Comparing Images Under Variable Illumination

Transcription

1 ( This paper appeared in CVPR 8. IEEE ) Comparing Images Under Variable Illumination David W. Jaobs Peter N. Belhumeur Ronen Basri NEC Researh Institute Center for Computational Vision and Control The Weizmann Institute Prineton, NJ 854 Yale University Rehovot, 76, Israel dwj@researh.nj.ne.om New Haven, CT 65 ronen@wisdom.weizmann.a.il belhumeur@yale.edu Abstrat We onsider the problem of determining whether two images ome from different objets or the same objet in the same pose, but under different illumination onditions. We show that this problem annot be solved using hard onstraints: even using a Lambertian refletane model, there is always an objet and a pair of lighting onditions onsistent with any two images. Nevertheless, we show that for point soures and objets with Lambertian refletane, the ratio of two images from the same objet is simpler than the ratio of images from different objets. We also show that the ratio of the two images provides two of the three distint values in the Hessian matrix of the objet s surfae. Using these observations, we develop a simple measure for mathing images under variable illumination, omparing its performane to other existing methods on a database of 45 images of individuals. Introdution A entral problem of visual objet reognition is to use information about an objet derived from sample images in order to reognize that objet under novel viewing onditions. In model-based approahes, it is typially assumed that training images are used to derive some definite information about the shape of an objet that aurately predits some properties of its appearane in any new image; in appearane-based vision an effort is made to represent the set of all images an objet an produe, either by sampling them or by generating a representation from a small set of training images. Both approahes enounter diffiulties when few training images are available, or when the images are taken under unontrolled onditions. A basi question arises: Given only a single training image of an objet how an one determine whether a test image is of the same objet taken under different onditions, or of a new objet altogether? This paper addresses this question for the ase where variation in appearane is due to illumination. In partiular, how does one look at intensity images of two ompletely different objets and determine that the differene between these images ould not be due to lighting variation, but must indiate a differene in objet identity or position? We show that given only two images, one annot determine with ertainty whether they arise from the same or different objets, for there always exists a ontinuous surfae with Lambertian refletane that ould have produed both images. While this result is demonstrated using point soures, it of ourse applies to more general lighting models that inlude point soures as a subset. Thus when two images appear to ome from different objets, this is beause it is unlikely, not impossible (under a Lambertian refletane model) that they ome from the same objet. Our hallenge then is to justify and quantify this unlikeliness. To do this, we must first haraterize what we an determine about an objet from two images. We do this for the ase of an objet with Lambertian refletane and lighting due to two different known point soures. We show that from the ratio of the two images, we an determine three omponents of the Hessian matrix that haraterizes the surfae of this objet. We then extrapolate from this result to the ase where the objet is illuminated by unknown light soures. To develop a measure for determining whether two images arise from the same objet, we examine properties of the ratio of the two images (the usefulness of this representation was previously pointed out in Wolff and Angelopoulou [3] and Fan and Wolff []). We show that in the ase where the objet is relatively simple the ratio of two images of the same objet must be even simpler than either of the individual images (where, as an example, we define simpliity based on the omplexity of the algebrai funtion needed to loally approximate the shape or image). However, the ratio of images produed by two different, but equally simple objets is more omplex for generi shapes and lighting onditions. We then use these insights to derive a simple loal measure of omparison. We show that these methods an be used to produe greater auray on a real task fae reognition under variable lighting onditions than widely used existing methods. Finally, we briefly onsider the ase of more general lighting onditions, when multiple light soures are present. We show the diffiulty of extrapolating from a small number of training images to the entire set of images an objet an produe. Speifially, Belhumeur and Kriegman [4] have shown that from as few as three images of an objet, where eah is produed by a single point light soure, one an determine the illumination one that desribes the set of all images that objet an produe with multiple light soures. We show that if the training images have multiple unknown light soures instead of point soures, it is not possible to determine the illumination one exatly.

