Analytic PCA Construction for Theoretical Analysis of Lighting Variability in Images of a Lambertian Object

We analyze theoretically the subspace best approximating images of a convex Lambertian object taken from the same viewpoint, but under different distant illumination conditions. Since the lighting is an arbitrary function, the space of all possible images is formally infinite-dimensional. However, previous empirical work has shown that images of largely diffuse objects actually lie very close to a five-dimensional subspace. In this paper, we analytically construct the principal component analysis for images of a convex Lambertian object, explicitly taking attached shadows into account, and find the principal eigenmodes and eigenvalues with respect to lighting variability. Our analysis makes use of an analytic formula for the irradiance in terms of spherical-harmonic coefficients of the illumination and shows, under appropriate assumptions, that the principal components or eigenvectors are identical to the spherical harmonic basis functions evaluated at the surface normal vectors. Our m...

• Type: Journal
• Year: 2002
• Where: PAMI
• Authors: Ravi Ramamoorthi
• Keywords: PAMI 2002, Principal Components, Principal Eigenmodes, Spherical Harmonics