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Image encoding with triangulation wavelets

We demonstrate some wavelet-based image processing applications of a class of simplicial grids arising in finite element computations and computer graphics. The cells of a triangular grid form the set of leaves of a binary tree and the nodes of a directed graph consisting of a single cycle. The leaf cycle of a uniform grid forms a pattern for pixel image scanning and for coherent computation of coefficients of splines and wavelets. A simple form of image encoding is accomplished with a 1D quadrature mirror filter whose coefficients represent an expansion of the image in terms of 2D Haar wavelets with triangular support. A combination the leaf cycle and an inherent quadtree structure allow efficient neighbor finding, grid refinement, tree pruning and storage. Pruning of the simplex tree yields a partially compressed image which requires no decoding, but rather may be rendered as a shaded triangulation. This structure and its generalization to n-dimensions form a convenient setting for wavelet analysis and computations based on simplicial grids.