all q captured images into MaxX. It is noted that a second matrix, or array, IdX maintains a running record correlating each entry within MaxX to its corresponding projector column within the projector pixel array. 
     In  FIG. 49 , the next part of the pseudo code of  FIG. 46  executes a similar process for storing the values of captured bright horizontal lines corresponding to projected horizontal lines. As before, each pixel location CY[i] of each newly captured image C Y  of a projected horizontal line (created by simultaneously turning ON all the projector pixels within one row of a projector pixel array) is compared to the current contents of MaxY. Whenever a current pixel value within captured image C Y  is brighter than the value currently stored in the correlating location within MaxY, the correlating location within matrix MaxY is overwritten with currently captured image&#39;s brighter pixel value. When comparing the first captured image, C 1 , to MaxY, this process results in MaxY storing a copy of the first captured picture C 1  since matrix MaxY was initially populated with all zero values. When comparing the second and subsequent captured images (C 2  to C p ) to the current values of MaxY, any previously stored bright horizontal line is retained, and the new bright horizontal line from the currently captured image is written to MaxY. Thus, by the time the present process is applied to captured image C p  of the last projected horizontal line, MaxY has stored all p horizontal lines from all p captured images into MaxY. It is noted that a second matrix, or array, IdY maintains a running record correlating each entry within MaxY to its corresponding projector row within the projector pixel array. 
     In  FIG. 50 , the last part of the pseudo code of  FIG. 46  compares each stored value of MaxX with its corresponding value stored in MaxY, and writes the smaller of the two (i.e. the dimmer value) to matrix Tval. This results in the storing of light footprint information in Tval. Again, matrix Tid maintains index information for Tval correlating each value within Tval to its corresponding projector pixel. Tid obtains its index information by combining the information of IdX and IdY. 
     In an effort to make the present techniques applicable to as many situations and equipment arrangements, as possible, we now turn our attention to a projector-camera system where the camera is of equal or smaller resolution as the projector. Before addressing some of the issues of such a set-up, however, it may be advantageous to provide a brief summary of some of the techniques discussed thus far. 
     Above, is discussed a non-parametric method for calibrating projector-camera systems and solving the so-called view projection problem: what image should a projector illuminate onto a scene so that a camera sees a desired view? Specifically, one first captures the light transport matrix T between an image p projected from a projector and an image c captured by a camera, such that c=T p. One then constructs a matrix T such that
 
 {hacek over (T)}r=Tr /(∥ Tr ∥) 2   , r= 1, 2, 3 , . . . , pq  
 
where {hacek over (T)}r is the r th  column of {hacek over (T)} and pq is the number of columns in T, which is equal to the number of projector image pixels in the projector. Each column of T refers to a vectorized camera image captured when the scene is lit by its corresponding projector pixel and all other projector pixels are turned OFF. Under the display constraint,
 
 {hacek over (T)}   T   =T   −1  
 
     and therefore,
 
p={hacek over (T)} T c
 
     The display constraint comes from the observation that in a typical projector camera setup for information display purposes, any two distinct light rays emitting from the projector will typically hit the camera sensor at distinct parts, i.e., there is usually little overlap in the camera sensor pixels hit by light from each of the rays. The display constraint thus implies that the columns of T are orthogonal to each other, which enables the normalization process described above in the derivation of {hacek over (T)} to lead to the inverse of T. 
     View Projection Matrix, {hacek over (T)} T , (i.e. the transpose of matrix {hacek over (T)}) solves the view projection problem such that given a desired view c, one can find an image p={hacek over (T)} T  c such that projecting p produces a scene which, when viewed from the reference camera location, has the same appearance as desired view c. 
     Also described above, are methods for efficiently capturing the light transport matrix T. Specifically, one may capture m+n stripe images by projecting a white stripe (or line) respectively using the m rows and n columns of the projector image plane, each in turn. Each column of T may then be synthesized by:
 
Tj≈MIN(C S1 , C S2 )
 
where Tj is the jth column of T. S 1  and S 2  are two sets of pixels respectively from a row and a column in the projector image plane that intersect at pixel j, i.e. S 1  ∩ S 2 ={j}. C S1  and C S2  are the sum of the columns of T corresponding to pixels in the row and the column, i.e. C S =Σ j∈S  Tj.
 
