Abstract:
An efficient computation of low-dimensional linear subspaces that optimally contain the set of images that are generated by varying the illumination impinging on the surface of a three-dimensional object for many different relative positions of that object and the viewing camera. The matrix elements of the spatial covariance matrix for an object are calculated for an arbitrary pre-determined distribution of illumination conditions. The maximum complexity is reduced for the model by approximating any pair of normal-vector and albedo from the set of all such pairs of albedo and normals with the centers of the clusters that are the result of the vector quantization of this set. For an object, a viewpoint-independent covariance matrix whose complexity is large, but practical, is constructed and diagonalized off-line. A viewpoint-dependent covariance matrix is computed from the viewpoint-independent diagonalization results and is diagonalized online in real time.

Description:
CROSS REFERENCE TO RELATED APPLICATION  
       [0001]    This application claims priority from U.S. Provisional Application No. 60/279,729 filed Mar. 28, 2001. 
     
    
     
       BACKGROUND OF THE INVENTION  
         [0002]    1. Field of the Invention  
           [0003]    This invention relates to computer vision and more specifically to the efficient computation of the low-dimensional linear subspaces that optimally contain the set of images that are generated by varying the illumination impinging on the surface of a three-dimensional object, such as a human head, for many different relative positions of that object and the viewing camera.  
           [0004]    2. Prior Art  
           [0005]    In a typical system for object recognition under varying viewpoint and illumination conditions, information about properties, such as the three-dimensional shape and the reflectance, of a plurality of objects  100 , such as human faces and/or heads, is stored in a database  101 . When such a system is in use, typically a plurality of queries  102 , in the form of images of objects taken from non-fixed viewpoint and illumination conditions, is matched against said database  101 . For each query image  103 , matching  104  is performed against all objects  100  in said database  101 .  
           [0006]    In order to match an individual object  105  against an individual query  103 , several steps are typical. First, the viewpoint of the query is estimated  106 . Second, a viewpoint-dependent illumination subspace  107  is generated, as outlined below, from: the three-dimensional data about each said object  100 , and the determined viewpoint  106 . Further, either the illumination condition is estimated, or a similarity score is generated, or both  108 .  
           [0007]    The scores from the matching of said query  103  to said plurality of objects  100  are finally compared against each other to determine the best match  109  resulting in object  105  being recognized.  
           [0008]    The set of images of a given object under all possible illuminations, but fixed viewpoint, will be called the illumination subspace of the object for that viewpoint. It has been both observed in practice, and argued in theory, that said illumination subspace is mostly enclosed in a low-dimensional subspace, with dimensionality M as low as Mε. See P. Hallinan, “A Low-Dimensional Representation of Human Faces for Arbitrary Lighting Conditions”,  IEEE Conference on Computer Vision and Pattern Recognition , pp. 995-999 (1994); and R. Basri et al., “Lambertian Reflectance and Linear Subspaces”, to appear in the  International Conference on Computer Vision , (July 2001).  
           [0009]    There are several object, and more specifically face, recognition algorithms that are based implicitly or explicitly on this fact. See Georghiades et al., “From Few to Many: Generative Models for Recognition Under Variable Pose and Illumination”,  Proceedings of the  4 th    International Conference on Automatic Face &amp; Gesture Recognition , pp. 264-270, (March 2000); R. Ishiyama et al., “A New Face-Recognition System with Robustness Against Illumination Changes”,  IAPR Workshop on Machine Vision Applications , pp. 127-131 (November 2000); and Basri, et al. supra.  
           [0010]    Estimation of a Gaussian Probability Density by the Karhunen-Loève Transform  
           [0011]    The method for finding a hierarchy of low-dimensional subspaces that optimally contain a given set of images, or more generally, snapshots, is the Karhunen-Loève Transform (KLT), which is known under a variety of names—Principal Component Analysis, Empirical Eigenfunctions, and Singular Value Decomposition (SVD)—and is closely related to Linear Regression and Linear Factor Analysis, among others. See M. Tipping et al., “Mixures of Probabilistic Principal Component Analysers”,  Neural Computation , Vol. 11, No. 2, pp. 443-482 (February 1999).  
           [0012]    The basic facts about the KLT are summarized in this section. Of importance for the disclosure further are the implementation choices, the optimality properties, and the requirements for computational time and storage space. Also, frequent references to the equations herein will be made below.  
           [0013]    A snapshot, such as an image, will be represented by the intensity values φ (x), where {x} is a pixel grid that contains V pixels. An ensemble of T snapshots will be denoted by {φ t (x)} tεT . Briefly, (see I. Joliffe, “Principal Component Analysis” (1986); and P. Penev, “Local Feature Analysis: A Statistical Theory for Information Representation and Transmission”, PhD Thesis, The Rockefeller University, (1998)) its KLT representation is  
                 φ   t          (   x   )       =       ∑     r   =   1     M            a   r   t          σ   r            ψ   r          (   x   )                   (   1   )                               
 
