Abstract:
A system and method for extracting structure from stereo that represents the scene as a collection of planar layers. Each layer optimally has an explicit 3D plane equation, a colored image with per-pixel opacity, and a per-pixel depth value relative to the plane. Initial estimates of the layers are made and then refined using a re-synthesis step which takes into account both occlusions and mixed pixels. Reasoning about these effects allows the recovery of depth and color information with high accuracy, even in partially occluded regions. Moreover, the combination of a global model (the plane) with a local correction to it (the per-pixel relative depth value) imposes enough local consistency to allow the recovery of shape in both textured and untextured regions.

Description:
BACKGROUND 
     1. Technical Field 
     The invention is related to a system and process for extracting 3D structure from plural, stereo, 2D images of a scene by representing the scene as a group of image layers characterized by estimated parameters including the layer&#39;s orientation and position, per-pixel color, per-pixel opacity, and optionally a residual depth map, and more particularly, to such a system and process for refining the estimates for these layer parameters. 
     2. Background Art 
     Extracting structure from stereo has long been an active area of research in the imaging field. However, the recovery of pixel-accurate depth and color information from multiple images still remains largely unsolved. Additionally, existing stereo algorithms work well when matching feature points or the interiors of textured objects. However, most techniques are not sufficiently robust and perform poorly around occlusion boundaries and in untextured regions. 
     For example, a common theme in recent attempts to solve these problems has been the explicit modeling of the 3D volume of the scene. The volume of the scene is first discretized, usually in terms of equal increments of disparity. The goal is then to find the so-called voxels which lie on the surfaces of the objects in the scene using a stereo algorithm. The potential benefits of these approaches can include, the equal and efficient treatment of a large number of images, the explicit modeling of occluded regions, and the modeling of mixed pixels at occlusion boundaries to obtain sub-pixel accuracy. However, discretizing space volumetrically introduces a huge number of degrees of freedom. Moreover, modeling surfaces by a discrete collection of voxels can lead to sampling and aliasing artifacts. 
     Another active area of research directed toward solving the aforementioned problems is the detection of multiple parametric motion transformations within image sequence data. The overall goal is the decomposition of the images into sub-images (or “layers”) such that the pixels within each sub-image move consistently with a single parametric transformation. Different sub-images are characterized by different sets of parameter values for the transformation. A transformation of particular importance is the 8-parameter homography (collineation), because it describes the motion of points on a rigid planar patch as either it or the camera moves. The 8 parameters of the homography are functions of the plane equations and camera matrices describing the motion. 
     While existing layer extraction techniques have been successful in detecting multiple independent motions, the same cannot be said for scene modeling. For instance, the fact that the plane equations are constant in a static scene (or a scene imaged by several cameras simultaneously) has not been exploited. This is a consequence of the fact that, for the most part, existing approaches have focused on the two frame problem. Even when multiple frames have been considered, it has primarily been solely for the purposes of using past segmentation data to initialize future frames. Another important omission is the proper treatment of transparency. With a few exceptions, the decomposition of an image into layers that are partially transparent (translucent) has not been attempted. 
     SUMMARY 
     The present invention relates to stereo reconstructions that optimally recover pixel-accurate depth and color information from multiple images, including around occlusion boundaries and in untextured regions. This is generally accomplished using an approach to the stereo reconstruction that represents the 3D scene as a collection of approximately planar layers, where each layer has an explicit 3D plane equation and a layer sprite image, and may also be characterized by a residual depth map. The layer sprite refers to a colored image with a defined per-pixel opacity (transparency). The residual depth map refers to a per-pixel depth value relative to the plane. The approach of segregating the scene into planar components allows a modeling of a wider range of scenes. To recover the structure of the scene, standard techniques from parametric motion estimation, image alignment, and mosaicing can be employed. 
     More specifically, the approach to the stereo reconstruction based on representing the 3D scene as a collection of approximately planar layers involves first estimating the desired parameters (e.g. plane equation, sprite image and depth map) and then refining these estimates. The estimating phase can be accomplished via any appropriate method, such as the methods disclosed in a co-pending application entitled STEREO RECONSTRUCTION EMPLOYING A LAYERED APPROACH by the inventors of this application and assigned to the common assignee. This application was filed on Mar. 20, 1998 and assigned Ser. No. 09/045,519. The full approach disclosed in the application, which is believed to provide the best estimate of the layer parameters, and so the structure of the 3D scene, includes: 
     (a) inputting plural 2D images as well as camera projection matrices defining the location and orientation of the camera(s) responsible for creating each image, respectively; 
     (b) assigning each pixel making up each 2D image to one of the plural layers; 
     (c) estimating a plane equation for each layer that defines the orientation and position of that layer in 3D space; 
     (d) estimating a sprite image for each layer characterized by a per-pixel color and a per-pixel opacity; 
     (e) estimating a residual depth map for each layer wherein each residual depth map defines the distance each pixel of the associated layer is offset from the estimated plane of that layer; 
     (f) re-estimating each layer&#39;s sprite image based on the residual depth map associated with the layer; 
     (g) re-assigning pixels assigned to a particular layer to another layer by using the estimates for the plane equation, sprite image, and residual depth map for each layer as a guide; 
     (h) iteratively repeating steps (c) through (g) for each layer until the change in the value of at least one layer parameter relative to its value in an immediately preceding iteration falls below a prescribed threshold assigned to the parameter; and 
     (i) outputting data representative of the plane equation, sprite image and residual depth map estimates for each layer. 
     Only the input, pixel assignment, plane equation and sprite image estimation, and output modules (less the residual depth map) are necessary to produce a useable layered representation of the scene. However, the accuracy of the layered representation can be progressively improved with the respective addition of each of the remaining modules, i.e. the depth map estimation, sprite image re-estimation, and pixel re-assignment and iteration modules. 
     The initial estimates of the layer parameters are refined in accordance with systems and methods embodying the present invention. The refining process is accomplished using a re-synthesis, which takes into account both occlusions and mixed pixels. Approximate knowledge of the 3D structure (camera matrices and plane equations) allows reasoning about the image formation process. Specifically, a forward (generative) model of image synthesis is used, as well as a process of measuring of how well the layers re-synthesize the input images. Optimizing this measure allows refinement of the layer sprite estimates, and, in particular, the estimate of their true colors and opacities. This approach results in the correct recovery of mixed pixels, a step which is necessary to obtain sub-pixel accuracy and to ensure robustness at occlusion boundaries. Once the layer sprite estimates are refined, the plane equations and residual depth maps (if employed) can be refined as well using the original estimation process, such as the one disclosed in the aforementioned co-pending application. 
     The layered approach to stereo reconstruction shares many of the advantages of the previously described volumetric approaches because the 3D information contained in the layers is used to reason about occlusion and mixed pixels. However, the layered approach according to the present invention offers a number of additional advantages, including: 
     A combination of the global model (the plane) and the local correction to it (the per-pixel depth map) that results in very robust performance and extremely accurate depth maps; 
     A layered approach that enables the recovery of scene structure in untextured regions because there is an implicit assumption that untextured regions are planar—this is not an unreasonable assumption, especially in man-made environments; 
     A form of the output (i.e., a collection of approximately planar regions with per-pixel depth offsets) that is more suitable than a discrete collection of voxels for many applications, including, view interpolation and interactive scene modeling for rendering and video parsing. 
    
