Accuracy of Spherical Harmonic Approximations for Images of Lambertian Objects under Far and Near Lighting

Various problems in Computer Vision become difficult due to a strong influence of lighting on the images of an object. Recent work showed analytically that the set of all images of a convex, Lambertian object can be accurately approximated by the low-dimensional linear subspace constructed using spherical harmonic functions. In this paper we present two major contributions: first, we extend previous analysis of spherical harmonic approximation to the case of arbitrary objects; second, we analyze its applicability for near light. We begin by showing that under distant lighting, with uniform distribution of light sources, the average accuracy of spherical harmonic representation can be bound from below. This bound holds for objects of arbitrary geometry and color, and for general illuminations (consisting of any number of light sources). We further examine the case when light is coming from above and provide an analytic expression for the accuracy obtained in this case. Finally, we show that low-dimensional representations using spherical harmonics provide an accurate approximation also for fairly near light. Our analysis assumes Lambertian reflectance and accounts for attached, but not for cast shadows. We support this analysis by simulations and real experiments, including an example of a 3D shape reconstruction by photometric stereo under very close, unknown lighting.