2 Bakground In model-based approahes to objet reognition, it has been assumed that one an onstrut a preise 3-D model of an objet to use for reognition. This is suitable for some appliations, but it has proven diffiult to build aurate 3-D models using only images taken in unontrolled irumstanes. This also raises questions about the suitability of approahes based on 3-D models as explanations of human vision (e.g., Marr [8], Ullman [8]). Another approah has been to desribe 3-D objets in terms of their invariant, or quasi-invariant properties. Suh desriptions of 3-D objets apture that portion of their struture that is apparent in all, or almost all images of the objets. For example, Biederman [6], based on earlier work in omputer vision (e.g., Lowe [7]), proposed that the human visual system desribes and retrieves 3-D objets based on non-aidental properties that an be deteted in images, regardless of viewpoint. Others in omputer vision have developed approahes to reognition based on invariants (for an overview, see []). Models of objets that are based on invariants an, by definition, be onstruted from a single image of an objet, although diffiulties have also arisen in applying suh approahes to general lasses of objets ([7, 8, ]). Perhaps most related to the topis in this paper is the work on image lightness (e.g., Horn [5]). Here the image of an objet is filtered in an attempt to remove or suppress the lighting effets in order to reover only the objet s surfae refletane. In [5] this is done in a three step proess of differentiation, thresholding, and integration. Yet this method is at best only loally reliable, as image noise and errors due to shadows are ompounded in the integration proess. Partly for these reasons, appearane-based methods of reognition have also been explored. In these methods, an objet is not desribed in terms of its 3-D properties, but rather in terms of the -D images that it produes. One approah to appearane-based reognition is to sample an objet s possible images, and then to ompare, in a lower dimensional image subspae, a novel image to the set of sampled images, using pattern reognition tehniques suh as nearest neighbors (e.g., [7, 6, ]). This works well when the training images densely sample the spae of images one hopes to reognize. In general, though, an objet an produe so many different images that it is not lear how to sample them all. Alternatively, some approahes attempt to predit the images an objet an produe from a small number of training images (e.g., [, 6,, 4]). This overomes the diffiulty of having to sample a great number of an objet s images. However, while this may not require full reovery of an objet s 3-D struture, it learly requires a lot of knowledge of 3-D struture to predit all its possible images (e.g., [5]). Suh information may not be available when an objet is viewed in only a few prior images under unontrolled onditions. We an ontrast these approahes with the one suggested in this paper in terms of the information they hope to extrat from an image. The model-based and invariant-based approahes hope to derive intrinsi properties of the 3-D objet struture. The seond set of appearane-based approahes hope to extrat a haraterization of the set of images an objet an produe, whih may be almost as ambitious. The first appearane-based approahes we disussed, in ontrast, extrat relatively little information from eah image. They essentially treat eah image as an isolated point of information. New images are ompared to these using a measure suh as Eulidean distane in a dimensionally redued image spae. On the other hand, the approah proposed here seeks to make omparisons between images that are derived from the nature of the imaging variability. We think of an image as providing onsiderable information about what other images an objet an produe, without neessarily providing any definite information about its 3-D struture. 3 Two Images Are Always Compatible We are interested in omparing two images to determine whether they ome from the same, unknown objet, with the same pose, but under different illumination onditions. So we ask first: Is it ever the ase that two images annot ome from the same objet? We show that the answer to this question is no. In fat, even if we assume that the lighting in eah sene is onstrained to be a known point soure at infinity, we an always onstrut an objet in a fixed pose that is onsistent with both images. We should point out that while our analysis in this setion orretly handles shadowing, it does not aount for interrefletions. We also show what aspets of a Lambertian objet s struture an be determined from two images with point soures. These results suggest a diretion we may take to gauge the likelihood that two images are produed by the same objet. We first assume that two images, and, ome from a Lambertian objet lit by two known point soures at infinity, and respetively. If given these limitations (known soures and Lambertian refletane), we an onstrut an objet that is onsistent with both images, we have poor prospets of ever telling with ertainty that two images, no matter how different they may appear, ould not ome from the same objet. We assume that the objet is viewed from the diretion, and therefore, that the depth of the surfae an be written as!"