Modeling Lambertian Surfaces Under Unknown Distant Illumination Using Hemispherical Harmonics

A surface reflectance function represents the process of turning irradiance signals into outgoing radiance. Irradiance signals can be represented using low-order basis functions due to their low-frequency nature. Spherical harmonics (SH) have been used to provide such basis. However the incident light at any surface point is defined on the upper hemisphere, full spherical representation is not needed. We propose the use of hemispherical harmonics (HSH) to model images of convex Lambertian objects under distant illumination. We formulate and prove the addition theorem for HSH in order to provide an analytical expression of the reflectance function in the HSH domain. We prove that the Lambertian kernel has a more compact harmonic expansion in the HSH domain when compared to its SH counterpart. We present the approximation of the illumination cone in the HSH domain where the non-negative lighting constraint is forced. Our experiments illustrate that the 1st order HSH outperforms 1st and 2nd order SH in the process of image reconstruction as the number of light sources grows. In addition, we provide empirical justification of the nonnecessity of enforcing the non-negativity constraint in the HSH domain, allowing us to use unconstrained least squares which is faster than the constrained one, while maintaining the same quality of the reconstructed images.