IEEE Conf. on Compute Vision and Patten Recognition, 1998. To appea. Illumination Cones fo Recognition Unde Vaiable Lighting: Faces Athinodoos S. Geoghiades David J. Kiegman Pete N. Belhumeu Cente fo Computational Vision and Contol Depatment of Electical Engineeing Yale Univesity New Haven, CT 06520-8267 Abstact Due to illumination vaiability, the same object can appea damatically dieent even when viewed in xed pose. To handle this vaiability, an object ecognition system must employ a epesentation that is eithe invaiant to, o models this vaiability. This pape pesents an appeaance-based method fo modeling the vaiability due to illumination in the images of objects. The method dies fom past appeaance-based methods, howeve, in that a small set of taining images is used to geneate a epesentation { the illumination cone { which models the complete set of images of an object with Lambetian eectance unde an abitay combination of point light souces at innity. This method is both an implementation and extension (an extension in that it models cast shadows) of the illumination cone epesentation poposed in [3]. The method is tested on a database of 660 images of 10 faces, and the esults exceed those of popula existing methods. 1 Intoduction An object's appeaance depends in lage pat on the way in which it is viewed. Often slight changes in pose and illumination poduce lage changes in an object's appeaance. While thee has been a geat deal of liteatue in compute vision detailing methods fo handling image vaiation poduced by changes in pose, few eots have been devoted to image vaiation poduced by changes in illumination. Fo the most pat, object ecognition algoithms have eithe ignoed illumination vaiation, o dealt with it by measuing some popety o featue of the image { e.g., edges o cones { which is, if not invaiant, at least insensitive to the vaiability. Yet, edges and cones do not contain all of the infomation useful fo ecognition. Futhemoe, objects which ae not simple polyheda o ae not composed of piecewise constant albedo pattens often poduce inconsistent edge and cone maps. Methods have ecently been intoduced which use low-dimensional epesentations of images of objects to pefom ecognition, see fo example [5, 11, 16]. These methods, often temed appeaance-based methods, die fom the featue-based methods mentioned above in that thei low-dimensional epesentation is, in a least-squaed sense, faithful to the oiginal image. Systems such as SLAM [11] and Eigenfaces [16] have demonstated the powe of appeaance-based methods both in ease of implementation and in accuacy. Yet these methods sue fom an impotant dawback: ecognition of an object (o face) unde a paticula pose and lighting can be pefomed eliably povided that object has been peviously seen unde simila cicumstances. In othe wods, these methods in thei oiginal fom have no way of extapolating to novel viewing conditions. The \illumination cone" method of [3] is, in spiit, an appeaance-based method fo ecognizing objects unde exteme vaiability in illumination. Howeve, the method dies substantially fom pevious methods in that a small numbe of images of each object unde small changes in lighting is used to geneate a epesentation, the illumination cone, of all images of the object (in xed pose) unde all vaiation in illumination. This pape focuses on issues fo building the illumination cone epesentation fom taining images and using it fo ecognition. While the stuctue of the set of images unde vaiable illumination was chaacteized in [3] and the elevant esults ae summaized in Sec. 2, no methods fo pefoming ecognition wee pesented. In this pape, such ecognition algoithms ae intoduced. Futhemoe, the cone epesentation is extended to explicitly model cast shadows poduced by objects which have non-convex shapes. This extension is non-tivial, equiing that the suface nomals fo the objects be ecoveed up to a shadow peseving genealized baselief (GBR) tansfomation. The eectiveness of these algoithms and the cone epesentation ae validated within the context of face ecognition { it has been obseved by Moses, Adini and Ullman that the vaiability in an image due to illumination is often geate than that due to a change in the peson's identity [10]. Figue 1 shows the vaiability fo a single individual. It has been obseved that methods fo face ecognition based on nding local image featues and using thei geometic elation ae geneally ineective [4]. Hence, faces povide an inteesting and useful class of objects fo testing the powe of the illumination cone epesentation. In this pape, we empiically compae these new methods to a numbe of popula techniques such as coelation [4] and Eigenfaces [9, 16] as well as moe ecently developed techniques such as distance to linea subspace [2, 5, 12, 13]; the latte technique has been shown to be much less sensitive to illumination
