Lambertian Reeectance and Linear Subspaces

Weizmann Institute of Science, Technical Report MCS00-21 NEC Research Institute Technical Report 2000-172R Ronen Basri David Jacobs Dept. of Computer Science NEC Research Institute The Weizmann Institute of Science 4 Independence Way Rehovot, 76100 Israel Princeton, NJ 08540 Abstract We prove that the set of all re ectance functions (the mapping from surface normals to intensities) produced by Lambertian objects under distant, isotropic lighting lies close to a 9D linear subspace. This implies that, in general, the set of images of a convex Lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low-dimensional linear subspace, explaining prior empirical results. We also provide a simple analytic characterization of this linear space. We obtain these results by representing lighting using spherical harmonics and describing the e ects of Lambertian materials as the analog of a convolution. These results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce non-negative lighting functions. Finally, we show a simple way to enforce non-negative lighting when the images of an object lie near a 4D linear space.