Inverse Rendering of Lambertian Surfaces Using Subspace Methods

We propose a vector space approach for inverse rendering of a Lambertian convex object with distant light sources. In this problem, the texture of the object and arbitrary lightings are both to be recovered from multiple images of the object and its 3D model. Our work is motivated by the observation that all possible images of a Lambertian object lie around a low-dimensional linear subspace spanned by the first few spherical harmonics. The inverse rendering can therefore be formulated as a matrix factorization, in which the basis of the subspace is encoded in a spherical harmonic matrix S associated with the object's geometry. A necessary and sufficient condition on S for unique factorization is derived with an introduction to a new notion of matrix rank called nonseparable full rank. A singular value decomposition-based algorithm for exact factorization in the noiseless case is introduced. In the presence of noise, two algorithms, namely, alternating and optimization based are proposed to deal with two different types of noise. A random sample consensus-based algorithm is introduced to reduce the size of the optimization problem, which is equal to the number of pixels in each image. Implementations of the proposed algorithms are done on a real data set.