document: 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 . 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 main contribution is in extending these results to the single-viewpoint case , showing how the principal eigenmodes and eigenvalues are affected when only a limited subset -LRB- the upper hemisphere -RRB- of normals is available and the spherical harmonics are no longer orthonormal over the restricted domain . Our results are very close , both qualitatively and quantitatively , to previous empirical observations and represent the first essentially complete theoretical explanation of these observations
doc_bio_tags: O O O O O O O B I O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
extractive_keyphrases: spherical harmonics lighting variability convex lambertian object surface normal vectors principal eigenmodes irradiance
abstractive_keyphrases: analytic principal component analysis five-dimensional subspace principal eigenvalues radiance