           [0014]    where M=min(T, V) is the rank of the ensemble, {σ r   2 } s the (non-increasing) eigenspectrum of the “spatial” 
               R        (     x   ,   y     )            =   Δ              1   T            ∑   t              φ   t          (   x   )              φ   t          (   y   )             =       ∑     r   =   1     M              ψ   r          (   x   )            σ   r   2            ψ   r          (   y   )                     (   2   )                               
 
           [0015]    and the “temporal” 
               C     t                   t   ′              =   Δ              1   V            ∑   x              φ   t          (   x   )              φ     t   ′            (   x   )             =       ∑     r   =   1     M            a   r   t          σ   r   2          a   r     t   ′                     (   3   )                               
 
           [0016]    covariance matrices, and {ψ r (x)} and {a r   t } are their respective orthonormal eigenvectors. When M=T&lt;V, the diagonalization of C (eqn. 3) is the easier.  
           [0017]    Notably, the storage of C requires O(T 2 ) storage elements, and of R, O(V 2 )—the dependence of the storage requirements on the size of the problem is quadratic. Analogously, the time to compute the eigenvalues and eigenvectors (eqn. 2) of C is O(T 3 ), and of R, O(V 3 )—the dependence of the computational time on the size of the problem is cubic. In practical terms, this means that solving a system that is ten times as large requires a hundred times the space and a thousand times the computational power.  
           [0018]    The average signal power of the ensemble is  
                 1     T                 V              ∑     x   ,   t                     φ   t          (   x   )            2         =       trR   ≡     trR   M            =   Δ            ∑     r   =   1     M          σ   r   2                 (   4   )                               
 
           [0019]    KLT is optimal in the sense that, among all N-dimensional subspaces (N&lt;M), the subset of eigenmodes {Ψ r } r=1   N  (eqn.2) span the subspace which captures the most signal power, trR N . See M. Loeve, “Probability Theory”, (1955); and I. Joliffe, supra. For a given dimensionality N, the reconstruction of the snapshot φ (x) is  
                 φ   N   t          (   x   )            =   Δ            ∑     r   =   1     N            a   r   t          σ   r            ψ   r          (   x   )                   (   5   )                               
 
           [0020]    With the standard multidimensional Gaussian model for the probability density P[φ] the information content of the reconstruction (eqn. 5) is  
                 -   log                     P        [     φ   N   t     ]         ∝       ∑     r   =   1     N                 a   r   t          2               (   6   )                               
 