vaiation than the fome. Howeve, these methods also beak down as shadowing becomes vey signicant. As we will see, the pesented algoithm based on the illumination cone outpefoms all of these methods on a database of 660 images. It should be noted that ou objective in this wok is to focus solely on the issue of illumination vaiation wheeas othe appoaches have been moe concened with issues elated to lage image databases, face nding, pose, and facial expessions. 2 The Illumination Cone In ealie wok, it was shown that fo an object with convex shape and Lambetian eectance, the set of all images unde an abitay combination of point light souces foms a convex polyhedal cone in the image space IR n. This cone can be constucted fom as few as thee images [3]. Hee we summaize the elevant esults. To begin, conside a convex object with a Lambetian eectance function which is illuminated by a single point souce at innity. Let x 2 IR n denote an image of this object with n pixels. Let B 2 IR n3 be a matix whee each ow of B is the poduct of the albedo with the inwad pointing unit nomal fo a point on the suface pojecting to a paticula pixel in the image. A point light souce at innity can be epesented by s 2 IR 3 signifying the poduct of the light souce intensity with a unit vecto in the diection of the light souce. A convex Lambetian suface with nomals and albedo given by B, illuminated by s, poduces an image x given by x = max(bs; 0); (1) whee max(:; 0) sets to zeo all negative components of the vecto Bs. The pixels set to zeo coespond to the suface points lying in an attached shadow. Convexity of the object's shape is assumed at this point to avoid cast shadows (shadows that the object casts on itself). While attached shadows ae dened by local geometic condition, cast shadows must satisfy a global condition. When no pat of the suface is shadowed, x lies in the 3-D subspace L given by the span of the matix B. It can be shown that the subset L 0 L having no shadows (i.e., falling in the non-negative othant 1 ) foms a convex cone [3]. The illumination subspace L slices though othe othants as well as the non-negative othant. Let L i be the intesection of the illumination subspace L with an othant i in IR n though which L passes. Cetain components of x 2 L i ae always negative and othes always geate than o equal to zeo. Since im- 1 By othant we mean the high-dimensional analogue to quadant, i.e., the set fxjx 2 IR n, with cetain components of x 0 and the emaining components of x < 0g. By nonnegative othant we mean the set fxjx 2 IR n, with all components of x 0g. age intensity is always non-negative, the image coesponding to points in L i is fomed by the pojection P i given by Equation 1. The pojection P i is such that it leaves the non-negative components of x 2 L i untouched, while the negative components of x become zeo. The pojected set P i (L i ) is also a convex cone. L intesects at most n(n 1) + 2 othants [3], and so the set of images ceated by vaying the diection and stength of a single light souce at innity is given by the union of at most n(n 1) + 2 convex cones, each of which is at most thee dimensional. If an object is illuminated by k light souces at in- nity, then the image is given by the supeposition of the images which would have been poduced by the individual light souces, i.e., x = kx i=1 max(bs i ; 0) (2) whee s i is a single light souce. It follows that the set of all possible images C of a convex Lambetian suface ceated by vaying the diection and stength of an abitay numbe of point light souces at innity is a convex cone. Futhemoe, it is shown in [3] that any image in the cone C (including the bounday) can be found as a convex combination of exteme ays given by whee x ij = max(bs ij ; 0); (3) s ij = b i b j : (4) The vectos b i and b j ae the ows of B with i 6= j. It is clea that thee ae at most m(m 1) exteme ays (images) fo m n independent suface nomals. Since thee is a nite numbe of exteme ays, the convex cone is polyhedal. 