Lambertian Reflectance and Linear Subspaces

We prove that the set of all Lambertian reflectance functions (the mapping from surface normals to intensities) obtained with arbitrary distant light sources 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 effects 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 nonnegative lighting functions. We also show a simple way to enforce nonnegative lighting when the images of an object lie near a 4D linear space. We apply these algorithms to perform face recognition by finding the 3D model that best matches a 2D query image.