           [0021]    Notably, this model is spherical—the KLT coefficients (eqn. 1) are of unit variance (eqn. 3), &lt;a r   t &gt;≡1, and each of the N dimensions contributes equally to the information that is derived from the measurement, although only the leading dimensions contribute significantly to the signal power.  
           [0022]    This is a manifestation of the fact that, in principle, even weak signals can be important if they are sufficiently rare and unexpected. In practice, however, signals are embedded in noise, which typically has constant power; and weak signals, even though important, may not be reliably detected.  
           [0023]    The situation is complicated frther by the fact that in practice every estimation is done from a finite sample (T&lt;∞, V&lt;∞). Nevertheless, the shape of the eigenspectrum of sample-noise covariance matrices is known—it is determined by the ratio V/T (See J. Silverstein, “Eigenvalues and Eigenvectors of Large-Dimensional Sample Covariance Matrices”,  Contemporary Mathematics , Vol. 50, pp. 153-159 (1986); and A. Sengupta et al., “Distributions of Singular Values for Some Random Matrices”,  Physical Review E , Vol. 60, No. 3, pp. 3389-3392, (September 1999), and this knowledge can be used to recover the true spectrum of the signal through Bayesian estimation. See R. Everson et al., “Inferring the Eigenvalues of Covariance Matrices from Limited, Noisy Data”,  IEEE Transactions on Signal Processing , Vol 48, No. 7, pp. 2083-2091, (2000). Although this can serve as a basis for principled choice for the dimensionality, N, also called model selection, in the context of face recognition, this choice is typically guided by heuristic arguments. See P. Penev et al., “The Global Dimensionality of Face Space”,  Proceedings of the  4 th    International Conference on Automatic Face &amp; Gesture Recognition , pp. 264-270 (March 2000).  
           [0024]    Viewpoint—Dependent Subspaces for Face Recognition  
           [0025]    At the heart of most high-level algorithms for computer vision is the question: How does an object look from a specific viewpoint and a specific illumination condition? 
           [0026]    Here we describe the standard method for finding the illumination subspace  107 —a small number of basis images  110  {ψ r (x)} (eqn. 2) which can be linearly admixed (eqn. 5) to approximate substantially all images of a given object under a fixed viewpoint, but any illumination condition.  
           [0027]    An illumination condition at a given point, x, on the surface of an object is defined by specifying the intensity of the light, L(x, n) that is coming to the point from the direction nεS 2 , where n is a normal vector on the unit sphere S 2  centered at that point. Typically, the assumption is made that the light sources are sufficiently far away from the object—the distance to them is much larger than the size of the object—and therefore all points on the surface see the same illumination, L(n).  
           [0028]    In order to calculate the illumination subspace, it is customary to first generate a finite set of T illumination conditions {L t (n)} tεT , and from them, a corresponding set of images {I t (x)} tεT  of the object the under a fixed viewpoint. When I t  (x) is identified with φ t  from (eqn. 2), the Illumination subspace hierarchy is determined by (eqn. 3), and an application-dependent cutoff, N (cf. eqn. 5), is chosen.  
           [0029]    There are two general ways to generate an image I t  (x) for a given illumination condition: to make an actual photograph of the physical object under the given illumination condition, or to use computer graphics techniques to render a 3D model of the object. See A. Georghiades, supra.; and R. Ishiyama et al., supra.  
           [0030]    Although the first method is more accurate for any given picture, it is very labor-intensive, and the number of pictures that can be done is necessarily small. Thus, the practical way to attempt the sampling of the space of illumination conditions is to use synthetic images obtained by computer rendering of a 3D model of the object.  
           [0031]    When doing computer rendering, very popular is the assumption that the reflectance of the surface obeys the Lambertian reflectance model (See J. Lambert, “Photometria Sive de Mensura et Gradibus Luminus, Colorum et Umbrae,” (1760)): the intensity of the reflected light I (x) at the image point x from a point light source, L which illuminates x without obstruction is assumed to be 
             I ( x )=α( x ) L•p ( x )  (7) 
           [0032]    where α(x) is the intrinsic reflectance, or albedo, p(x) is the surface normal at the grid point x, and .•. denotes the dot product of the two vectors.  
           [0033]    Since the Lambertian model (eqn. 7) is linear, the sampling of the space of illumination conditions can be chosen to be simple—the I t  is rendered with a single point-light source illuminating the object from a pre-specified direction—n t : 
             L   t ( n )=l t δ( n−n   t )  (8) 
           [0034]    where l t  is the intensity of the light that comes from this direction. Moreover, the typical assumption is that light from any direction is equally probable, and l t  is taken to be a constant, typically unity.  
           [0035]    Sufficient sampling of the illumination space is expensive. Currently, the space of images of a given object under varying illumination conditions but a fixed pose (camera geometry) is sampled by sampling the illumination conditions. These samples are used to estimate the full space. The current approach suffers from the following problems:  
           [0036]    sampling is expensive: either by varying the light physically and taking pictures, or by rendering a 3-dimensional model;  
           [0037]    therefore, A) dense sampling is not performed, leading to an inaccurate model, or B) dense sampling is performed, which makes the system unacceptably slow.  
           [0038]    Therefore, currently the subspace approach cannot be practically applied to certain face recognition tasks.  
           [0039]    Dense sampling of the illumination space consumes impractically large time and memory. In order to make the model of the subspace more accurate, denser and denser sampling of the illumination conditions is necessary. This entails quadratic growth of the necessary storage space and cubic growth in the time of finding the enclosing linear subspace. As the accuracy of the model increases, the complexity becomes limited by the number of pixels in the image; this complexity is impractically large.  
           [0040]    Direct computation of the covariance matrix is expensive. In the regime where the complexity of the model is limited by the number of pixels in the images, an amount of computation quadratic in the number of pixels is necessary to calculate the elements of the covariance matrix. This is only practical if the individual computation is fast.  
           [0041]    The computations for a given 3D object need to be done for arbitrary viewpoint. In typical situations, many 3D objects need to be matched to a given query, for multiple queries. The viewpoints for which the respective enclosing linear subspaces need to be calculated are not known in advance, and change with every query/model pair. When the time for finding an illumination subspace for a given viewpoint is large, practical application of the method is impossible.  
         SUMMARY OF THE INVENTION  
         [0042]    The disclosed method involves a series of steps.  