    
     In addition to the just described benefits, other advantages of the present invention will become apparent from the detailed description which follows hereinafter when taken in conjunction with the drawing figures which accompany it. 
     DESCRIPTION OF THE DRAWINGS 
     The specific features, aspects, and advantages of the present invention will become better understood with regard to the following description, appended claims, and accompanying drawings where: 
     FIG. 1 is a diagram depicting a general purpose computing device constituting an exemplary system for implementing the present invention. 
     FIG. 2 is a diagram graphically depicting the basic concepts of the layered stereo reconstruction approach according to the present invention. 
     FIG. 3 is a block diagram of an overall process for stereo reconstruction according to the present invention. 
     FIG. 4 is a block diagram of a process for accomplishing the plane equation estimation module of the overall process of FIG.  3 . 
     FIG. 5 is a block diagram of a process for accomplishing the layer sprite image estimation module of the overall process of FIG.  3 . 
     FIG. 6 is a block diagram of a process for accomplishing the residual depth map estimation module of the overall process of FIG.  3 . 
     FIG. 7 is a block diagram of a process for accomplishing the pixel re-assignment module of the overall process of FIG.  3 . 
     FIG. 8 is a block diagram of a process for accomplishing the layer refinement module of the overall process of FIG.  3 . 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     In the following description of the preferred embodiments of the present invention, reference is made to the accompanying drawings which form a part hereof, and in which is shown by way of illustration specific embodiments in which the invention may be practiced. It is understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the present invention. 
     FIG.  1  and the following discussion are intended to provide a brief, general description of a suitable computing environment in which the invention may be implemented. Although not required, the invention will be described in the general context of computer-executable instructions, such as program modules, being executed by a personal computer. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Moreover, those skilled in the art will appreciate that the invention may be practiced with other computer system configurations, including hand-held devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, network PCs, minicomputers, mainframe computers, and the like. The invention may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in both local and remote memory storage devices. 
     With reference to FIG. 1, an exemplary system for implementing the invention includes a general purpose computing device in the form of a conventional personal computer  20 , including a processing unit  21 , a system memory  22 , and a system bus  23  that couples various system components including the system memory to the processing unit  21 . The system bus  23  may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. The system memory includes read only memory (ROM)  24  and random access memory (RAM)  25 . A basic input/output system  26  (BIOS), containing the basic routine that helps to transfer information between elements within the personal computer  20 , such as during start-up, is stored in ROM  24 . The personal computer  20  further includes a hard disk drive  27  for reading from and writing to a hard disk, not shown, a magnetic disk drive  28  for reading from or writing to a removable magnetic disk  29 , and an optical disk drive  30  for reading from or writing to a removable optical disk  31  such as a CD ROM or other optical media. The hard disk drive  27 , magnetic disk drive  28 , and optical disk drive  30  are connected to the system bus  23  by a hard disk drive interface  32 , a magnetic disk drive interface  33 , and an optical drive interface  34 , respectively. The drives and their associated computer-readable media provide nonvolatile storage of computer readable instructions, data structures, program modules and other data for the personal computer  20 . Although the exemplary environment described herein employs a hard disk, a removable magnetic disk  29  and a removable optical disk  31 , it should be appreciated by those skilled in the art that other types of computer readable media which can store data that is accessible by a computer, such as magnetic cassettes, flash memory cards, digital video disks, Bernoulli cartridges, random access memories (RAMs), read only memories (ROMs), and the like, may also be used in the exemplary operating environment. 
     A number of program modules may be stored on the hard disk, magnetic disk  29 , optical disk  31 , ROM  24  or RAM  25 , including an operating system  35 , one or more application programs  36 , other program modules  37 , and program data  38 . A user may enter commands and information into the personal computer  20  through input devices such as a keyboard  40  and pointing device  42 . Of particular significance to the present invention, a camera  55  (such as a digital/electronic still or video camera, or film/photographic scanner) capable of capturing a sequence of images  56  can also be included as an input device to the personal computer  20 . The images  56  are input into the computer  20  via an appropriate camera interface  57 . This interface  57  is connected to the system bus  23 , thereby allowing the images to be routed to and stored in the RAM  25 , or one of the other data storage devices associated with the computer  20 . However, it is noted that image data can be input into the computer  20  from any of the aforementioned computer-readable media as well, without requiring the use of the camera  55 . Other input devices (not shown) may include a microphone, joystick, game pad, satellite dish, scanner, or the like. These and other input devices are often connected to the processing unit  21  through a serial port interface  46  that is coupled to the system bus, but may be connected by other interfaces, such as a parallel port, game port or a universal serial bus (USB). A monitor  47  or other type of display device is also connected to the system bus  23  via an interface, such as a video adapter  48 . In addition to the monitor, personal computers typically include other peripheral output devices (not shown), such as speakers and printers. 
     The personal computer  20  may operate in a networked environment using logical connections to one or more remote computers, such as a remote computer  49 . The remote computer  49  may be another personal computer, a server, a router, a network PC, a peer device or other common network node, and typically includes many or all of the elements described above relative to the personal computer  20 , although only a memory storage device  50  has been illustrated in FIG.  1 . The logical connections depicted in FIG. 1 include a local area network (LAN)  51  and a wide area network (WAN)  52 . Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet. 
