Adaptive multivariate approximation using binary space partitions and geometric wavelets

The binary space partition (BSP) technique is a simple and efficient method to adaptively partition an initial given domain to match the geometry of a given input function. As such, the BSP technique has been widely used by practitioners, but up until now no rigorous mathematical justification for it has been offered. Here we attempt to put the technique on sound mathematical foundations, and we offer an enhancement of the BSP algorithm in the spirit of what we are going to call geometric wavelets. This new approach to sparse geometric representation is based on recent developments in the theory of multivariate nonlinear piecewise polynomial approximation. We provide numerical examples of n-term geometric wavelet approximations of known test images and compare them with dyadic wavelet approximation. We also discuss applications to image denoising and compression.