     Recalling that the display constraint dictates that the non-zero entries in the distinct columns of T do not line up on the same row. This means that on each row of T, there will only be one non-zero entry. In a real world setting, due to sensor noise many of the unlit entries of T will not be exactly zero. In this case, one can impose the display constraint on a system by identifying the maximum value in each row and designating that entry as the one non-zero entry for the entire row, and setting to zero all the other entries in that row. 
     A collection of these maximum values may be gathered by constructing a first maximum intensity image and a corresponding first maximum index image from all of the row-stripe images, and constructing a second maximum intensity image and corresponding second maximum index image from all of the column-stripe images. The first and second maximum intensity images store the maximum intensity value at each pixel across the row-stripe or column-stripe images, respectively, and the first and second maximum index images store at each pixel the index of the row-stripe or column-stripe images where the maximum value is captured. Since the indices of the row-stripe and column-stripe images directly correspond to the row and column indices in the projector image plane, the maximum index at each camera pixel is the index of the projector pixel it captures. In other words, it determines the indices of the non-zero entries in T. 
     Note that when the camera resolution is much greater than the projector resolution, it is likely that a single turned ON projector pixel may create a light footprint on the camera sensor pixel array (i.e. multiple camera sensor pixels capture light from the single projector pixel). In this case, the maximum indices at these camera pixels are identical. The values of the non-zero entries in T are then determined by the corresponding maximum intensity images. When the resolution of the camera is higher than that of the projector, the display constraint holds because each of the projector pixels would light an area that is captured by a patch (i.e. group or footprint) of camera pixels. 
     This brings us to the problem being addressed by the present embodiment. When the camera resolution is close to, or lower, than that of the projector (i.e. when the camera resolution is preferably not greater than 1.1 times that of the projector resolution), the area on a camera sensor array lit by a couple of neighboring projector pixels may be captured by a single camera pixel, and thus the display constraint does not exactly hold. This may occur in low-budget display systems where low-end cameras are used. In such settings, light individually, and separately, emitted from a group of neighboring projector pixels may fall, in turn, onto the same, single, camera sensor pixel. In effect, this single camera sensor pixel is a many-to-one camera sensor pixel since light from several projector pixels hit this single camera sensor pixel. Since in the construction of the maximum images described above only the highest light intensity value is stored at each camera sensor pixel location, for each many-to-one camera sensor pixel, the maximum images will store only the light intensity value of the brightest projector pixel among a corresponding group of neighboring projector pixels. Therefore, only the projector pixel having the highest intensity value among the group of neighboring projector pixels gets its intensity value stored, and only it contributes a non-zero value to the construction of a light transport matrix T. Consequently, light intensity information of the other projector pixels within the corresponding group of neighboring projector pixels is lost, and is therefore missing from the constructed light transport matrix T. An example of this is shown in  FIG. 51 . 
     In  FIG. 51 , a partial view of a light transport matrix T shows in white those columns that have at least one non-zero value, NZ, which means that image information was captured and retained for that column&#39;s corresponding projector pixel. Projector pixels whose light intensity values were lost (due to them being part of a group of projector pixels in a many-to-one relationship with a single camera sensor pixel) correspond to columns that have all zero values within light transport matrix T. These all-zero columns are shown as darkened columns, and their corresponding projector pixels are effectively, “missing projector pixels”. As is explained above, there may be other reasons why a projector pixel may have all zeroes in its corresponding column in light transport matrix T, such as if the projector pixel was outside the field-of-view of the camera, but those issues are addressed below. 