#. By writing in this form we are ensuring that desribes an integrable surfae, i.e., that the surfae normals of are onsistent with a true surfae. Let the albedo of the objet be written as a funtion $ %!&'# also. Then, the surfae normals of the objet are ( ). %!"#/ 5 %!"#/ and we have the two equations - $ %!&'# ( 34 ( - $ %!&'# 67 8, Our problem is to determine whih funtions and $ may satisfy these equations. As Wolff and Angelopoulou [3] and Fan and Wolff [] have pointed out, we an deal with image pairs more simply by taking their ratio, sine this auses the effet of albedo to anel. However, they go on to use the ratio image for stereo mathing and for reonstruting a surfae from three

3 M M O M images, quite different purposes than ours. Nayar and Bolle [3] use the ratio of two regions in the same image for yet another purpose, to anel the effets of lighting, under the assumption that the regions ome from oplanar portions of the objet. Taking these ratios, and defining : ; )<, we have whih implies %:>? ;: < %:>? = 5 (Throughout this setion, for simpliity we assume that neither image is zero at any point, so that the ratio is welldefined). Sine is our only unknown, this is a first order, partial differential equation with variable oeffiients. We an solve it using well-known methods (see, for example, Zauderer [3]). In brief, we may divide the image into harateristi urves. Along eah harateristi urve, we hange variables so that is a funtion of a single variable. Then we may find the value of, along a harateristi urve, up to an unknown initial ondition, by integrating along this urve. As a simple example of this method, onsider the ase of A B and C B. Then we have D: The harateristi urves in this ase are horizontal lines aross the image. The value of %!&'# is given by!"'#5 E # GF : %H#JI4H () where we have no obvious soure of knowledge available to provide the initial ondition '#. We denote this initial ondition # as KL %#. This shows that we an reover the value of up to an unknown initial ondition given by K. Note that we have ;: ;K NMPO :>I7H # Thus, we an reover diretly from the ratio image. We annot reover, however, sine K is unknown. Moreover, even if we did know K, any straight-forward reovery of from a real image would be extremely unstable, sine we would have to numerially integrate : along adjaent harateristi urves, and then take its derivative. Taking further partial derivatives, we have Q: R S Q: S QK S :>I7H Thus, we an reover S by taking derivatives of the ratio image, but again we annot reover S. For this example, then, we see that we an use two images with known light soures to reover three omponents of the Hessian matrix of the surfae of the objet. Moreover, these equations always have a solution for, whih is given expliitly as Equation. We an also note that for these light soures, there are no shadows. The soure # asts no shadows on the surfae beause is monotonially inreasing along eah harateristi urve, sine, and therefore T: are, by nature, non-negative. Sine the seond light soure is also the viewing diretion, it annot ast shadows on any visible objet point. Note also that for any given that satisfies Equation, we an hoose $ to satisfy the equations given by the two images (suh an $ may have values greater than one. To avoid this, we must sale the intensity of eah light soure by an appropriate onstant). For other lighting onditions, we get similar results. In general, the slope of the harateristi urve is :U >) %:> V 7 These harateristis are not straight lines, but vary their diretion as a funtion of :. Assuming general lighting, so that there is no value of : that satisfies both the equations :> WG X Y: ZG, this diretion is always unique and well-defined. In this ase, the harateristi urves an never interset, and so there always exists a surfae that satisfies the ratio image s PDE (see [3]). Again, there is a whole family of these solutions, one for any funtion that provides an initial ondition. Similar issues have been onsidered in work on photometri stereo. However, the photometri stereo work addresses settings in whih the reonstrution problem is not underonstrained. For example, Coleman and Jain [] disuss reovery of struture for textured shapes with speularities, using four images. Onn and Brukstein [4] show how to use integrability to reover struture from two images when the sene has a uniform albedo. And Fan and Wolff [] onsider reovery of struture and albedo from three images. In ontrast, we have shown that given two images of an objet with unknown struture and albedo, there is always a large family of solutions. In fat, for any pair of point light soures there is a family of possible solutions. We have shown that given known light soures, we an determine two independent omponents of the Hessian of the surfae at any position, but not the third. The diretion in whih we an determine these omponents may vary throughout the image, depending both on the light soures and the ratio image. Finally, even for unknown lighting onditions, the ratio image : still provides information albeit impreise information about the loal nature of the objet s surfae. 