     When used in a LAN networking environment, the personal computer  20  is connected to the local network  51  through a network interface or adapter  53 . When used in a WAN networking environment, the personal computer  20  typically includes a modem  54  or other means for establishing communications over the wide area network  52 , such as the Internet. The modem  54 , which may be internal or external, is connected to the system bus  23  via the serial port interface  46 . In a networked environment, program modules depicted relative to the personal computer  20 , or portions thereof, may be stored in the remote memory storage device. It will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between the computers may be used. 
     The exemplary operating environment having now been discussed, the remaining part of this description section will be devoted to a description of the program modules embodying the invention and the testing of these modules. Unless stated otherwise, the following description employs homogeneous coordinates for both 3D world coordinates x=(x, y, z, 1) T  and for 2D image coordinates u=(u, v, 1) T . In addition, it is assumed that when indexing 2D images, the homogeneous coordinates are converted into 2D pixel indices via conventional methods before accessing the array containing the image. Referring now to FIG. 2, suppose K images I 1 (u 1 ), I 2 (u 2 ), . . . , I K (u K ) captured by K cameras with projection matrices P 1 , P 2 , . . . , P K . In what follows, the image coordinates u k  will be dropped unless they are needed to explain a warping operation explicitly. 
     The basic concept is that the world can be reconstructed as a collection of L approximately planar layers. A sprite image of a layer is denoted by L l (u l )=(α l   · r l , α l   · g l , α l   · b l , α l ) where r l =r l (u l ) is the red band, g l =g l (u l ) is the green band, b l =b l (u l ) is the blue band, and α l =α l (u l ) is the opacity of pixel u l . Each layer is also associated with a homogeneous vector n l , which defines the plane equation of the layer via n l   T X=0, and optionally a per-pixel residual depth offset Z l (u l ). Thus, the scene is represented by L sprite images L l  on planes n l   T =0 with depth offsets Z l . 
     Techniques for estimating the layer parameters n l , L l , and Z l , typically require an assumption of boolean valued opacities where a pixel in an image is characterized as being either completely opaque or completely transparent. The aforementioned co-pending application is an example where such an assumption is made for estimation purposes. However, the techniques embodying the present invention use these estimates of the structure as an input into a re-synthesis process that refines the layer sprites L l , to including real opacity values α l . 
     By way of an example, the methods employed in the previously-identified co-pending application will be described herein to show how the layer parameters L l , n l , and Z l  are first estimated prior to employing the refinement techniques associated with the present invention. The estimation of the layer parameters can be subdivided into a number of steps for respectively estimating L l , n l , and Z l . To compute these quantities, we need to add auxiliary boolean mask images B kl . The boolean masks B kl  denote the pixels in image I k  which are images of points in layer L l . In other words, the mask images B kl  define which pixels are in a particular layer. Since we are assuming boolean opacities, B kl =1 if and only if L l  is the front-most layer which is opaque at that pixel in image I k . Hence, in addition to L l , n l , and Z l , we also need to estimate the boolean masks B kl . Once we have estimated these masks, we can compute masked input images M kl =B kl ™I k . 
     FIG. 3 illustrates a high-level structure of a preferred layered stereo reconstruction approach employing refinement of the estimated layer parameters L l , n l , and Z l , using real opacity values α l . The estimation steps associated with the aforementioned co-pending application have been included as the preferred method for estimating the layer parameters L l , n l , and Z l . However, other layer estimation techniques could also be employed if desired. Referring to FIG. 3, the input consists of the aforementioned collection of images I k  taken with cameras of known geometry P k . The goal is to estimate the layer sprites L l , the plane vectors n l , and preferably the residual depths Z l . Given any three of L l , n l , Z l , and B kl , there are techniques for estimating the remaining one. The approach generally consists of first initializing these quantities. Then, each of these quantities is in turn iteratively estimated while fixing the other three. After good initial estimates of the layers are obtained, the refinement process embodying the present invention is begun in which real valued opacities α l  are used and the layer sprites are refined, including the opacities. In the following discussion, each of the steps in the layer parameter estimation process of the previously-identified co-pending application is discussed. Then, the subject process involving an image formation model and a re-synthesis procedure used to refine each of the layer sprites is discussed. 
     Additionally, in the remainder of this specification, the description refers to various individual publications identified by an alphanumeric designator contained within a pair of brackets. For example, such a reference may be identified by reciting, “reference [WA93]” or simply “[WA93]”. Multiple references will be identified by a pair of brackets containing more than one designator, for example, [WA93, SA96, BJ96]. A listing of the publications corresponding to each designator can be found at the end of this Detailed Description section. 
     1. Computation of Layers Using Boolean Opacities 
     1.1 Initialization of Layers 
     Initialization of the layers is a process by which the pixels forming the images are initially assigned to a particular layer, thereby segmenting the images. The Boolean mask images B kl , and so the masked input images M kl  are defined from this initial assignment of pixels. There are a number of methods of initializing the layers. One way is to initialize a large number of small random layers, which are then allowed to grow and merge until a small number of larger layers remain which accurately model the scene. Such an approach was employed in references [WA93, SA96, BJ96]. A second approach would be to use a sequential method in which a dominant motion is first extracted and then the residual regions are processed recursively. This method was employed in references [IAH95, SA96]. A third possibility would be to perform a color or intensity segmentation in each image as used in reference [ASB94]. The resulting segments could be matched and used as initial layer assignments. A fourth approach consists of first applying a conventional stereo algorithm to get an approximate depth map. A plane fitting algorithm could then be used to initialize the layers. A final method would be to get a user to initialize the layers manually. In many applications, such as model acquisition [DTM96] and video parsing [SA96], the goal is a semi-automatic process where limited user input is acceptable. 
     1.2 Estimation of Plane Equations 
     Referring to FIG. 4, in order to compute the plane equation vector n l , it is necessary to map points in the masked images M kl , derived as previously discussed (step  400 ), onto the plane n l   T X=0. If X is a 3D world coordinate of a point and u k  is the image of X in camera P k : 
     