     The point is that one cannot recover the non-zero entries in light transport matrix T corresponding to the missing projector pixels. In other words, since each column in light transport matrix T refers to a camera image lit by a respective projector pixel, each captured camera image corresponding to a missing projector pixel is effectively lost. Thus, the columns of T corresponding to missing projector pixels only zero-value entries. Equivalently the corresponding rows in the view projection matrix {hacek over (T)} T  would have only zero-value entries. This means that a projector image computed by p={hacek over (T)} T  c would have “holes”, i.e. darkened areas, each of which corresponds to one of the missing projector pixels whose intensity value was set to zero due to it not being the brightest pixel within the group of neighboring projector pixels. To display desirable images, one needs to fill in these holes in the images before projecting them on a screen. 
     To fill in the holes, one straightforward method would be to post-process the projector image computed by view projection, e.g. using a median filtering method to estimate the color of the missing projector pixels based on the colors of the neighboring pixels. However, it is computationally expensive to filter each frame when projecting a video. Furthermore, such filtering often runs in the color space only and does not consider the spatial relationship between the neighboring projector pixels and the corresponding camera pixels. 
     Therefore, the present embodiment proposes a method that directly fills in the missing information in the view projection matrix {hacek over (T)} T  (or alternatively in the original or modified (by imposition of the display constraint) light transport matrix T). The present embodiment naturally takes into account the spatial relationship and does not introduce more computation than the original view projection process since the filling in of holes is done in the calibration process before view projection. 
     With reference to  FIG. 52 , a partial projector pixel array and partial light transport matrix T illustrate the relationship between missing projector pixels and all-zero columns in T. Columns in T that have all-zero values are again shown darkened. Each projector pixel px 1 , px 2 , etc. is identified as a circle, and is labeled NZ if it corresponds to column within T that has at least one non-zero value. Missing projector pixels, which correspond to all-zero columns in T are labeled M, and are shown as darken circles. Since the view projection matrix {hacek over (T)} T  is the transpose of the normalized light transport matrix T (preferably after application of the display constraint), both matrices indicate which projector pixels have stored light transport information and which projector pixels have no corresponding light transport information. Therefore, the same M and NZ projector pixel identifiers may be extracted from view projector matrix {hacek over (T)} T , or from the modified version (by imposition of the display constraint) of T. For ease of illustration, however,  FIG. 52  illustrates the use of light transport matrix T. 
     The missing information of M-labeled projector pixels is preferably filled by a row-scanning operation on the projector pixel array, followed by a column-scanning operation on the projector pixel array. The first step is to identify all the projector pixels within the projector pixel array whose light transport information may have been lost due to not being the brightest projector pixel within a group of neighboring projector pixels. Recalling that each column of T (or row of {hacek over (T)} T ) corresponds to a respective projector pixel, this first step consists of labeling (for example with an M-label) each projector pixel whose corresponding column in T (or row of {hacek over (T)} T ) has all-zero values, or entries (i.e. has no non-zero entries). If a column in T has all-zero entries, then its corresponding projector pixel either was outside of the field-of-view of the camera (i.e. was not included in the image captured by the camera), or was missed (i.e. lost) in the maximum image process described above. Thus, such an M-labeled projector pixel becomes a “candidate pixel” for a projector pixel whose captured light transport information was lost due to a single camera sensor pixel capturing light transport information from multiple projector pixels (for example, from a group of neighboring projector pixels). 
