[BANANA] LA/Opt seminar Wed Feb 6 (Gitta Kutyniok)

[Michael A. Saunders]

   Linear Algebra and Optimization Seminar (CME510)
   iCME, Stanford University
   http://icme.stanford.edu/seminars/seminars.php

   4:15pm Wed Feb 6, 2008
   Rm 317 Wallenberg Hall (Bldg 160)

   L_1-MINIMIZATION AND THE GEOMETRIC SEPARATION PROBLEM

   Prof. Gitta Kutyniok
   Heisenberg-Fellow of the German Research Foundation
   Visiting Dept of Statistics, Stanford University
   kutyniok at stanford.edu

Consider an image mixing two (or more) geometrically distinct
but spatially overlapping phenomena - for instance, pointlike
and curvelike structures in astronomical imaging of galaxies.
This raises the Problem of Geometric Separation, taking a single
image and extracting two images: one containing just the
pointlike phenomena and one containing just the curvelike
phenomena.  Although this seems impossible (as there are two
unknowns for every datum), suggestive empirical results have
been obtained by Jean-Luc Starck and collaborators.

We develop a theoretical approach to the Geometric Separation
Problem in which a deliberately overcomplete representation is
chosen made of two frames.  One is suited to pointlike
structures (wavelets) and the other suited to curvelike
structures (curvelets or shearlets).  The decomposition
principle is to minimize the l_1-norm of the analysis (rather
than synthesis) frame coefficients.  This forces the pointlike
objects into the wavelet components of the expansion and the
curvelike objects into the curvelet or shearlet part of the
expansion.  Our theoretical results show that at all
sufficiently fine scales, nearly-perfect separation is achieved.

Our analysis has two interesting features.  First, we use a
viewpoint deriving from microlocal analysis to understand
heuristically why separation might be possible and to organize a
rigorous analysis.  Second, we introduce some novel technical
tools: cluster coherence (rather than the now-traditional
singleton coherence) and l_1-minimization in frame settings
(including those where singleton coherence within one frame may
be high).

Our general approach applies in particular to two variants of
geometric separation algorithms.  One is based on frames of
radial wavelets and curvelets and the other uses orthonormal
wavelets and shearlets.

This is joint work with David Donoho (Stanford University).