4 Determining the Simpliity of Interpretations We annot tell that two images must ome from different objets. We now turn instead to determining whether in some ases, explaining the images with a ommon objet would require an unlikely oinidene. This approah to image interpretation has been applied to other vision problems by, for example, Rok [5], Lowe [7], and Freeman []. Speifially, we begin by showing that the ratio of two images from the same objet is generally simpler than either of the individual images, while the ratio of images from different objets is generially more omplex than either image. As previously noted, the image of an objet is $ < < ()

4 \ \ \ ba ba and the ratio of two images from the same objet is Q: 4 > However, suppose our seond image &[ is of a different objet, whose surfae is desribed by the funtion R KL %!&'#, and whose albedo is desribed by the funtion!"#. Then we have as the ratio image [ ]: [ 7 K = K ^_$ K K `, (3) (4) In many instanes Equation 3 desribes a simpler ratio image than Equation 4, unless the extra multipliative term ^_$ K K happens to anel the rest of the equation. Consider the ase in whih loal surfae pathes of and K, and their respetive albedo funtions $ \ and are well approximated by seond degree polynomials in! and #. In this ase, the ratio image : from Equation 3 is a rational funtion with both a linear numerator and denominator. However, the ratio of images from different objets : [ (Equation 4) is the produt of an algebrai funtion and a rational funtion with third degree polynomials in the denominator and numerator. The degree of a polynomial (or parameters in an algebrai expression) needed to approximate a funtion is often used as a measure of that funtion s simpliity. By that measure, in this ase we see that the ratio of images from the same objet is far simpler than the ratio of images from two different objets. This simpliity generally translates into simpler properties, suh as fewer extrema and, in many ases, less overall variability in the ratio image. Similar reasoning holds in other ases as well. Of speial interest is the ase in whih the surfaes and K are loally planar. In this ase, the ratio of two images from the same objet is onstant. However, the ratio of images from different objets is only onstant if their albedos are idential up to a sale fator,in whih ase the differenes in the albedo patterns annot in prinipal be diserned unless the magnitude of the lighting is known. We an now relate these results to those in the previous setion in a brief, intuitive form. We showed that the shape of an objet lit by point soures ould be derived by integrating the ratio image along its harateristi urves. These results suggest that we an attempt to measure the likelihood that two images ome from the same objet by measuring the simpliity of the ratio image. 5 Experiments Using a Simple Comparison Method Our results suggest a number of ways of attempting to measure whether the differene between two images is due to a differene in lighting or in objet struture. In this setion, we experiment with only the simplest of these on the task of reognizing faes under variable illumination onditions. A simple measure of the omplexity of the ratio image is the integral of the magnitude of its squared gradient. This measures the smoothness of the ratio image. Suh measures have often been used in vision, for example in interpolating surfaes by minimizing the urvature of the interpolation (some early methods are reviewed in [8]). This measure has the advantage of being loal, and therefore, the analysis we have done assuming point light soures and low degree polynomial surfaes must hold only loally to apply. Now notie that the squared magnitude of the gradient of the ratio image dfe Vg has two signifiant disadvantages: first, it is asymmetri in and, and seond, it behaves poorly for regions of image whih are in shadow. To orret for the asymmetry, we instead use the geometri mean dfe Vg dfe g To orret for the behavior in the shadowed regions, we weight the measure by the min h " Thus to ompare images and, we simply integrate this quantity over the image region to get FF min S " d e d e g g I7!.I7# With straight-forward algebrai manipulation, one an show the surprising similarity of this measure to a measure simply omparing image edges. In some sense, we have ome full irle, using a Lambertian model for image formation to justify an edge-based measure of omparison. Yet, this measure has two important differenes from simple edge-based mathing: first, this measure does not make any hard deisions about the presene or