     This is illustrated in  FIG. 53 , where a partial view of an exemplary single row of a projector pixel array is illustratively shown with thirty-three projector image pixels. It is to be understood that the row of a projector pixel array will typically have many more projector image pixels. The projector pixel array is subjected to a row-scanning process, in which each row of projector pixels is scanned and each projector pixel is labeled M or NZ, as determined from its corresponding column in light transport matrix T or corresponding row in view projection matrix {hacek over (T)} T . For example, an M-labeled projector pixel (i.e. a missing projector pixel) corresponds to corresponds a column in T having all-zero entries, and an NZ-labeled projector pixel (i.e. a non-zero projector pixel) corresponds a column in T have at least one non-zero entry. The M-labeled projector pixels within each row are grouped into M-segments (for example, MS 1  to MS 5 ) and the NZ-labeled projector pixels are grouped into NZ-segments (for example, NZS 1  to NZS 4 ). 
     Each M-segment between two NZ-segments is initially determined to consist of “candidate pixels” (i.e. projector pixels whose captured light transport information was lost) and identified as a “candidate row-pixel segment”. For example, in  FIG. 53 , M-segments MS 2 , MS 3 , and MS 4  would receive this initial determination status of “candidate row-pixel segment”. However, M-segments MS 1  and MS 5  would not be identified as candidate row-pixel segments since they are at the beginning and end of the row, respectively, and thus are not between two NZ-segments. If the length of (i.e. number of projector pixels within) such a candidate row-pixel segment is longer than a predefined threshold number of projector pixels (preferably 6 to 8 projector pixels), its initial determination status of “candidate row-pixel segment” is revoked and its ultimate determination is left for a column-scanning process. For example, M-segment MS 4  consists of 9 projector pixels, so the final designation of each of its constituent projector pixels as “candidate pixels” is indeterminate, and will be determined during a subsequent column-scanning operation that follows the present row-scanning operation. Such candidate row-pixel segments that are longer than the predefined threshold number of projector pixels in the row direction are usually elongated segments comprised of projector pixels that are also part of corresponding candidate column-pixel segments whose length along the column direction is typically much shorter. 
     However, if the candidate row-pixel segment is not longer than the predefined threshold, then non-zero light transport values are assigned to the projector pixels that make up the candidate row-pixel segment. The assigned non-zero light transport values are determined using the non-zero value of the projector pixel at each end of the candidate row-pixel segment, as is explained below. 
     If each projector pixel is captured by a single camera pixel, each of the non-zero valued projector pixels at each end of the candidate row-pixel segment should have been captured by neighboring camera pixels. Thus, one can simply divide the candidate row-pixel segment into two halves and directly copy the values of the two end-non-zero pixels into their contiguous halves. For example if this process were applied to MS 2  in  FIG. 53 , then M-segment MS 2  would be divided into a first half of two candidate projector pixels adjoining NZS 1 , and a second half of two candidate projector pixels adjoining NZS 2 . Assuming that the entry value of the projector pixel in NZS 1  is NZ 1 , then this same value of NZ 1  would be copied into the first half of two candidate projector pixels. Similarly, if the value of the projector pixel within NZS 2  that adjoins an end of MS 2  is NS 2 , then this same value of NS 2  would be copied into the second half of two candidate projector pixels. 