3 Constucting the Illumination Cone Equations 3 and 4 suggest a way to constuct the illumination cone fo each individual: gathe thee o moe images of the face unde vaying illumination without shadowing and use these images to estimate the thee-dimensional illumination subspace L. One way of estimating this is to nomalize the images to be of unit length, and then use singula value decomposition (SVD) to estimate the best thee-dimensional othogonal basis B in a least squae sense. Note that the basis B dies fom B by an unknown linea tansfomation, i.e., B = B A whee A 2 GL(3); fo any light souce, x = Bs = (B A)(A 1 s). Nonetheless fom B, the exteme ays dening the illumination cone C can be computed using Equations 3 and 4. This method, intoduced in [3], was named the illumination subspace method. The st poblem that aises with the above pocedue is with the estimation of B. Fo even a convex object whose Gaussian image coves the Gauss
Subset 1 Subset 2 Subset 3 Subset 4 Subset 5 Figue 1: Example images fom each subset of the Havad Database used to test the algoithms. sphee, thee is only one light souce diection (the viewing diection) fo which no point on the suface is in shadow. Fo any othe light souce diection, shadows will be pesent. Fo faces, which ae not convex, shadowing in the modeling images is likely to be moe ponounced. When SVD is used to estimate B fom images with shadows, these systematic eos can bias the estimate of B signicantly. Theefoe, altenative ways ae needed to estimate B that take into account the fact that some data values should not be used in the estimation. The next poblem is that usually m, the numbe of independent nomals in B, can be lage (moe than a thousand) hence the numbe of exteme ays needed to completely dene the illumination cone can un in the millions. Theefoe, we must appoximate the cone in some fashion; in this wok, we choose to use a small numbe of exteme ays (images). In [3] it was shown empiically that the cone is at (i.e., elements lie nea a low dimensional linea subspace), and so the hope is that a subsampled cone will povide an appoximation that leads to good ecognition pefomance. In ou expeience, aound 60-80 images ae sucient, povided that the coesponding light souce diections s ij ae moe o less unifom on the illumination sphee. The esulting cone C is a subset of the object's tue cone C. An altenative appoximation to C can be obtained by diectly sampling the space of light souce diections athe than geneating the samples though Eq. 4. While the esulting images fom the exteme ays of the epesentation C and lie on the bounday of C, they ae not necessaily exteme ays of C. Again C is a subset of C. The last poblem comes fom the fact that faces ae non-convex, and so cast shadows cove signicant potions of the face unde exteme illumination (See the images fom Subsets 4 and 5 in Fig. 1). The image fomation model (Eq. 1 used to develop the illumination cone does not account fo cast shadows. Fo the light souce diections of the exteme ays given by Equation 4, we can pedict which pixels will be in cast shadows. It has been shown [1, 17] that fom multiple images whee the light souce diections ae unknown, one can only ecove a Lambetian suface up to a thee-paamete family given by the genealized baselief (GBR) tansfomation. This family scales the elief (attens o extudes) and intoduces an additive plane. Consequently, when computing s ij fom B, the light souce diection dies fom the tue light souce by a GBR tansfomation. Since shadows ae peseved unde these tansfomation [1], images synthesized fom a suface whose nomal eld is given by B unde light souce s ij will have coect shadowing. Thus, in constucting the exteme ays of the cone, we st econstuct a suface (a height function) and then use ay-tacing techniques to detemine which points lie in a cast shadow. It should be noted that the vec-