           [0043]    First, for any given object  105  in said database  101 , the matrix elements of the spatial covariance matrix  201  are calculated for an arbitrary pre-determined distribution of illumination conditions. We disclose a method for such a calculation in which the matrix element for any given pair of pixels, for the most common assumption of illumination conditions, only depends on a scalar parameter—the planar angle between the surface normals at these pixels. Moreover, we disclose a method for the efficient approximation and tabulation of this dependency  206 .  
           [0044]    Second, for any given 3D object  105 , we reduce the maximum complexity of the model by first constructing the set  205  of all pairs of an albedo and a normal vector—one pair from each point on the surface of the object—and then approximating every such pair with the corresponding centers of the clusters that are the result of vector quantization  202  of this set.  
           [0045]    Third, for any 3D object  105 , we construct and diagonalize off-line a viewpoint-independent covariance matrix  201  whose complexity is large, but practical  203 . Further, we store only the leading subspace of that matrix  204 .  
           [0046]    Fourth, for any given viewpoint of a 3D object  105 , the complexity of said viewpoint-dependent illumination subspace  107  is bounded from above by some small number, which is much larger than the actual complexity, but much smaller than the number of clusters from the previous step. A viewpoint-dependent covariance matrix  301  is computed from the viewpoint-independent diagonalization results  204  and is diagonalized online in real time  302 . The leading subspace  303  of this covariance matrix is an approximation of the true solution  107 .  
           [0047]    The present invention provides the following advantages that solves those problems with prior art methods.  
           [0048]    The model is accurate because the illumination conditions are, in effect, sampled infinitely densely.  
           [0049]    The complexity is reduced in an optimal way by vector-quantizing the set of actual normal vectors.  
           [0050]    The calculation of the individual matrix element is fast, because it involves only the calculation of a cosine and a lookup of a table.  
           [0051]    Sufficient precision is maintained throughout all viewpoint-independent calculations, which need to be done only once per 3D object, and therefore can be done off line. The real-time, viewpoint-dependent computations are performed in a low-dimensional space, which makes them practically feasible.  
           [0052]    The invention allows the efficient computation of the low-dimensional linear subspaces that mostly contain the set of images that are generated by varying the illumination impinging on the surface of a 3-dimensional object, such as a human head, for many different viewpoints—relative positions of that object and the viewing camera.  
           [0053]    This is necessary in order to use in practice any object-recognition method that utilizes a low-dimensional viewpoint-dependent linear subspace.  
           [0054]    Apart from the primary use of the disclosed method as a necessary step in face recognition, it makes practically feasible the use of a low-dimensional viewpoint-dependent linear subspace for: the estimation of the lighting conditions of an image; the estimation of the pose of the 3D object in that image; the recognition of an object within a class of objects. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0055]    [0055]FIGS. 1A and 1B are general flowcharts of the prior-art matching process;  
         [0056]    [0056]FIG. 2 is a flowchart of the viewpoint-independent steps or the present invention; and  
         [0057]    [0057]FIG. 3 is a flowchart of the viewpoint-dependent steps of the present invention. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0058]    Fast Calculation of an Element of the Spatial Covariance Matrix for a Given Continuous Distribution of Illumination Conditions  
         [0059]    The results of any high-level computer-vision algorithm, such as determination of the light sources and recognition of a 3-dimensional object, critically depend on the degree of correctness of the low-dimensional subspace  107  in which the images of the target object  105  are believed to be contained.  
         [0060]    In order to build increasingly correct approximations of said sub space  107 , increasingly finer samplings of the illumination conditions are necessary. This is, however, a problem, since for every sample point a new rendering of the object needs to be made, and the rendering operation is very expensive. Also, since the direction of the illumination is parameterized by the points on a sphere—a two-dimensional surface—the number of samples on that surface, T, grows quadratically, T=O((1/d) 2 ), with the sampling quality—the inverse of the distance between the samples, d—and hence the storage space is O((1/d) 4 ), and the computational requirements, O((1/d) 6 ). That is, to increase the accuracy of the sampling ten times, the storage requirement is increased ten thousand times, and the required computational power, one million times.  
         [0061]    In order to solve the problem with the density of sampling of the illumination conditions, we disclose the following method for calculating the elements of the spatial covariance matrix  201  (eqn. 2) under infinitely dense sampling of space of illumination conditions (cf. eqn. 8).  
         [0062]    According to (eqn. 2) the same subspace can be found by either diagonalizing the “temporal” covariance matrix, C, or the “spatial” one, R. We disclose below a method to calculate, according to (eqn. 2), the element R(x, y) for infinitely dense sampling of the illumination conditions.  
         [0063]    When the image I is identified with φ, and the point-source basis is used (eqn. 8), the individual element of the sum in (eqn. 2) that corresponds to a light source L, with intensity L≡∥L∥ and coming from direction n≡L/L, is 
           I   L ( x ) I   L ( y )=α( x )α( y ) L•p ( y )≡α( x )α( y ) D ( L,p,q )  (9) 
         [0064]    Where D(L, p, q) depends on the light, and only on the value of the normals p and q, but not on which particular points of the image they appear at. Therefore, the matrix element R(x, y) is viewpoint-independent.  
         [0065]    In order to calculate D(L, p, q), we express the vectors in a coordinate system in which the z-axis is perpendicular to the plane defined by p and q, and the x-axis is along the bisector of the angle between them,  2 θ. In these coordinates 
           p= ( c,+s, 0) 
           q= ( c,−s, 0)  (10) 
           L= ( L   x   ,L   y   ,L   z ) 
         [0066]    where c=cos θ, and s=sin θ; then L•p=cL x +sL y , L•q=cL x −sL y , and finally 
           D ( L,p,q )= c   2   L   x   2 −(1 −c   2 ( L   y   2   =L^  2( c   2   n   x   2 −(1 −c   2 ) n   y   2 )≡ L   2   C ( n,c )  (11) 
         [0067]    where L is the light intensity and n is the unit vector in the light direction. Notably, the result does not depend on the vectors p and q individually, but only on the cosine of the half-angle between them, c; this forms the foundation of the method for fast calculation of the matrix element R(x, y). Also note this result (eqn. 11) does not depend on the z-component of the light.  
         [0068]    We now calculate the matrix element R(x, y) from (eqn. 2). In the limit of infinitely dense sampling, the sum  
         1   T          ∑   t                           
 
