Nonlinear Approximation of Spatiotemporal Data Using Diffusion Wavelets

Austrian Research Centers GmbH smart systems Division Donau City Str. 1, 1220 Vienna, Austria www.smart-systems.at Motivation ● Recent concept of Diffusion Wavelets (Coifman and Maggioni, 2006) allows construction of wavelet bases for functions defined on other than , such as certain domains, manifolds and graphs ● In this work: study the use of classical wavelet algorithms, lifted to a graph based setting, concentrating on an algorithm for nonlinear approximation of 2d+time image data Introduction ● Diffusion Wavelets: particular instance of spectral graph theory ● In place of the usual dilation on : diffusion operator on the data (given as a weighted graph) as scaling tool, dividing its spectrum (set of eigenvalues) into sets of different `frequencies’, obtaining a multiresolution analysis and an ONB for functions on the graph ● Diffusion Wavelet coefficients code structural similarity of the data encoded as a graph → Nonlinear approximation via thresholding on the coefficients just like in the classical wavelet algorithms (compression, denoising), obtaining a structure-preserving compression of the data Outlook ● Nonlinear approximation on a graph: can be seen as a first step towards structural spatiotemporal wavelet segmentation ● In order to obtain a true segmentation: first idea is to use a Hidden Markov Model on the coefficient tree, classifying the coefficients as ‘large’ or ’small’ and following them across resolution levels ● Segmentation method we work on: could be used as a way of a combined spatial and motion segmentation of a whole image sequence → instance of ’offline’ tracking method (similar to the normalized cut tracking, involving a true multiresolution analysis on the data) n R