Surface reflectance models based on characteristic functions

Surface reflectance functions (SRFs) and spectral power distributions (SPDs) of illuminants are typically modeled as elements in an N-dimensional linear function subspaces. Each SRF and SPD is represented by an N-vector and the mapping between SRF and SPD functions and an N-dimensional vector assigns N-dimensional "color" codes representing surface and light information. The N basis functions are chosen so that SRFs and SPDs can be accurately reconstructed from their N-dimensional vector codes. Typical rendering applications assume that the resulting mapping is an isomorphism where vector operations of addition, scalar multiplication and component-wise multiplication on the N-vectors can be used to model physical operations such as superposition of lights, light-surface interactions and inter-reflection. When N is small, this implicit isomorphism can fail even though individual SPDs and SRFs can still be accurately reconstructed by the codes. The vector operations do not mirror the physical. However, if the choice of basis functions is restricted to characteristic functions (that take on only the values 0 and 1) then the resulting map between SPDs/SRFs and N-vectors is an isomorphism that preserves the physical operations needed in rendering. The restriction to bases composed of characteristic functions can only reduce the goodness of fit of the linear function subspace to actual surfaces and lights. We will investigate how to select characteristic function bases of any dimension N (number of basis functions) and evaluate how accurately a large set of Munsell color chips can approximated as a function of dimension.