     However, since a low-resolution camera is often of low quality as well, a single projector pixel may occasionally become associated with (or captured by) a group of neighboring camera pixels due to camera noise and to the maximum image process, itself. In this case, each of the two non-zero projector pixels at either end of a candidate row-pixel segment (i.e. each end-non-zero projector pixel) may respectively correspond to (i.e. have been captured by) multiple camera pixels. In other words, each end-non-zero projector pixel may correspond to a separate group of camera sensor pixels. Thus, straightforward distributive copying of the light transport values of the two end-non-zero projector pixels to either half of the candidate row-pixel segment would not be the best approach. In such cases, each group of camera sensor pixels corresponding to each end-non-zero projector pixel should be split and distributed to the candidate row-pixel segment. Otherwise, the image could be blurred or even locally distorted due to the mis-assigned (i.e. incorrectly assigned) spatial relationship between the projector pixels and the camera sensor pixels. 
     To resolve this issue, one may make an assumption of local continuity of the view projection matrix. That is, one may assume that projector pixels within a candidate row-pixel segment (i.e. a local window (i.e. span of M-labeled projector pixels) between two end-non-zero projector pixels (i.e. NZ-labeled projector pixels)) are also captured by camera pixels. This assumption holds generally for practical display system settings where there is little discontinuity on a display surface. 
     In  FIG. 54 , a section of a row of projector pixels includes a candidate row-pixel segment MS 6  between a first NZ-segment (NZS 5 ) and second NZ-segment (NZS 6 ). The last projector pixel PXL_ 1  within NZS 5  adjoins MS 6 , and the first projector pixel PXL_ 2  within NZS 6  adjoins MS 6 . Also shown is a partial view of a camera sensor pixel array, and seven representative camera sensor pixels, Cmr 1  to Cmr 7 . It is to be understood that the camera sensor pixel array in actuality would includes many more camera sensor pixels, typically arranged in array fashion of rows and columns. However, for ease of explanation, only those camera sensor arrays that captured light from projector pixels PXL_ 1  and PXL_ 2  are shown. 
     Projector pixels PXL_ 1  and PXL_ 2  may each correspond to a respective group of camera sensor pixels, as determined from a column of matrix T, or row of matrix {hacek over (T)} T . Therefore, the camera sensor pixels corresponding to (i.e. who captured light transport information from) PXL_ 1  and PXL_ 2  are identified and collected into respective groups, G 1  and G 2 . That is, camera sensor pixels within group G 1  (i.e. Cmr 1  to Cmr 4 ) provide non-zero light transport information (i.e. captured non-zero light intensity values) for projector pixel PXL_ 1 , and the camera sensor pixels within group G 2  (i.e. Cmr 5  to Cmr 7 ) provide non-zero light transport information for projector pixel PXL_ 2 . Light transport information (i.e. light intensity information) for the missing MS 6  projector pixels between PXL_ 1  and PXL_ 2  may be obtained from the identified two groups of camera sensor pixels, G 1  and G 2 . 
     The centers, o 1  and o 2 , of the two groups of camera pixels, G 1  and G 2 , can be computed from each group&#39;s camera pixels&#39; 2D coordinates (i.e. their x-y coordinates). Since each of o 1  and o 2  is identified by a set of x-y coordinates, each is defined by a vector [x y] (i.e. a matrix having one row and two columns). Because o 2  and o 1  are 2D vectors representing the camera image coordinates (row, column) T , a vector, V 12 , connecting the two centers can be identified as V 12 =o 2 −o 1 . Vector V 12  is then normalize, and use as the principle axis (i.e. it becomes the relative position of a new, local x-y coordinate system with o 1  preferably at its origin). The distance of each camera sensor pixel in G 1  and G 2  to vector V 12  is determined by projecting each camera pixel within the two groups (G 1  and G 2 ) onto the local x-axis (i.e. onto vector V 12 ) by,
 