to eld B estimated via SVD may not be integable, and so pio to econstucting the suface up to GBR, integability of B is enfoced. This leads to the following steps fo constucting a epesentation of the illumination cone of an individual fom a set of images taken unde unknown lighting. Details of these steps ae given below. 1. Estimate B fom taining images. 2. Enfoce integability of B. 3. Reconstuct the suface up to GBR. 4. Fo a set of light souce diections that unifomly sample the sphee, synthesize exteme ays (images) of the cone that account fo cast and attached shadows. 3.1 Estimating B Using singula value decomposition diectly on the images leads to a biased estimate of B due to shadows. In addition, potions of some of the images fom the Havad database wee satuated. Both shadows fomed unde a single light souce and satuations can be detected by thesholding and labeled as \missing" { these pixels do not satisfy the linea equation x = Bs. Thus, we need to estimate the 3-D linea subspace B with known missing values. Dene the data matix fo c images of an individual to be X = [x 1 : : : x c ]. If thee wee no shadowing, X would be ank 3 and we could use SVD to decompose X into X = B S whee S is a 3c matix of the light souce diection fo all c images. To estimate a basis B fo the 3-D linea subspace L fom image data with missing elements, we have implemented a vaiation of the algoithm pesented by Shum, Ikeuchi, and Reddy [14]; see also the methods of Tomasi and Kanade [15] and Jacobs [8]. The oveview of this method is as follows: without doing any ow o column pemutations sift out all the full ows (with no invalid data) of matix X to fom a full sub-matix X. ~ Pefom SVD on X ~ and get an estimate of S. Fix S and estimate each of the ow of B independently using least squaes. Then, x B and estimate each of the light souce diection s i independently. Repeat last two steps until estimates convege. The inne wokings of the algoithm ae given as follows: Let b i be the ith ow of B, let x i be the ith ow of X. Let p be the indices of non-missing elements in x i, and let x p i be the ow obtained by taking only the non-missing elements of x i, and let S p similaly be the submatix of S consisting of ows with indices in p. Then, each ow of B is given by b i = (S p ) y (x p i )T whee (S p ) y is the pseudo-invese of S p. With the new estimate of B at hand, we now let x j be the jth column of X, let p be the indices of non-missing elements in x j, and let x p j be the column obtained by taking only the non-missing elements of x j. Let B p similaly be the submatix of B consisting of ows with indices in p. Then, the light souce diections ae given by, s j = (B p ) y (x p ) j Afte the new set of light souce S has been calculated, the last two steps can be epeated until the estimate of B conveges. The algoithm is quite well behaved conveging to the global minimum within 10-15 iteations. Though it is possible to convege to a local minimum, we neve obseved this in simulation o in pactice. 3.2 Enfocing Integability To pedict cast shadows, we must econstuct a suface and to do this, the vecto eld B must coespond to an integable nomal eld. Since no method has been developed to enfoce integability duing the estimation of B, we enfoce it aftewads. That is, given B computed as descibed above, we estimate a matix A 2 GL(3) such that B A coesponds to an integable nomal eld; the development follows [17]. Conside a continuous suface dened as the gaph of a function z(x; y), and let b be the coesponding nomal eld scaled by an albedo (scala) eld. The integability constaint fo a suface is z xy = z yx whee subscipts denote patial deivatives. In tun, b must satisfy: b1 b 3 y = b2 b 3 x To estimate A such that b T (x; y) = b T (x; y)a coesponds to a suface, we expand this out. Letting the columns of A be denoted by A 1 ; A 2 ; A 3 yields (b T A 3 )(b T x A 2) (b T A 2 )(b T x A 3) = (b T A 3 )(b T y A 1 ) (b T A 1 )(b T y A 3 ) which can be expessed as b T S 1 b x = b T S 2 b y (5) whee S 1 = A 3 A T 2 A 2A T 3 and S 2 = A 3 A T 1 A 1A T 3. S 1 and S 2 ae skew-symmetic matices, and so they each have thee degees of feedom. Equation 5 is linea in the six elements of S 1 and S 2. Fom the estimate of B obtained using the method in Section 3.1, discete appoximations of the patial deivatives (b x and b y) ae computed, and then SVD is used to solve fo the six elements of S 1 and S 2. In [17], it was shown that the elements of S 1 and S 2 ae cofactos of A, and a simple method fo computing A fom the cofactos was pesented. This pocedue only detemines six degees of feedom of A. The othe thee coespond to the genealized bas elief (GBR) tansfomation [1] and can be chosen abitaily since GBR peseves integability. The suface coesponding to B A dies fom the tue suface by GBR, i.e., z (x; y) = z(x; y) + x + y fo abitay ; ; with 6= 0.