         [0069]    is replaced by an integral over all possible light directions, weighted by their respective intensities  
                     R        (     x   ,   y     )       =                  α        (   x   )            α        (   y   )              ∫     n   ∈     S   2                  L   2          (   n   )            D        (     n   ,   p   ,   q     )                         =                  α        (   x   )            α        (   y   )            E        (       u   x     ,     u   y     ,   c     )                       (   12   )                               
 
         [0070]    where u x  and u y  are the unit vectors of the special coordinate system defined by p and q (eqn. 10). Depending on the symmetry of the assumptions for the illumination conditions, E(u x , u y , c) depends on either three, two, or one parameter. Notably, it does not depend on the 3D object itself, and can be pre-calculated once for the whole database of objects.  
         [0071]    The most common case, which is also the most symmetric, allows a further simplification. With the assumption that there is no preferred direction in the illumination—light could come from any direction equally likely (cf. eqn. 8)—L(n) is a constant, and the integral (eqn. 12) no longer depends on the special coordinate system (eqn. 10); it depends only on c. In that case, it can be calculated in advance with great precision and stored in a one-dimensional table; this makes its use later essentially free. The final formula for the matrix element is then 
           R ( x,y )=α( x )α( y ) E ( c ( p ( x ), q ( y )))  (13) 
         [0072]    where the only operations involved in the calculation are table lookups, one cosine function, and two multiplications.  
         [0073]    Notably, (eqn. 12) is much more generally applicable than the point-light-source illumination conditions that were assumed in its derivation. Suppose there is an a priori distribution of illumination conditions—the probability density that l is the light intensity that comes from a particular direction nεS 2  on the unit sphere, S 2 , is P[l(n)]; this density integrates to a distribution, σ, which induces the measure dσ. Then, in the limit of infinitely dense sampling, the sum  
         1   T          ∑   t                           
 
         [0074]    is replaced by and integral over all possible illumination conditions, weighted by their respective probabilities  
                     R        (     x   ,   y     )       =                  α        (   x   )            α        (   y   )              ∫     l   ∈       (     R   +     )       S   2                     σ            ∫     n   ∈     S   2                l        (   n   )            D        (     n   ,   p   ,   q     )                             =                  α        (   x   )            α        (   y   )              ∫     n   ∈     S   2                D        (     n   ,   p   ,   q     )              ∫     L   ∈       (     R   +     )       S   2                     σ                     l        (   n   )                             =                  α        (   x   )            α        (   y   )              ∫     n   ∈     S   2                D        (     n   ,   p   ,   q     )            L        (   n   )                           (   14   )                               
 
