Towards Efficient and Compact Phenomenological Representation of Arbitrary Bidirectional Surface Reflectance

Abstract

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.

Analytic Bilinear Appearance Subspace Construction for Modeling Image Irradiance under Natural Illumination and Non-Lambertian Reflectance

“…The average reconstruction error as a function of the signal-to-noise (SNR) ratio in dB. Surface reflectance is represented using (a) spherical harmonics basis [3], (b) hemispherical Zernike-based basis [5] and (c) the isotropic hemispherical harmonics (HSH)-based Helmholtz reflectance basis [8]. One can observe that bilinear subspace still capture appearance even with low SNR levels.…”

“…We fit the BRDF measurements up-to reflectance order P = 8 to (1) spherical harmonics basis [3], (2) hemispherical Zernike-based basis [5] and (3) the isotropic version of hemispherical harmonics (HSH)-based Helmholtz reflectance basis 2 [8]. The main difference between the three types of basis is modeling the dependency of the surface BRDF w.r.t.…”

“…1. Representing the illumination by its spherical harmonics (SH) coefficients l m n as in [3,5] and the surface reflectance by its coefficients a q pr in the basis of [7], [5] or [8], the image irradiance can be defined as,…”

Modeling image irradiance under natural illumination and isotropic surface reflectance

“…In this paper, we derive the special case of the basis of [9] in case of isotropic surfaces 1 , where the surface appearance remains the same under rotations of the tangential plane about the surface normal. We prove that the number of reflectance basis are reduced drastically when compared to the anisotropic case [9].…”

“…In this paper, we are after deriving an analytic form for the image irradiance basis which spans the subspace representing the set of images under natural illumination conditions of non-Lambertian surfaces; taking into account that hemispherical functions present discontinuities at the boundary of the hemisphere when represented in the spherical domain [8], demanding more spectrum coefficients for accurate representation. Recently, Elhabian et al [9] addressed the physical compliance of reflectance hemispherical basis by defining a tensor product of hemispherical harmonics (HSH) to provide a compact and accurate representation for arbitrary reflectance, while satisfying the Helmholtz's law of reciprocity.…”