3.3 Geneating the Height Function Having estimated the matix B and then enfocing integability, we now calculate the height function z(x; y) of the face so that cast shadows can be pedicted. Note that the econstuction of the height is not Euclidean, but a epesentative element of the obit unde a GBR tansfomation. Fo each nomal b i the deivatives of z(x; y) with espect to x and y ae given by the following equations p = @z @x = b i1 b i3 ; q = @z @y = b i2 b i3 : In ode to nd z(x; y), we use the vaiational appoach pesented in [7]. A suface z(x; y) is t to the given components of the gadient p and q by minimizing the functional Z Z (z x p) 2 + (z y q) 2 dx dy; whose Eule equation educes to 2 z = p x + q y. We need to constain the solution of the Eule equation, and this is achieved by the following natual bounday conditions (z x ; z y ) n = (p; q) n whee n = ( dy=ds; dx=ds) is a nomal vecto to the bounday cuve @, and s is the ac-length along the bounday [7]. Thus, the component of (z x ; z y ) nomal to the chosen bounday cuve must match the nomal component of (p; q). An iteative scheme using a discete appoximation of the Laplacian can be used to geneate a height function of the face [7]. Once the height function has been detemined, it is a simple matte to modify the illumination cone epesentation to incopoate cast shadows. Using aytacing techniques, we can detemine the cast shadow egions and coect the exteme ays of C. Figue 2 demonstates the pocess of constucting the cone C. Figue 2.a shows the taining images fo one individual in the database. Figue 2.b shows the columns of the matix B. Figue 2.c shows the econstuction of the suface up to a GBR tansfomation. The left column of Fig. 2.d shows sample images in the database; the middle column shows the closest image in the illumination cone without cast shadows; and the ight column shows the closest image in the illumination cone with cast shadows. 4 Recognition The cone C can be used in a natual way fo face ecognition, and in expeiments descibed below, we compae thee ecognition algoithms to the poposed method. Fom a set of face images labeled with the peson's identity (the leaning set) and an unlabeled set of face images fom the same goup of people (the test set), each algoithm is used to identify the peson in the test images. Fo moe details of the compaison algoithms, see [2]. We assume that the face has been located and aligned within the image. a. b. c. d. Figue 2: The gue demonstates the pocess of constucting the cone C. a) the taining images. b) matix B. c) econstuction up to a GBR tansfomation. d) sample images fom database (left column); closest image in illumination cone without cast shadows (middle column); and closest image in illumination cone with cast shadows (ight column). The simplest ecognition scheme is a neaest neighbo classie in the image space [4]. An image in the test set is ecognized (classied) by assigning to it the
label of the closest point in the leaning set, whee distances ae measued in the image space. If all of the images ae nomalized to have zeo mean and unit vaiance, this pocedue is equivalent to choosing the image in the leaning set that best coelates with the test image. Because of the nomalization pocess, the esult is independent of light souce intensity. As coelation methods ae computationally expensive and equie geat amounts of stoage, it is natual to pusue dimensionality eduction schemes. A technique now commonly used in compute vision { paticulaly in face ecognition { is pincipal components analysis (PCA) which is populaly known as Eigenfaces [5, 11, 9, 16]. Given a collection of taining images x i 2 IR n, a linea pojection of each image y i = W x i to an f-dimensional featue space is pefomed. A face in a test image x is ecognized by pojecting x to the featue space, and neaest neighbo classication is pefomed in IR f. The pojection matix W is chosen to maximize the