       
           u   k   =P   k   X   (1) 
       
     
     where equality is in the 2D projective space ρ 2 . Since P k  is of rank 3, it can be written: 
     
       
           X=P   k   *u   k   +sp   k   (2) 
       
     
     where P k *=P T   k (P k p T   k ) −1  is the pseudoinverse of P k , s is an unknown scalar, and p k  is a vector in the null space of P k , i.e. P k p k =0. If X lies on the plane n l   T X=0 then: 
       n   l   T   P   k   *u   k   +sn   l   T   p   k =0  (3) 
     Solving this equation for s, substituting into Equation (2), and rearranging using the fact that equality is up to a scale factor in ρ 2 , yields: 
     
       
           X= (( n   l   T   p   k ) I−p   k   n   l   T ) P   k   *u   k   (4) 
       
     
     The importance of Equation (4) is that it facilitates the mapping of a point u k  in image M kl  onto the point on plane n l   T X=0, of which it is an image. Afterwards this point is mapped onto its image in another camera P k′ : 
     
       
           u   k′   =P   k′ (( n   l   T   p   k ) P   k   *u   k   ≡H   l   kk′   u   k   (5) 
       
     
     where H l   kk ′ is a homography (collineation of ρ 2 ) Equation (5) describes the image coordinate warp between the two images M kl  and M k′l , which would hold if all the masked image pixels were images of world points on the plane n l   T X 0. Using this relation, all of the masked images can be warped onto the coordinate frame of one distinguished image, M 1l  (step  402 ), without loss of generality, as follows: 
     
