The visual appearance of real-world surfaces is the net result of surface reflectance characteristics when exposed to illumination. Appearance models can be constructed using phenomenological models which capture surface appearance through mathematical modeling of the reflection process. This yields an integral equation, known as reflectance equation, describing the surface radiance, which depends on the interaction between the incident light field and the surface bidirectional reflectance distribution function (BRDF). The BRDF is a function defined on the cartesian product of two hemispheres corresponding to the incident and outgoing directions; the natural way to represent such a hemispherical function is to use hemispherical basis. However, due to their compactness in the frequency space, spherical harmonics (SH) have been extensively used for this purpose. In this paper, we address the geometrical compliance of hemispherical basis for representing surface BRDF. We propose a tensor product of the hemispherical harmonics (HSH) to provide a compact and efficient representation for arbitrary BRDFs, while satisfying the Helmholtz reciprocity property. We provide an analytical analysis and experimental justification that for a given approximation order, our proposed hemispherical basis provide better approximation accuracy when compared to Zernike-based basis, while avoiding the high computational complexity inherited from such polynomials. We validate our proposed Helmholtz HSH-based basis functions on Oren-Nayar and Cook-Torrance BRDF physical models.