         [0075]    which is identical to (eqn. 12); L(n) is interpreted now as the average light intensity that comes from direction n.  
         [0076]    In summary, for any assumed or estimated distribution of illumination conditions, not necessarily from point light sources, the matrix element R(x, y) (eqn. 2) does not depend on the object in question, and therefore can be pre-calculated once for database  101  and stored  206  for successive use (eqn. 12). Also, in the most complicated case, a 3-dimensional table is necessary, which is practical. Moreover, in the most common case, only a one-dimensional table is needed (eqn. 13), which requires essentially zero computational and storage resources in practice.  
         [0077]    Monte-Carlo Calculation of the Integrals Involved  
         [0078]    In order for (eqn. 12) to be useful in practice, there has to be a practical way to evaluate the integral. In the simplest case, when the illumination conditions are isotropic (eqn. 13), the integral is relatively simple 
           E ( c )=∫ nεS     2     ∩{C(n,c)&gt;0}   c   2   n   x   2 −(1 −c   2 ) n   y   2   (15) 
         [0079]    and can be evaluated analytically. Here, the condition C(n, c)&gt;0 (eqn. 11) ensures that both points are simultaneously visible from the given direction, (n). In general, since the assumed light-source distribution L(n) can be anything, even something that has been experimentally measured and does not have an analytic form to begin with, it is of interest to be able to calculate (eqn. 12) numerically. Because the result is object-independent, this can be done just once for the whole database  101 , and stored  206 .  
         [0080]    A feasible approach for the evaluation of (eqn. 15) is the Monte Carlo (MC) procedure for evaluating integrals. In this case, the same procedure is iterated over many random realizations ξεX. First a random vector is generated, r ξ ε[0, 1] 3 , that is uniformly distributed in the unit cube. The subset that is also in the unit sphere, {r ξ |||r ξ ||&lt;1}, is hence uniformly distributed there. Then, the normalized random vectors {n ξ ≡r ξ /||r ξ ||||r ξ ||&lt;1} are distributed uniformly on the surface of the unit sphere, S 2 . This procedure for generating uniformly distributed normals is relatively efficient—a large fraction, 4π/(3*8)=52% of the vectors r ξ  can be used to produce useful normal vectors, n ξ .  
         [0081]    The next step is to discard those light directions for which one or both of the surface points are not visible. This is achieved by the test for the condition C(n, c)&gt;0. For the rest of the directions, from which both surface points are visible, the integrand is evaluated and accumulated.  
         [0082]    Another computational saving can be achieved, which is grounded in the fact that the joint-visibility condition, C(n, c), is a monotonically increasing function of c. This allows one to find, for any random light direction, n ξ , the value c ξ |C(n ξ , c ξ )=0, which defines the border of the joint-visibility, and then update all intermediate results for the integrals {E(c)|c&gt;c ξ }. Thus, the integrals (eqn. 15) are calculated simultaneously, in the same MC procedure.  
         [0083]    Finally the results are tabulated and stored  206  for the subsequent calculation of R(x, y) for said plurality of objects  101 .  
         [0084]    Optimal Dimensionality Reduction By Vector Quantization  
         [0085]    An efficient method is disclosed above to calculate the element R(x, y) of the spatial covariance matrix  201  (eqn. 2) when the space of illumination conditions is sampled infinitely densely. In order for that method to be useful in practice, the covariance matrix R needs to be diagonalized in reasonable time. In a typical picture, there are V=480×640˜300,000 pixels. Even if the object occupies only a fraction of the picture, say 10%, this still leaves too large a matrix to be subsequently diagonalized. Here we disclose a method to reduce the dimensionality of the problem and at the same time optimally retain the illumination subspace.  
         [0086]    This method is based on the fact that when the elements of a matrix are perturbed slightly, then the perturbations in the eigenvectors and eigenvalues are also small, and are bounded if the perturbation in the matrix elements is bounded. See R. Everson et al., supra. Below we disclose a method for finding a family of optimally small perturbations that decrease the dimensionality of the system, and thus, the time to diagonalize it.  
         [0087]    Indeed, when the normal vector to the surface p(x) at the point x is close to some reference normal {circumflex over (q)}, and also its albedo α(x) is close to a reference albedo {circumflex over (α)} q , then the substitution of α(x) and p(x) with {circumflex over (α)} q  and {circumflex over (q)}, respectively, in (eqn. 13) would lead to a small perturbation of R(x, y). Then, if many image points have albedos and normals that are close to the same reference albedo and normal-vector pair, substituting all of them will lead to a series of small perturbations in the matrix elements of R, but will also reduce the degrees of freedom in the problem.  