       
         ( H   l   1 k   o M   kl )( u   i )≡ M   kl ( H   l   1 k   u   1 )  (6) 
       
     
     Here, H l   1 k o M kl  is the masked image M kl , warped into the coordinate frame of M 1l . 
     It should be noted that it would also be possible to add an extra 2D perspective coordinate transformation in the just-described method (not shown in FIG.  4 ). Suppose H is an arbitrary homography. Each masked image could be warped onto H o H l   1 k o M kl (u 1 )≡M kl (HH l   1k U 1 ). The addition of the homography H can be used to remove the dependence of the above on one distinguished image. For example, H could be chosen to optimize some function of the homographies HH l   1 k  which measures how close they are to identity. 
     The property used to compute n l  is the following: assuming the pixel assignments to layers B kl  are correct, the world is piecewise planar, and the surfaces are Lambertian, the warped images H l   1 k o M kl  should agree with each other and with M 1l  where they overlap. There are a number of functions which could possibly be used to measure the degree of consistency between the warped images (step  404 ), including least squares measures [BAHH92] and robust measures [DP95, SA96]. In both cases, the goal is the same, find the plane equation vector n l  which maximizes the degree of consistency (step  406 ), which is usually a numerical minimum. Typically, this extremum is found using some form of gradient decent, such as the Gauss-Newton method, and the optimization is performed in a hierarchical (i.e. pyramid based) fashion to avoid local extrema. To apply this standard approach, simply derive the Jacobian of the image warp H l   1 k  with respect to the parameters of n l  as described in reference [SS98]. This can be computed in a straightforward manner from Equation (5) because the camera matrices P k  are assumed to be known. It is noted that the plane defined by n l  can turn out to be of any orientation and need not be a frontal plane. 
     1.3 Estimation of Layer Sprites Images 
     Referring to FIG. 5, before the layer sprite images L l  can be computed, 2D coordinate systems must be chosen for the planes. Such coordinate systems can be specified by a collection of arbitrary (rank  3 ) camera matrices Q l . An interesting choice for Q l  is one for which the null space of Q l  is perpendicular to the plane, and for which the pseudo-inverse maps the coordinate axes onto perpendicular vectors in the plane (i.e., a camera for which the plane is a frontal image plane). However, it is often undesirable to use a fronto-parallel camera since this may unnecessarily warp the input images. 
     Once Q l  has been specified, it can be shown, by the same argument used in Equations (4) and (5), that the image coordinates u k of the point in image M kl  which is projected onto the point u l  on the plane n l   T X=0 is given by: 
     
       
           u   k   =P   k ((n l   T q l ) I−q   l   n   l   T ) Q   l   *u   l   ≡H   l   k   u   k   (7) 
       
     
     where Q l * is the pseudo-inverse of Q l  and q l  is a vector in the null space of Q l . The homography H l   k  warps the coordinate frame of the plane backward onto that of image M kl . The homography can also be used to warp the image M kl  forward onto the plane (step  500 ), the result of which is denoted H l   k o M kl . After the masked image has been warped onto the plane, the layer sprite image (with boolean opacities) can be estimated by “blending” the warped images:                L   l     =       ⊕     k   =   1     K            H   k   l     ∘                M   kl                 (   8   )                                
     where ⊕ is the blending operator (step  502 ). 
     There are a number of ways the blending could be performed. One simple method would be to take the mean of the color or intensity values. A refinement would be to use a “feathering” algorithm, where the averaging is weighted by the distance of each pixel from the nearest invisible pixel in M kl , as used in reference [SS97]. Alternately, robust techniques could be used to estimate L l  from the warped images. The simplest example of such a technique is the median, but many others exist. 
     There is a possibility that the images I k  could have different exposure characteristics owing to differences in the cameras used to capture the images or differences in the camera settings between images if the same camera was used to take the disparate images. The difference in the exposure between images will skew the colors and opacities exhibited by corresponding image points. One possibility for addressing this exposure problem is to simply ignore it. The previously-described blending operation will tend to smooth out the disparities between the images. In some applications, this smoothing will produce sprite images reasonably close to the actual layer sprite image. However, if proactive measures are needed to ensure the sprite images exhibit an acceptable resolution, it is possible to monitor the bias and gain characteristics of the camera or cameras, and subject the images to a correction procedure to compensate for any disparity between the camera characteristics. Such corrective procedures are well known and so will not be described in detail in this specification. Once the images have been corrected, the blending process can proceed as described above. 
     One unfortunate effect of the blending in Equation (8) is that most forms of averaging tend to increase image blur, particularly when there are a large number of images. Part of the cause is non-planarity in the scene (which will be modeled in the next section), however image noise and resampling error also contribute. Therefore, this blurring effect should be compensated for (step  504 ) before proceeding to the residual depth estimation module (if employed). One simple method of compensating for this effect is to “deghost” the sprite images as described in reference [SS98]. Another solution is to explicitly use an image enhancement technique such as one of those described in references [IP92, MP94, CB97]. These latter techniques have the added advantage of estimating a sprite with greater resolution than that of the input images. 
     1.4 Estimation of Residual Depth 
     In general, the scene will not be exactly piecewise planar. To model any non-planarity, it can be assumed that the point u l  on the plane n l   T X=0 is displaced slightly. Referring to FIG. 6, it will be assumed the point is displaced in the direction of the ray through u l  defined by the camera matrix Q l , and that the distance it is displaced is Z l (u l ), measured in the direction normal to the plane (step  600 ). In this case, the homographic warps used previously are not applicable. However, using a similar argument to that in Section 1.2 and 1.3, it can be shown (see also [KAH94, DTM96]) that: 
     