absene of an edge, and seond, it normalizes the response by the loal image intensity. The measure proposed here is probably most losely related to the log filter desribed in Ballard and Brown [] followed by a gradient highpass filter. We experimented with this measure on the task of fae reognition under variable illumination. We used a publily available database of faes onstruted by Hallinan [4]. From this database we used 45 images of individuals. The images were divided into four subsets in whih the lighting diretions within the subsets were 5i (Subset ), 3i (Subset ), 45i (Subset 3), and 6i (Subset 4) from the amera s optial axis. Figure shows images of one fae from eah of these subsets. Faes lit at a 5i angle (Subset ) were used as a training set, and then tested using images in whih the lighting had greater eentriity. To disount the effets of improper alignment, eah image was aligned manually and ropped as shown in Figure. Figure shows the omparative performane of the method proposed here and three ompeting methods. These results show that the squared magnitude of the gradient of the ratio image works dramatially better than simple orrelation, or orrelation after projeting onto the

5 5i 3i 45i 6i 8 7 PCA () Correlation 6 Error Rates (%) Gradient of Ratio Illumination Cones Subset (5) Subset (3) Subset 3 (45) Subset 4 (6) Lighting Diretion Subset (Degrees) Method Error Rate (%) 5j 3j 45j 6j Subset Subset Subset 3 Subset 4 Correlation PCA () Gradient of Ratio Illumination Cones.... Figure : Pitures from the Harvard Fae Database. The pitures are of the same individual lit by varying the diretion of a point light soure. The angle of the light soure with the optial axis (5, 3, 45, and 6 degrees) is the same in eah olumn. Figure : A measure of omparison based on the magnitude of the gradient of the ratio image signifiantly outperforms both orrelation and prinipal omponent analysis (PCA) on database of 45 faes of individuals taken under extreme variation in lighting onditions. twenty prinipal omponents of the training images (as desribed in [7]). However, it does not perform as well as the illumination ones method whih builds a representation for the set of possible images from a small set of training images [4, 3]. (Adini, Moses, and Ullman [] have also reported experiments on fae reognition under variable illumination. We have not yet been able to ompare these methods on their database.) In omparing this measure to others, one should note that it does not attempt to ombine information from a number of training images to build up as representation of a fae, as do methods suh as Fisherfaes [3], the linear subspae method [3], or the illumination ones method [3]. When enough training images are available to well haraterize the entire set of images that a fae an produe, we expet that one should ahieve better performane with these methods, and our experiments seem to support this. However, our method simply ompares a new image independently to eah previously seen image. We feel that this type of approah, and indeed the results of our paper in general, are most suited to the situation in whih one does not have enough prior information about an objet to attempt to aurately haraterize its possible appearanes. 6 Diffiulties with Representation We have onsidered an approah to reognition in whih two images are ompared to judge whether they appear to ome from the same objet. An alternative approah is to use a number of training images to build a representation of the set of all images that an objet an produe. This has been done effetively to aount for viewpoint variation (e.g., []), and to aount for lighting variation when the training images are eah lit with a single point soure ([, 6, 4]). The illumination ones method [3] tested in the previous setion is one suh method. We now onsider the diffiulties in generating a representation of an objet s possible images when the training images are taken under unontrolled onditions, ontaining multiple light soures. Belhumeur and Kriegman show how to build an exat representation of the images that a polyhedral objet an produe when lit with multiple soures, whih they all the illumination one. Their method requires at least three images of the objet eah illuminated by a single light soure. It is also evident from their results that given many images of the objet, eah of whih ontains multiple light soures, taking the onvex ombination of these images produes a subset of the illumination one whih an provide a good approximation to it. It is not lear, however, whether it is possible to build the illumination one exatly using training images that ontain multiple light soures. In this setion we show that this is not possible. We take this as one indiation of the potential diffiulty of building a omplete representation of an objet s possible images using a small number of training images taken under unontrolled viewing onditions. For suh situations, it remains valuable to develop methods for diretly omparing images.

