Applied Math Seminar
Spring Quarter 2007
3:15 p.m.
Sloan Mathematics Corner
Building 380, Room 380-C

Friday, April 20, 2007

Gabriel Peyre
Ceremade, Universite Paris-Dauphine

Adaptive Sparse Representation of Geometric Textures


Abstract:

In this talk, I will show how adapted dictionaries can be trained to approximate and generate geometric texture patterns. I will restrict myself to two different approaches, namely bandlets dictionaries and dictionaries learned from data.

Bandlets dictionaries are well suited to approximate locally parallel textures that exhibit a strong anisotropy. Each basis is parameterized by a geometric flow that follows closely the texture patterns. The resulting basis vectors generate a tight frame of elongated atoms that can sparsify turbulent textures. A sparse modeling over a well chosen bandlet dictionary can be used for various tasks such as texture synthesis, compressed sensing decoding or morphological component separation.

Rather that using ad-hoc geometrical constraints for textures, one can try to infer the model from a set of exemplar input textures. A way to perform this modeling is to impose a sparsity assumption on the set of patches extracted from the texture. This sparsity can be optimized by learning a dictionary to optimally represent the input patches. The resulting model can smoothly interpolate between aggressive compression routinely used in computational harmonic analysis and realistic texture synthesis required by computer graphics applications. This texture model also finds applications in problems such as structure+texture decomposition or image inpainting.