       
           u   k   =H   l   k   u   l   +Z   l (u l ) t   kl   (9) 
       
     
     where H l   k =P k ((n l   T q l )I−q l n l   T )Q l * is the planar nomography of Section 1.3, t kl =P k q l  is the epipole, and it is assumed that the plane equation vector n l =(n x , n y , n z , n d ) T  has been normalized so that n 2   x+ n 2   y+ n 2   z =1. Equation (9) can be used to map plane coordinates u l  backwards to image coordinates u k , or to map the image M kl  forwards onto the plane (step  602 ). The result of this warp is denoted by (H l   k , t kl , Z l ) o M kl , or W l   k o M kl  for more concise notation. 
     To compute the residual depth map Z l , an optimization of the same (or a similar) consistency metric as that used in Section 1.2 to estimate the plane equation could be employed (steps  604  and  606 ). Doing so is essentially solving a simpler stereo problem. In fact, almost any stereo algorithm could be used to compute Z l . The one property the algorithm should have is that it favors small disparities. 
     1.5 Re-Estimation of Layer Sprite Images 
     Once the residual depth offsets have been estimated, the layer sprite images should be re-estimated using:                L   l     =         ⊕     k   =   1     K                       (       H   k   l     ,     t   kl     ,     Z   l       )     ∘                M   kl         =       ⊕     k   =   1     K                       W   k   l     ∘                M   kl                   (   10   )                                
     rather than Equation (8). This re-estimation of the layer sprite images using the just computed residual depth offsets provides a more accurate representation of the per-pixel color values of each sprite image. When the images I k  are warped onto the estimated plane of a layer, points actually on the plane will register exactly. However, corresponding image points in the images that are offset from the estimated plane will warp onto the plane in different locations, thereby reducing the accuracy of the blended layer sprite image. Thus, re-estimating the layer sprite images by first compensating for the parallax effect between points offset from the estimated plane and so bringing these points into alignment in the warped images, will increase the resolution of the layer sprite image estimates. Equation (10) accomplishes this alignment and blending process. 
     1.6 Re-Assiqnment of Pixels to Layers 
     The initial assignments of pixels to a particular layer were made using one of the methods previously described in section 1.1. However, the initial assignment process represented a somewhat crude estimate of what layer each pixel should belong and was done simply to provide a starting point for estimating the plane equations, sprite images and residual depths (i.e., n l , L l , and Z l , respectively). The Boolean masks B kl  contain this initial assignment information and allowed the computation of the masked image M kl  using M kl =B kl ·I k . It will now be described how to re-estimate the pixel assignments from the estimates of n l , L l  and Z l  to obtain a more accurate grouping of the pixels to the various layers. 
     One possible method would be to update the pixel assignments by comparing the warped images W l   k o M kl  to the layer sprite images L l . However, if these images were compared, it would not be possible to deduce anything about the pixel assignments outside of the current estimates of the masked regions. Thus, it is preferable to make the Boolean mask B kl  “grow”, thereby allowing for pixel assignments outside the current estimated masked regions to be considered. Referring to FIG. 7, the Boolean mask B kl  is allowed to “grow” by comparing W l   k o I k  with:                Full                   L   l       =       ⊕     k   =   1     K                       W   k   l     ∘                I   k                 (   11   )                                
     where Z l  is enlarged so that it declines to zero outside the masked region (step  700 ). The danger with working with warped versions of the complete input images W l   k o I k  is that unoccluded pixels will be blended together with occluded pixels and result in poor estimates of the “full” layer sprites Full L l . One possible solution to this is a use of a robust blending operator, such as the median. Another solution might be to weight masked pixels more than unmasked pixels during the blend. A final possibility would be to grow the current estimate of B kl  by a small prescribed number of pixels (e.g., 5 pixels), and then recompute W l   k o M kl . Such a strategy would still allow the layer masks to grow slightly on each iteration, but, it is believed, limit the influence of occluded pixels in the blend. 
     Given the full layer sprites Full L l , a preferred approach to pixel re-assignment is as follows. First compute a measure P kl (u l ) of the likelihood that the pixel W l   k o I k (u 1 ) is the warped image of the pixel u l  in the full layer sprite Full L l (step  702 ). Next, P kl  is warped back into the coordinate system of the input image I k (step  704 ) to yield: 
     