         [0088]    Thus, a solution to the large-dimensionality problem is to find a set of Q pairs of reference values  207  for the albedo and normals, {({circumflex over (α)} q , {circumflex over (q)})} qεQ  such that, for a given value of Q, the perturbations are smallest. This is exactly the setting of a variety of vector quantization algorithms—given a set of vectors, {(α(x), p(x))} xεV  in our case  205 , these algorithms cluster them together in Q clusters, and find their centroids  207 , {({circumflex over (α)} q , {circumflex over (q)})} qεQ , such that the average distance from the vectors to the nearest respective cluster-centroid, q(x)≡q x , is minimal.  
         [0089]    There are many algorithms for VQ, and most are suitable as a step in the solution of our problem. We highlight two of them here as preferred practice: the Linde-Buzo-Gray algorithm, which benefits from a straightforward and conceptually clean implementation and has been found to perform well in practice, and the Deterministic Annealing (DA) algorithm which has better speed, convergence, and optimality properties, albeit with a more complicated implementation. See Y. Linde et al., “An Algorithm for Vector Quantizer Design”,  IEEE Transactions on Communications , Vol. 28, pp. 84-95 (1980); and K. Rose, “Deterministic Annealing for Clustering, Compression, Classification, Regression, and Related Optimization Problems”,  Proceedings of IEEE , Vol. 86, No. 11, pp. 2210-2239 (1998).  
         [0090]    In summary, for any object  105  in said database  101 , the set is generated of all of pairs of normals and albedo  205 . Then, for any desired complexity, Q, this set is clustered into Q clusters  207 , and the original matrix elements R(x, y) are perturbed, according to (eqn. 13), to 
           R ( x,y )→ R   Q ( x,y )(= R   Q ( q   x   ,q′   y )≡{circumflex over (α)} q     x   {circumflex over (α)} q′     y     E ( c ({circumflex over (q)} x   ,{circumflex over (q)}′   y ))  (16) 
         [0091]    Notably, the rank of the V×V matrix R Q (q x , q′ y ) is the same as that of the Q×Q matrix  208  with elements {square root}{square root over (N q N′)} q R Q (q, q′), where N q  is the number of image points that are clustered together in the cluster q; this rank is at most Q, which makes the diagonalization practical, for moderately large values of Q. Moreover, the eigenvalues of these two matrices are the same, and their eigenvectors are simply related—once the smaller set is known  209 , the larger set  210  is obtained by expansion in the number of elements and resealing of their values. Therefore, the diagonalization of R Q  (eqn. 16) requires at most O(Q 3 ) computational power, which is much better thanO(V 3 ). It is practical for relatively large values of Q, on the order of 10,000. Notably, since this calculation has to be done only once per object  105 , as opposed to once for every query  103  in said plurality of queries  102 , it can be performed off line and the results stored  302 . Also, since typically Q&lt;&lt;V, the storage requirements, —for O(Q 2 ) elements, —are reasonable in practice.  
         [0092]    Optimal Viewpoint-Independent Dimensionality Reduction  
         [0093]    A method is disclosed to reduce the dimensionality of the spatial covariance matrix  201  (eqn. 2), and at the same time preserve most of its eigen-structure  210 . Although vector quantization with Q˜10,000 and the subsequent diagonalization of a system of that size is feasible, it cannot be performed in real time, nor, many times for every query  103 —once for every different target object in said database  101 . In this section we disclose a method to speed up the calculation of the viewpoint-dependent illumination subspace  107 .  
         [0094]    The method is based on the fact that the rendering process has two steps. On one hand, features on the 3D surface of the object are mapped to locations on the 2D image plane of the camera. That is, any property φ(u, v)≡φ(u) that is associated with the surface point u≡(u, v) is mapped to a property on the image plane {tilde over (φ)}(x), through the viewpoint-dependent warp x(u). This warp  304  is entirely a geometric entity, and depends only on the viewpoint.  
         [0095]    On the other hand, the surface properties φ interact with the light, and possibly the viewpoint, to give rise to image features {tilde over (φ)}. Therefore, to the extent that the surface properties do not interact with the viewpoint, all calculations can be made in the viewpoint-independent coordinate system (u, v), and, at a later stage, warped to the image plane in a viewpoint-dependent manner. Hence, any basis of the viewpoint-dependent illumination subspace is a warp of a viewpoint-independent basis, defined on the surface of the object. Since the warps preserve neither the lengths of the vectors in the basis, nor the angles between them, a warp of a viewpoint-independent eigenbasis hierarchy (eqn. 2) does not necessarily result in a viewpoint-dependent eigenbasis hierarchy. Nevertheless, for any warp, there is region on the surface that maps more or less linearly to most of the area on the image plane. Hence, if the viewpoint-independent eigenbasis describes well that region on the surface, its warp will describe well the image. This is the foundation for the following method.  
         [0096]    For every object in said database  101 , generate  201  and diagonalize  210  its vector-quantized viewpoint-independent covariance matrix  208 , R Q , and from it, determine the eigenbasis hierarchy defined on the surface of the object  
               R        (     u   ,   v     )       ≈       ∑     r   =   1     Q              ψ   r          (   u   )            σ   r   2            ψ   r          (   v   )                   (   17   )                               
 