6 $ y s We show that the illumination one annot be onstruted from a set of images that ontain multiple, unknown lights, by showing that a muh simpler problem is not solvable. We show that even if one knows the 3- D struture of a onvex-shaped Lambertian objet exatly, one annot in general determine the albedo of the objet points exatly, even from a large set of images. Let kl be a faet on a polyhedral objet, let 7l be the orresponding intensity produed in the image,. Denote kl s surfae normal pointing inward towards the objet as m l. Let the albedo at this objet faet be $ l. Assume that there are n light soures, denoted q. Then, with a Lambertian surfae, we have =l, q rsut op nwv<!& $ l m lx Belhumeur and Kriegman [4] have shown that with this lighting model, the set of all images that a Lambertian objet an produe forms a onvex one in the spae, IRy, of all images, where eah oordinate of the spae is the intensity value of a different pixel in the image. Let z denote the onvex one of images that ould be produed by this objet if the albedo of all its points were equal and set to unity; we refer to this as the onstant albedo objet. We an then desribe the set of images the atual objet an produe as follows. Let { be a diagonal matrix, with, denoting the albedo. Then the diagonal entries $ atual objet an produe po all images of the form {- suh that 8}Xz is a olumn vetor desribing one of the images that the onstant albedo objet ould produe. This tells us that if is a olumn vetor whose entries are the pixel intensities of an image of the atual objet, we must have ~ ;{- > 8}Xzw { }ƒz z is a onvex polytope that is defined by a set of bounding half-planes, whih all pass through the origin. We an define eah half-plane by a normal vetor, so that a point, k, is inside the half-plane when kc ˆ. So This tells us that -}ƒzš R A ˆ ' 3 op { " Œ ˆ 3 po This is a series of inequalities that are linear in the inverse of the objet albedos, sine the image and the illumination one (i.e., the C ) of the onstant albedo objet are both known. This tells us that a single image of an objet that has known surfae normals onstrains the albedos of the objet to lie inside a onvex polytope in the spae of all possible inverse albedos. Suppose we have many images of the same objet available. The true albedos of the objet lie inside the intersetion of a set of onvex polytopes in albedo spae. The intersetion of these polytopes gets smaller and smaller as we have more images available, onstraining the possible objet albedos. However, the true albedos do not lie on the boundary of any of these onvex polytopes unless a point in the objet has a light soure lying in its tangent plane. Hene, the intersetion of these onvex polytopes is still an open set in albedo spae, and the albedos of the objet are not uniquely determined. On the other hand, given images from objets with the same struture but different albedos, these onvex ones in inverse albedo spae an be non-interseting, revealing that the objets are different. These results illustrate the following point: it may be impossible to determine a omplete representation of the images that an objet an produe, using multiple images taken under unontrolled onditions. However, we still may be able to tell whether a new image is onsistent with one or more previous images we have seen. We have shown this to be true for the simple ase of a onvex objet with known struture but unknown albedo. The results in this setion are losely related to Forsyth s [] olor onstany algorithm. That work dealt with a very different problem, that of determining the olor of pathes of a planar sene from a single image in whih the spetrum of the illumination is unknown. However, our derivation is similar. In Forsyth s ase the appearane of all possible olor hips under a known light soure plays the role that is played by the illumination one of a known model in our ase. For Forsyth, the appearane of eah path of uniform olor in the sene plays the role that eah image plays in our derivation. Forsyth uses these to onstrain the unknown illuminant funtion in a sene in muh the same way that we onstrain the unknown albedo. The key differene is that for olor onstany one derives a onvex onstraint from every different olor in the sene, where in our ase there is a omparable onstraint produed by every image, so that many images may be required to narrow down the solution. 7 Conlusions Model-based reognition methods have ahieved onsiderable suess when they have adequate prior knowledge to build a preise objet model that aptures its 3-D struture. However, in some appliations, this prior knowledge is not available. It is also an open question whether human objet reognition routinely funtions with suffiient prior knowledge of objets to onstrut and use preise 3-D models. For this reason, we argue that the information that an image provides about an objet may be best thought of as information about what other images are likely, or unlikely to ome from the same objet. We have made this onrete for the ase of illumination variation. We have shown that it an be diffiult to exatly reover the properties of an objet, suh as its albedo, from images in whih the lighting onditions are unknown. At the same time, we have shown that there may be onsiderable information about whether two images either do or do not ome from the same objet. Therefore we may be able to use previously seen images of an objet to reognize it in new images under variable lighting onditions, even without using these prior images to perform any sort of expliit or impliit reonstrution of any objet properties. Stritly speaking, this is true only when has volume in IRŽ. If it doesn t we must restate our argument to fous on only the subset of objet points that have distint surfae normals. Our basi argument still holds, however.