       
         Warped  P   kl =( W   l   k ) −1   o P   k   (12) 
       
     
     This warping tends to blur P kl , but the blurring is believed to be acceptable since it is desired to smooth the pixel assignment anyway. In fact, it may be advantageous to smooth P kl  even more by, for example, using an isotropic smoother such as a Gaussian. Other, more directed, smoothing methods could include performing a color segmentation of each input image and only smoothing within each segment as described in [ASB94]. Alternatively, P kl  might be smoothed less in the direction of the intensity gradient since strong gradients often coincide with depth discontinuities and hence layer boundaries. These latter two methods may produce a better result as it is believed they would tend to smooth P kl  where it is most needed, rather than in a general fashion as would be the case with the aforementioned Gaussian smoothing procedure. 
     The pixel re-assignment can now be computed by choosing the best possible layer for each pixel (step  706 ):                  B   kl          (     u   k     )       =              1             if                 Warped                     P   kl          (     u   k     )         =       min     l   ′            Warped                     P     kl   ′            (     u   k     )                                  0       otherwise                   (   13   )                                
     There are a number of possible ways of defining P kl . Perhaps the simplest is the residual intensity difference [SA96]: 
     
       
           P   k1   =∥W   l   k   o I   k −Full  L   l ∥  (14) 
       
     
     Another possibility is the residual normal flow magnitude:                P   kl     =                  W   k   l     ∘                I   k       -     Full                   L   l                       ∇   Full                     L   l              .             (   15   )                                
     Locally estimated variants of the residual normal flow have been described in [IP92, IRP92, IAH95]. A third possibility would be to compute the optical flow between W l   k o I k  and Full L l  and then use the magnitude of the flow for P kl . 
     Once the pixel re-assignment process is complete and updated Boolean masks B kl  have been computed via equation (13), the estimation steps described in sections 1.2 through 1.6 can be iteratively repeated until the change from one iteration to the next fall below a prescribed threshold. This prescribed threshold can relate to one or any combination of the layer parameters n l , L l , Z l  and B kl . For example, if the change in the Boolean mask image B kl  from the immediately preceding iteration falls below a certain percentage change threshold, it might be assumed that the then current estimates for the other parameters (i.e., n l , L l , and Z l ) are as accurate as possible for the initial estimation phase of the stereo reconstruction process. Alternately, one of the other layer parameters, or a combination of any or all of the parameters could be put to a similar threshold test. In these alternate methods, if the change between iterations in some number or all the tested parameters falls below their associated prescribed thresholds, then the iterative process would be stopped and the then current layer parameters accepted as the final estimate for the initial phase of the process. The actual threshold or thresholds employed will depend on the application and the degree of accuracy required. The number and value of the threshold(s) can be readily determined given the resolution desired for the particular application in which the present invention is being employed, and so will not be elaborated upon herein. 
     2. Layer Refinement by Re-Synthesis 
     In this section, it will be shown how the estimates of the layer sprites images, as well as the layer plane equations and residual depths (if employed), can be further refined, assuming that the opacities α l  associated with the sprite images are now represented by real values rather than the Boolean values used previously. The process begins by formulating a generative model of the image formation process. Afterwards, a proposed measure of how well the layers re-synthesize the input images is described, and it is shown how this measure can be minimized to refine the complete layer sprite estimates. 
     2.1 The Image Formation Process 
     Referring to FIG. 8, the first phase of the layer refinement process is to formulate a forward model of the image formation process using image compositing operations such as described in [Bli94], i.e. by painting the sprites one over another in a back-to-front order. The basic operator used to overlay the sprites is the over operator: 
     
       
           F⊙B≡F+( 1−α F ) B   1   (16) 
       
     
     where F and B are the foreground and background sprites, and α F  is the opacity of the foreground. Given this, the generative model consists of the following steps: 
     a. Using the known camera matrices, plane equations, and residual depths, warp each layer backwards onto the coordinate frame of image I k  using the inverse of the operator in Section 1.4, (step  800 ). 
     This yields the un-warped sprite image: 
     
       
           U   kl =(W l   k ) −1   o L   l   (17) 
       
