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L_1-Minimization and the Geometric Separation ProblemPDE and Applied Math Seminar
|Speaker: ||Gitta Kutyniok, Stanford|
|Location: ||1143 MSB|
|Start time: ||Thu, May 8 2008, 2:00PM|
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 already been obtained.
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. Firstly, we use a viewpoint deriving
from microlocal analysis to understand heuristically why separation might be
possible and to organize a rigorous analysis. Secondly, 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
algorithms. One is based on frames of radial wavelets and curvelets and the
uses orthonormal wavelets and shearlets.
This is joint work with David Donoho (Stanford University)