7 We have also used these insights in a reognition system. We have shown that the simpliity of the ratio of two images provides a good indiation of whether they ome from the same objet; we measure this simpliity by looking at the gradient of the ratio image. This approah is simple and loal, but provides good results. Referenes [] Y. Adini, Y. Moses, S. and Ullman, 7. Fae Reognition: The Problem of Compensating for Changes in Illumination Diretion, IEEE Trans. PAMI (7):7 73. [] D. Ballard and C. Brown, 8, Computer Vision, Prentie Hall, Englewood Cliffs, New Jersey. [3] P. Belhumeur, J. Hespanha, and D. Kriegman, 7. Eigenfaes vs. Fisherfaes: Reognition Using Class Speifi Linear Projetion, IEEE Trans. PAMI (7):7 7. [4] P. Belhumeur and D. Kriegman, 6. What is the Set of Images of an Objet Under All Possible Lighting Conditions?, IEEE Conf. on Comp. Vis. and Pat. Re.:7 77. [5] P. Belhumeur, D. Kriegman, and A. Yuille, 7. The Bas-Relief Ambiguity CVPR:6 66. [6] I. Biederman, 85, Human Image Understanding: Reent Researh and a Theory, Computer Graphis, Vision, and Image Proessing, (3):-73. [7] J. Burns, R. Weiss, and E. Riseman,, The Non-Existene of General-Case View-Invariants, Geometri Invariane in Computer Vision, edited by J. Mundy, and A. Zisserman, MIT Press, Cambridge. [8] D. Clemens and D. Jaobs,, Spae and Time Bounds on Model Indexing, IEEE Transations on Pattern Analysis and Mahine Intelligene, 3():7-8. [] E. Coleman and R. Jain, 8, Obtaining 3-Dimensional Shape of Textured and Speular Surfaes Using Four-Soure Photometry, CGIP 8(4):3 38. [] J. Fan and L. Wolff, 7, Surfae Curvature and Shape Reonstrution from Unknown Multiple Illumination and Integrability, Computer Vision and Image Understanding 65(): [] W. Freeman, 4, The Generi Viewpoint Assumption in a Framework for Visual Pereption, Nature 368: [] D. Forsyth,. A Novel Algorithm for Color Constany, Int. J. of Comp. Vis. 5():5 36. [3] A. Georghiades, D. Kriegman, and P. Belhumeur, 8. Illumination Cones for Reognition Under Variable Lighting: Faes, IEEE Conf. CVPR. [4] P. Hallinan, 4. A Low-Dimensional Representation of Human Faes for Arbitrary Lighting Conditions, IEEE Conf. CVPR:5. [5] B. Horn, 74, Determining Lightness from an Image, CGIP, 3(4):77. [6] M. Kirby, and L. Sirovih,, The appliation of the Karhunen-Loeve proedure for the haraterization of human faes, IEEE transations on Pattern Analysis and Mahine Intelligene, ():3-8. [7] D. Lowe, 85, Pereptual Organization and Visual Reognition, Kluwer Aademi Publishers, The Netherlands. [8] D. Marr, 8, Vision, W.H. Freeman and Company, San Franiso. [] Y. Moses, 3. Fae reognition: generalization to novel images, Ph.D. Thesis, Weizmann Institute of Siene. [] Y. Moses and S. Ullman,, Limitations of Non Model-Based Reognition Shemes, Seond European Conferene on Computer Vision:8-88. [] J. Mundy and A. Zisserman (eds.),, Geometri Invariane in Computer Vision, MIT Press, Cambridge. [] H. Murase and S. Nayar, 5. Visual learning and reognition of 3D objets from appearane. Int. Journal of Computer Vision, 4():5 5. [3] S. Nayar and R. Bolle, forthoming, Refletane Based Objet Reognition, Int. J. of Comp. Vis. [4] R. Onn, and F. Brukstein,, Integrability Disambiguates Surfae Reovery in Two-Image Photometri Stereo, Int. J. of Comp. Vis. 5():5 3. [5] I. Rok, 83. The Logi of Pereption, MIT Press, Cambridge, MA. [6] A. Shashua, 7. On Photometri Issues in 3D Visual Reognition from a Single D Image. IJCV, (/):. [7] M. Turk, and A. Pentland,. "Eigenfaes for Reognition", Journal of Cognitive Neurosiene, 3, [8] S. Ullman, 8, Aligning Pitorial Desriptions: An Approah to Objet Reognition, Cognition 3(3):3-54. [] S. Ullman and R. Basri,, Reognition by Linear Combinations of Models, IEEE Trans. PAMI, 3():-7. [3] L. Wolff and E. Angelopoulou, 4, Eur. Conf. on Comp. Vis.: [3] E. Zauderer, 83. Partial Differential Equations of Applied Mathematis, John Wiley and Sons.