     
     Note that the real valued opacities are warped along with the color values. 
     b. Composite the un-warped sprite images in back-to-front order:                S   k     =         ⊙     l   =   1     L          U   kl       =           U   kl                ⊙     ...                ⊙     U   kl                 (   18   )                                
     to obtain the synthesized image S k . If the stereo reconstruction problem has been solved, S k  should match the input I k . 
     This last step can be re-written as three simpler steps: 
     (i) Compute a visibility V kl  for each un-warped sprite image (step  802 ) [SG97]:                V   kl     =         V     k        (     l   -   1     )              (     1   -     α     k        (     l   -   1     )           )       =       ∏       l   ′     =   1       l   -   1                       (     1   -     α     kl   ′         )                 (   19   )                                
      where α kl  is the alpha channel of U kl , and V kl =1. 
     (ii) Compute the masked layer images, M kl =V kl  U kl (step  804 ). 
     (iii) Sum up the masked images, S k =Σ L   l=1  M kl (step  806 ). 
     In these three sub-steps, the visibility map makes the contribution of each sprite pixel to each image explicit. 
     2.2 Minimization of Re-Synthesis Error 
     As mentioned previously, if the reconstructed layer estimates are accurate, the synthesized image S k  should be similar to the input image I k . If not, the second phase of the layer refinement process is employed. Specifically, the prediction error:                 =       ∑   k            ∑     u   k                       S   k          (     u   k     )       -       I   k          (     u   k     )              2                 (   20   )                                
     can be used to refine the layer estimates (step  808 ) using a simple technique such as gradient descent. However, rather than trying to optimize over all of the parameters (L l , n l  and Z l ) simultaneously, it is preferable to only adjust the sprite colors and opacities L l , and then re-run the previous estimation steps described in sections 1.2 through 1.5 to refine n l  and Z l (step  810 ). 
     In solving Equation (20), the derivatives of the cost function C with respect to the colors and opacities in L l (u l ) can be computed using the chain rule [SG97]. In more detail, the visibility map V kl  mediates the interaction between the un-warped sprite U kl , and the synthesized image S k1  and is itself a function of the opacities in the un-warped sprites U kl . For a fixed warping function W l   k , the pixels in U kl  are linear combinations of the pixels in sprite L l . This dependence can either be exploited directly using the chain rule to propagate gradients, or alternatively the derivatives of C with respect to U kl  can be warped back into the reference frame of L l [SG97]. 
     While the invention has been described in detail by specific reference to preferred embodiments, it is understood that variations and modifications thereof may be made without departing from the true spirit and scope of the invention. 
     References 
     [ASB94] S. Ayer, P. Schroeter, and J. Bigün. Segmentation of moving objects by robust parameter estimation over multiple frames. In 3rd  ECCV , pages 316-327, 1994. 
     [BAHH92] J. R. Bergen, P Anandan, K. J. Hanna, and R. Hingorani. Hierarchical model-based motion estimation. In 2nd  ECCV , pages 237-252, 1992. 
     [BJ96] M. J. Black and A. D. Jepson. Estimating optical flow in segmented images using variable-order parametric models with local deformations.  PAMI , 18(10):972-986, 1996. 
     [Bli94] J. F. Blinn. Jim Blinn&#39;s corner: Compositing, part 1: Theory.  IEEE Computer Graphics and Applications,  14(5):83-87, September 1994. 
     [CB97] M. C. Chiang and T. E. Boult. Local blur estimation and super-resolution. In  CVPR  &#39;97, pages 821-826, 1997. 
     [DP95] T. Darrell and A. P. Pentland. Cooperative robust estimation using layers of support.  PAMI,  17(5):474-487, 1995. 
     [DTM96] P. E. Debevec, C. J. Taylor, and J. Malik. Modeling and rendering architecture from photographs: A hybrid geometry and image-based approach. In  SIGGRAPH &#39; 96, pages 11-20, 1996. 
     [IAH95] M. Irani, P. Anandan, and S. Hsu. Mosaic based representations of video sequences and their applications. In 5th  ICCV,  pages 605-611, 1995. 
     [IP92] M. Irani and S. Peleg. Image sequence enhancement using multiple motions analysis. In  CVPR &#39; 92, pages 216-221, 1992. 
     [IRP92] M. Irani, B. Rousso, and S. Peleg. Detecting and tracking multiple moving objects using temporal integration. In 2nd  ECCV,  pages 282-287, 1992. 
     [KAH94] R. Kumar, P. Anandan, and K. Hanna. Direct recovery of shape from multiple views: A parallax based approach. In 12th  ICPR,  pages 685-688, 1994. 
     [MP94] S. Mann and R. W. Picard. Virtual bellows: Constructing high quality stills from video. In 1st  ICIP,  pages 363-367, 1994. 
     [SA96] H. S. Sawhney and S. Ayer. Compact representations for videos through dominant and multiple motion estimation.  PAMI,  18(8):814-830, 1996. 
     [SG97] R. Szeliski and P. Golland. Stereo matching with transparency and matting. In  ICCV &#39; 98, pages 517-524, 1998. 
     [SS97] R. Szeliski and H. -Y. Shum. Creating full view panoramic image mosaics and texture-mapped models. In  SIGGRAPH &#39; 97, pages 251-258, 1997 
     [SS98] H. -Y. Shum and R. Szeliski. Construction and refinement of panoramic mosaics with global &amp; local alignment. In  ICCV &#39; 98, pages 953-958, 1998. 
     [WA93] J. Y. A. Wang and E. H. Adelson. Layered representation for motion analysis. In  CVPR &#39; 93, pages 361-366, 1993.