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Recent work by some of the authors presented a novel construction of a multiresolution analysis on manifolds and graphs, acted upon by a given symmetric Markov semigroup {T^t}t0, for which T^t has low rank for large t. This includes important classes of diffusion-like operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. The dyadic powers of an operator are used to induce a multiresolution analysis, analogous to classical Littlewood-Paley and wavelet theory, while associated wavelet packets can also be constructed. This extends multiscale function and operator analysis and signal processing to a large class of spaces, such as manifolds and graphs, with efficient algorithms. Powers and functions of T (notably its Green's function) are efficiently computed, represented and compressed. This construction is related and generalizes certain Fast Multipole Methods, the wavelet representation of Calderon-Zygmund and pseudo-differential operators, and also relates to algebraic multigrid techniques. The original diffusion wavelet construction yields orthonormal bases for multiresolution spaces {V_j}. The orthogonality requirement has some advantages from the numerical perspective, but several drawbacks in terms of the space and frequency localization of the basis functions. Here we show how to relax this requirement in order to construct biorthogonal bases of diffusion scaling functions and wavelets. This yields more compact representations of the powers of the operator, better localized basis functions. This new construction also applies to non self-adjoint semigroups, arising in many applications.

©2005 COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.