Determining Generative Models of Objects Under Varying Illumination: Shape and Albedo from Multiple Images Using SVDand Integrability

  • Authors:

  • A. L. Yuille;D. Snow;R. Epstein;P. N. Belhumeur

  • Affiliations:

  • Smith-Kettlewell Eye Research Institute, 2318 Fillmore St., San Francisco, CA 94115;Smith-Kettlewell Eye Research Institute, 2318 Fillmore St., San Francisco, CA 94115;Smith-Kettlewell Eye Research Institute, 2318 Fillmore St., San Francisco, CA 94115;Department of Electrical Engineering, Yale University

  • Venue:
  • International Journal of Computer Vision
  • Year:
  • 1999

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Abstract

We describe a method of learning generative models of objects from a set of images of the object under different,and unknown, illumination. Such a model allows us to approximate the objects‘ appearance under a range of lightingconditions. This work is closely related to photometric stereo with unknown light sources and, in particular, to the useof Singular Value Decomposition (SVD) to estimate shape and albedo from multiple images up to a linear transformation(Hayakawa, 1994). Firstly we analyze and extend the SVD approach to this problem. We demonstrate that it applies toobjects for which the dominant imaging effects are Lambertian reflectance with a distant light source and a backgroundambient term. To determine that this is a reasonable approximation we calculate the eigenvectors of the SVD on a set ofreal objects, under varying lighting conditions, and demonstrate that the first few eigenvectors account for most of thedata in agreement with our predictions. We then analyze the linear ambiguities in the SVD approach and demonstrate thatprevious methods proposed to resolve them (Hayakawa, 1994) are only valid under certain conditions. We discuss alternativepossibilities and, in particular, demonstrate that knowledge of the object class is sufficient to resolve this problem.Secondly, we describe the use of surface consistency for putting constraints on the possible solutions. We prove that thisconstraint reduces the ambiguities to a subspace called the generalized bas relief ambiguity (GBR) which is inherent in theLambertian reflectance function (and which can be shown to exist even if attached and cast shadows are present (Belhumeuret al., 1997)). We demonstrate the use of surface consistency to solve for the shapeand albedo up to a GBR and describe, and implement, a variety of additionalassumptions to resolve the GBR. Thirdly, we demonstrate an iterative algorithm thatcan detect and remove some attached shadows from the objects thereby increasing theaccuracy of the reconstructed shape and albedo.