The Bas-Relief Ambiguity

  • Authors:

  • Peter N. Belhumeur;David J. Kriegman;Alan L. Yuille

  • Affiliations:

  • Center for Comp. Vision and Control, Dept. of Electrical Engineering, Yale University, New Haven CT 06520, USA;Beckman Institute, Department of Computer Science, University of Illinois, Urbana-Champaign, Urbana, IL 61801, USA;Smith-Kettlewell Eye Research Institute, San Francisco, CA 94115, USA

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

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Abstract

When an unknown object with Lambertian reflectance is viewedorthographically, there is an implicit ambiguity in determining its3-d structure: we show that the object‘s visible surface f(x, y)is indistinguishable from a “generalized bas-relief”transformation of the object‘s geometry, \bar f(x, y) = λf(x, y) + μx + νy, and a corresponding transformation on the object‘s albedo. For each image of the object illuminated by an arbitrary number of distant light sources, there exists anidentical image of the transformed object illuminated by similarlytransformed light sources. This result holds both for theilluminated regions of the object as well as those in cast andattached shadows. Furthermore, neither small motion of the object,nor of the viewer will resolve the ambiguity in determining theflattening (or scaling) λ of the object‘s surface. Implications of this ambiguity on structure recovery and shape representation are discussed.