scatte of all pojected samples. It has been shown that when f equals the numbe of taining images, the Eigenface and Coelation methods ae equivalent (See [2, 11]). One poposed method fo handling illumination vaiation in PCA is to discad fom W the thee most signicant pincipal components; in pactice, this yields bette ecognition pefomance [2]. A thid appoach is to model the illumination vaiation of each face as a thee-dimensional linea subspace L as descibed in Section 2. To pefom ecognition, we simply compute the distance of the test image to each linea subspace and choose the face coesponding to the shotest distance. We call this ecognition scheme the Linea Subspace method [1]; it is a vaiant of the photometic alignment method poposed in [13], and elated to [6, 12]. While this is expected to model the vaiation in intensity when the suface is completely illuminated, it does not model shadowing. Finally, given a test image x, ecognition using illumination cones is pefomed by st computing the distance of the test image to each cone, and then choosing the face that coesponds to the shotest distance. Since each cone is convex, the distance can be found by solving a convex optimization poblem. In paticula, the non-negative linea least squaes technique contained in Matlab was used in ou implementation, and this algoithm has computational complexity O(n e 2 ) whee n is the numbe of pixels and e is the numbe of exteme ays. 5 Expeimental Results To test the eectiveness of these ecognition algoithms, we pefomed a seies of expeiments on a database fom the Havad Robotics Laboatoy in which lighting had been systematically vaied [5, 6]. In each image in this database, a subject held his/he head steady while being illuminated by a dominant light souce. The space of light souce diections, Subset 1 Subset 2 Subset 3 Subset 4 Subset 5 Figue 3: The highlighted lines of longitude and latitude indicate the light souce diections fo Subsets 1 though 5. Each intesection of a longitudinal and latitudinal line on the ight side of the illustation has a coesponding image in the database. which can be paameteized by spheical angles, was then sampled in 15 incements. See Figue 3. Fom this database, we used 660 images of 10 people (66 of each). We extacted ve subsets to quantify the eects of vaying lighting. Sample images fom each subset ae shown in Fig. 1. Subset 1 (espectively 2, 3, 4, 5) contains 30 (espectively 90, 130, 170, 210) images fo which both of the longitudinal and latitudinal angles of light souce diection ae within 15 (espectively 30 ; 45 ; 60 ; 75 ) of the camea axis. All of the images wee copped within the face so that the contou of the head was excluded. Fo the Eigenface and coelation tests, the images wee nomalized to have zeo mean and unit vaiance, as this impoved the pefomance of these methods. Fo the Eigenface method, we used twenty pincipal components { ecall that pefomance appoaches coelation as the dimension of the featue space is inceased [2, 11]. Since the st thee pincipal components ae pimaily due to lighting vaiation and since ecognition ates can be impoved by eliminating them, eo ates ae also pesented when pincipal components fou though twenty-thee ae used. Fo the cone expeiments, we tested two vaiations: in the st vaiation (cones-attached), the epesentation was constucted ignoing cast shadows and so exteme ays wee geneated diectly fom Eq. 3. In the second vaiation (Cones-cast), the epesentation was constucted as descibed in Section 3. Mioing the extapolation expeiment descibed in [2], each method was tained on samples fom Subset 1 and then tested using samples fom Subsets 2, 3, 4 and 5. (Note that when tested on Subset 1, all methods pefomed without eo). Figue 4 shows the esult fom this expeiment. 6 Discussion Fom the esults of this expeiment, we daw the following conclusions: The illumination cone epesentation outpefoms all of the othe techniques.
