Sparse Representation in Redundant Systems

CSIC Building (#406), Seminar Room 4122.
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On Adaptive Representations & Smoothness Spaces in R^d

Dr. Shai Dekel

Research & Development at RealTimeImage

Abstract:   In our setup, adaptivity is obtained by replacing the Euclidian metric by adaptive quasi-metrics. We derive the quasi-metrics from multiresolution structures of local geometric elements such as ellipsoids. For example, in the bivariate case, away from curve singularities, the ellipses are disks and the quasi-metrics are locally Euclidian. But in the vicinity of curve singularities, the ellipses are long and thin and the quasi-metrics are, locally, warps of the Euclidian metric. We can also construct quasi-metrics using higher order local elements that are `banana shaped'. Using the framework of spaces of homogeneous type, we obtain geometric wavelet representations in L<sub>p</sub> (R<sup>d</sup>), 1 < p < ∞, and smoothness spaces. The wavelets will be essentially supported on the local geometric elements. Adaptivity implies `sparser' representations and higher smoothness for functions with geometric structure, such as curve/surface singularities.