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We develop new mathematical results based on the spherical harmonic convolution framework for reflection from a curved surface. We derive novel identities, which are the angular frequency domain analogs to common spatial domain invariants such as reflectance ratios. They apply in a number of canonical cases, including single and multiple images of objects under the same and different lighting conditions. One important case we consider is two different glossy objects in two different lighting environments. Denote the spherical harmonic coefficients by $B_{lm}^{light,{material}}$, where the subscripts refer to the spherical harmonic indices, and the superscripts to the lighting (1 or 2) and object or material (again 1 or 2). We derive a basic identity, $B^{\rm 1,1}_{lm}$$B^{\rm 2,2}_{lm}$ = $B^{\rm 1,2}_{lm}$$B^{\rm 2,1}_{lm}$, independent of the specific lighting configurations or BRDFs. While this paper is primarily theoretical, it has the potential to lay the mathematical foundations for two important practical applications. First, we can develop more general algorithms for inverse rendering problems, which can directly relight and change material properties by transferring the BRDF or lighting from another object or illumination. Second, we can check the consistency of an image, to detect tampering or image splicing.

A Theory of Spherical Harmonic Identities for BRDF/Lighting Transfer and Image Consistency