E o a t e 70 60 50 40 30 20 10 Eigenface (20) Coelation Eigenface (20) w/o fist 3 Linea Subspace Cones Attached Cones Cast 0 Subset 1 (15) Subset 2 (30) Subset 3 (45) Subset 4 (60) Lighting Diection Subset (Degees) Extapolating fom Subset 1 Eigenface (20) Coelation Eigenface (20) w/o fist 3 Linea Subspace Cones Attached Cones Cast Method Eo Rate (%) Subset Subset Subset Subset 2 3 4 5 Coelation 8.9 40.8 65.9 84.1 Eigenface 8.9 45.4 67.1 84.1 Eigenface 6.7 33.8 55.3 78.1 w/o 1st 3 Linea subspace 0.0 3.8 22.4 50.7 Cones-attached 0.0 2.3 17.1 44.8 Cones-cast 0.0 0.0 10.0 37.0 Figue 4: Extapolation: When each of the methods is tained on images with nea fontal illumination (Subset 1), the gaph and coesponding table show the elative pefomance unde exteme light souce conditions. When cast shadows ae included in the illumination cone, eo ates ae impoved. Fo vey exteme illumination (Subset 5), the Coelation and Eigenface methods completely beak down, and exhibit esults that ae slightly bette than chance (90% eo ate). The cone method pefoms signicantly bette, but cetainly not well enough to be usable in pactice. At this point, moe expeimentation is equied to detemine if ecognition ates can be impoved by eithe using moe sampled exteme ays o by impoving the image fomation model. The expeiment descibed above was limited to the available dataset fom the Havad Robotics Laboatoy. To pefom moe extensive expeimentation, we ae constucting a geodesic lighting ig that suppots 64 xenon stobes. Using this ig, we will be able to modify the illumination at fame ates and gathe an extensive image database coveing a boade ange of lighting conditions including multiple souces. The speed of acquisition will also pemit us to eadily obtain images of a lage numbe of individuals. We will then pefom moe extensive expeimentation with this newly gatheed database. Refeences [1] P. Belhumeu, D. Kiegman, and A. Yuille. The baselief ambiguity. In Poc. IEEE Conf. on Comp. Vision and Patt. Recog., pages 1040{1046, 1997. [2] P. N. Belhumeu, J. P. Hespanha, and D. J. Kiegman. Eigenfaces vs. Fishefaces: Recognition using class specic linea pojection. IEEE Tans. Patten Anal. Mach. Intelligence, 19(7):711{720, 1997. Special Issue on Face Recognition. [3] P. N. Belhumeu and D. J. Kiegman. What is the set of images of an object unde all possible lighting conditions. In Poc. IEEE Conf. on Comp. Vision and Patt. Recog., pages 270{277, 1996. [4] R. Bunelli and T. Poggio. Face ecognition: Featues vs templates. IEEE Tans. Patten Anal. Mach. Intelligence, 15(10):1042{1053, 1993. [5] P. Hallinan. A low-dimensional epesentation of human faces fo abitay lighting conditions. In Poc. IEEE Conf. on Comp. Vision and Patt. Recog., pages 995{999, 1994. [6] P. Hallinan. A Defomable Model fo Face Recognition Unde Abitay Lighting Conditions. PhD thesis, Havad Univesity, 1995. [7] B. Hon and M. Books. The vaiational appoach to shape fom shading. Compute Vision, Gaphics and Image Pocessing, 35:174{208, 1992. [8] D. Jacobs. Linea tting with missing data: Applications to stuctue-fom-motion and to chaacteizing intensity images. In CVPR97, pages 206{212, 1997. [9] L. Siovitch and M. Kiby. Low-dimensional pocedue fo the chaacteization of human faces. J. Optical Soc. of Ameica A, 2:519{524, 1987. [10] Y. Moses, Y. Adini, and S. Ullman. Face ecognition: The poblem of compensating fo changes in illumination diection. In Euopean Conf. on Compute Vision, pages 286{296, 1994. [11] H. Muase and S. Naya. Visual leaning and ecognition of 3-D objects fom appeaence. Int. J. Compute Vision, 14(5{24), 1995. [12] S. Naya and H. Muase. Dimensionality of illumination in appeaance matching. IEEE Conf. on Robotics and Automation, 1996. [13] A. Shashua. On photometic issues to featue-based object ecognition. Int. J. Compute Vision, 21:99{ 122, 1997. [14] H. Shum, K. Ikeuchi, and R. Reddy. Pincipal component analysis with missing data and its application to polyhedal object modeling. PAMI, 17(9):854{867, Septembe 1995. [15] C. Tomasi and T. Kanade. Shape and motion fom image steams unde othogaphy: a factoization method. Intenational Jounal of Compute Vision, 9(2):134{154, 1992. [16] M. Tuk and A. Pentland. Eigenfaces fo ecognition. J. of Cognitive Neuoscience, 3(1), 1991. [17] A. Yuille and D. Snow. Shape and albedo fom multiple images using integability. In Poc. IEEE Conf. on Comp. Vision and Patt. Recog., pages 158{164, 1997.