 v   i =( x   i   −o   1 ) T   *V   12  
 
where v i  is a scalar representing the location on the local x-axis onto which the camera sensor pixel is projected, and x i  is the coordinate of the i th  camera sensor pixel in the two groups. One then computes the minimum and maximum values of the v i &#39;s and evenly divides the range between the two values into S+2 intervals, where S is the length, i.e. pixel number, of the candidate row-pixel segment. The additional two intervals are to account for the two end-non-zero projector pixels PXL_ 1  and PXL_ 2 . Based on the local continuity assumption, each of the intervals corresponds to one of the projector pixels in the candidate row-pixel segment plus PXL_ 1  and PXL_ 2 . The two groups of camera sensor pixels are then split and distributed to the projector pixels based on which intervals on the principal axis their respective projection falls into. More specifically, if the projection of a camera sensor pixel x i  falls into the interval I s , this camera sensor pixel is re-assigned to the corresponding projector pixel p s , i.e. replaces the zero value of the corresponding index in the column of T related to p s  with the non-zero value of this camera sensor pixel. The values of PXL_ 1  and PXL_ 2  are preferably also re-assigned according to their interval location.
 
     This process is run on all the candidate row-pixel segments to fill the missing projector pixels within each candidate row-pixel segment along each row. Therefore, after running this process along a row, the M-labeled projector pixels within the row&#39;s candidate row-pixel segments may be reassigned as NZ-labeled projector pixels, as required. The process is then run along each column (i.e. the column-scanning operation) to identify column-pixel segments in a manner analogous to how row-pixel segments were identified, and the missing information within M-labeled projector pixels of column-pixel segments is filled in a manner analogous to how missing information in row-pixel segments was filled in. 
     For example, column-pixel segments are identified as consecutive M-labeled projector pixels along a column between two NZ-labeled projector pixels. Missing information for the M-labeled projector pixels within each column-pixel segment is obtained from the light transport information corresponding to the two NZ-labeled projector pixels adjoining each end of the column-pixel segment. 
     If light from each projector pixel was captured by only one camera projector pixel, then the column-pixel segment may be divided in half, and each projector pixel within each half may be assigned the NZ value of its adjoining NZ-labeled projector pixel. 
     Alternatively, if each projector pixel may be captured by a group of camera pixels, then two groups of camera pixels corresponding to the two end-NZ-labeled projector pixels of a column-pixel segment are identified. The center of the first group of camera pixels is identified from the camera pixels&#39; 2D coordinates. Similarly, the center of the second group of camera pixels is similarly identified from the camera pixels&#39; 2D coordinates. A vector connecting the two centers is identified as V 12 =o 2 −o 1 , where in the present column scanning process, o 2  and o 1  now identify the centers of two groups of camera pixels corresponding to the two end-NZ-labeled projector pixels of a column-pixel segment. As before, V 12  is preferably normalized and made the new principle axis with o 1  at its origin. The distance to normalized vector V 12  of each camera sensor pixel (within the two groups) is obtained by projecting each camera sensor pixel onto this new principle axis. That is, a scalar representation, v i , of the location within the principle axis V 12  onto which each camera sensor pixel is projected is determined, as is explained above following the relationship v i =(x i −o 1 ) T *V 12 , where x i  is the coordinate of the i th  camera sensor pixel in the two groups. As was the case in the row canning process, in the present column scanning process, one computes the minimum and maximum values of the ui&#39;s and evenly divides the range between the two values into S+2 intervals, where S is now the length, i.e. pixel number, of the candidate column-pixel segment. The additional two intervals are to account for the two end-non-zero projector pixels at either end of the column-pixel segment. 
     Based on the local continuity assumption, each of the intervals corresponds to one of the projector pixels in the candidate column-pixel segment plus the two projector pixels at either end of the column-pixel segment. The two groups of camera sensor pixels are then split and distributed to the projector pixels based on which intervals on the principal axis their respective projection falls into. More specifically, if the projection of a camera sensor pixel x i  falls into interval I s , this camera sensor pixel is re-assigned to the corresponding projector pixel p s , i.e. replaces the zero value of the corresponding index in the column of T corresponding to p s  with the non-zero value of this camera sensor pixel. The values the two projector pixels at either ends of the column-pixel segment are preferably also re-assigned according to their corresponding interval location. 
     After the two scanning operations (i.e. the row scans and the column scans), all the “holes” in T will have been filled, and the view projection matrix without missing projector pixels is thus constructed. Seamless view projection can then be achieved even when low-quality and low-resolution cameras are used for calibration. This technique is therefore called resolution scalable view projection. 
     While the invention has been described in conjunction with several specific embodiments, it is evident to those skilled in the art that many further alternatives, modifications and variations will be apparent in light of the foregoing description. Thus, the invention described herein is intended to embrace all such alternatives, modifications, applications and variations as may fall within the spirit and scope of the appended claims.