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Beyond Lambert Reconstructing Surfaces with Arbitrary BRDFs Sebastian Magda David J Kriegman Todd Zickler Peter N Belhumeur Beckman Institute Computer Science University of Illinois Urbana Champaign Urbana IL 61801 Center for Comp Vision Control EE and CS Yale University New Haven C T 06520 8267 Abstract W e address a n open and hitherto neglected problem i n computer vision how to reconstruct the geometry of objects with arbitrary and possibly anisotropic bidirectional reflectance distribution functions B R D F s Present reconstruction techniques whether stereo vision structure f r o m motion laser range finding etc make explicit or implicit assumptions about the B R D F Here we introduce two methods that were developed by re examining the underlying image formation process the methods make n o assumptions about the object s shape the presence or absence of shadowing o r the nature of the B R D F which m a y vary over the surface The first method takes advantage of Helmholtz reciprocity while the second method exploits the fact that the radiance along a ray of light is constant I n particular the first method uses stereo pairs of images in which point light sources are co located at the centers of projection of the stereo cameras The second method is based on double covering a scene s incident light field the depths of surface points are estimated using a large collection of images in which the viewpoint remains fixed and a point light source illuminates the object Results f r o m our implementations lend empirical support to both techniques 1 Introduction We address an open problem in computer vision how t o reconstruct the shape geometry of an object with an arbitrary unknown bidirectional reflectance distribution function BRDF 8 Our solutions stand in contrast t o existing methods which assume either implicitly or explicitly that the BRDF of points on the object s surface are Lambertian approximately Lambertian or of some known parametric form A BRDF at a point on the surface is the ratio of the outgoing radiance t o the incident irradiance The BRDF can be represented as a positive 4 D function p i 6 where i is the direction of an incident light ray and 6 is the direction of the outgoing ray The coordinates of and 6 are usually expressed with respect D Kriegman and S Magda were supported by NSF ITR CCR 00 86094 and DAAH04 95 1 0494 P N Belhumeur was supported by a Presidential Early Career Award 11s 9703134 NSF KDI 9980058 NIH RO1 EY 12691 01 and an NSF ITR 0 7695 1143 0 01 10 00 0 2001 IEEE 391 Figure 1 The measured intensity of one pixel as a function of light source position Images were acquired as a point source was moved over a quarter of a sphere to a coordinate system attached t o the tangent plane The BRDF is not an arbitrary function since from the second law of thermodynamics it sltisfies Helmholtz s reciprocity condition p i e p 6 i SI Th IS symmetry essentially says that the fraction of light coming from direction i and emitted in direction e is the same as that coming from 6 and emitted in direction I In computer vision and computer graphics models are used t o simplify the BRDF In computer vision Lamberts Law is the basis for most reconstruction techniques And in computer graphics it would not be an exaggeration t o say that more than 99 99 of rendered images use a Phong reflectance model which is composed of an ambient term a diffuse Lanibertian term and an ad hoc specular term 18 While the isotropic Phong model captures the reflectance properties of plastics over a wide range of conditions it does not effectively capture the reflectance of materials such as metals and ceramics particularly when they have rough random surfaces or a regular surface structure e g parallel grooves Much less common are a nuniber of physics based parametric models l a 17 211 yet each of these only characterizes a limited class of surfaces So a recent alternative is to measure the BRDF and represent it by a suitable set of basis functions ll As a simple empirical illustration of the complexity of the BRDF s of real surfaces consider the two views of a surface plot shown in Fig 1 For a ceramic figurine this plot shows the measured intensity of one pixel as an isotropic point source is moved over a quarter of a sphere a t approximately a constant distance from the surface While the deep rectangular cutout dark part is attributable t o self shadowing n0t e that the rest of surface lacks the characteristic lobes in reflectance models such as Phong For a surface whose BRDF is not a function of 6 i e Lambertian the image intensity of a surface point will be the same irrespective of the viewing direction This constant brightness assumption is the basis for establishing correspondence in dense stereo and motion methods Yet for objects with a general and unknown BRDFs this constant brightness assumption is violated Thus establishing correspondences between images gathered from different viewpoints under constant lighting is difficult if not impossible Methods for computing optical flow e g Horn and Schunck 8 also assume constant brightness Similarly nearly all photometric stereo methods assume that the BRDF is Lambertian 13 19 221 is completely known a priori or can be specified using a small number of parameters usually derived from limited physical models 7 9 16 201 In these methods multiple images under varying lighting but fixed viewpoint are used to estimate a field of surface normals which is then integrated to produce a surface When the BRDF varies across the surfaces there is insufficient information t o reconstruct both the geometry and the BRDF Naturally with only a single image shapefrom shading methods are even more limited In 14 a hybrid method with controlled lighting and object rotation was used to estimate both the structure and a non parametric reflectance map though the BRDF must be isotropic and uniform across the surface Even the effectiveness of structured light methods such as triangulation based light stripers and laser range finders depends upon the BRDF While it is no longer necessary to paint an object matte white t o obtain effective range scans from light stripers specularities and interreflections tend t o cause erroneous depth readings for metallic objects Similarly when the surface is specular and there is little backscatter there may be insufficient return for a laser range finder t o estimate depth There are numerous other reconstruction techniques yet their

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