         [0097]    Further, choose a cutoff, N, such that the average residual power, tr(R N ) (eqn. 4), is sufficiently small. Ratios of residual to noise power in the [0.1-10] range work well in practice. Keep only the first N eigensurfaces and store them for subsequent use  204 . Because this computation is viewpoint-independent, it can be done off line, once per object  105  in said database  101 , and the results, stored  204 . Typically, even though Q˜10,000 is practical as an off line computation, only the first N˜100 eigen-surfaces need to be preserved.  
         [0098]    Finally, at the viewpoint-dependent stage—when a query  103  at a particular viewpoint needs to be matched to the objects in said database—warp  304  the eigen-surfaces  204  (eqn. 17) to a basis  301  of the viewpoint-dependent illumination subspace  107   
         {tilde over (ψ)} r ( x )={tilde over (ψ)} r ( x ( u ))  (18) 
         [0099]    These final warps can be implemented as lookups in a table, which is very fast.  
         [0100]    Efficient Generation of Many Viewpoint-Dependent Subspaces from the Viewpoint-Independent One  
         [0101]    An efficient method is disclosed to generate a low-dimensional basis of the viewpoint-dependent illumination subspace  107  (eqn. 18) from the pre-computed viewpoint-independent hierarchy of eigen-surfaces  204  (eqn. 17). Although this can be done for a dimensionality as low as N≈100, which is the recommended practice, reduction of N much further is not possible, because all areas of the surface need to be represented sufficiently well. On the other hand, there is theoretic (see R. Basri et al., supra.) and experimental evidence that the matching can be performed in a space with dimension M, 4≦M≦9. See P. Hallinan, supra.; A. Georghiades et al., supra.; R. Ishiyama et al., supra.; and R. Basri et al., supra. M is the final dimensionality of the viewpoint-dependent subspace in which recognition is performed. The value of M can be up to 20. In the preferred embodiment, M is between 4 and 9, but values up to 11 have been used. In addition, the value of N should be such that 2M≦N≦8M. We disclose an efficient method to find the leading M-dimensional viewpoint-dependent eigen-subspace of the N-dimensional viewpoint-dependent non-eigen-subspace.  
                 R        (     x   ,   y     )       =       ∑     p   =   1     M                ψ   _     p          (   x   )              σ   _     p              ψ   _     p          (   y   )                  
        where        
                σ   _     p              ψ   _     p          (   x   )         =       ∑     r   =   1     N            U   pr              ψ   ~     r          (   x   )                     (   19   )                               
 
         [0102]    where {overscore (σ)} p  and U pr  are determined by the eigenvalue decomposition of an N×N matrix B rs  where  
         B   rs     =         ∑   x                ψ   ~     r          (   x   )            σ   r          σ   s              ψ   ~     s          (   x   )           =       U   pr          σ   p   2          U     p                 s                                 
 
         [0103]    Since this viewpoint-dependent step requires a diagonalization of a matrix of dimensionality N≈100, it is very fast and can be performed on line, many times, once for each trial match between the query  103  and the objects in said database  101 .  
         [0104]    This allows the off-line computation and storage of the results  204  of the most time-consuming steps in the generation of the illumination subspace  107 , once for every object  105  in said database  101 , and their subsequent use for efficient fast on-line matching  306  of said plurality of queries  102 .  
         [0105]    While there has been shown and described what is considered to be preferred embodiments of the invention, it will, of course, be understood that various modifications and changes in form or detail could readily be made without departing from the spirit of the invention. It is therefore intended that the invention be not limited to the exact forms described and illustrated, but should be constructed to cover all modifications that may fall within the scope of the appended claims.