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  \textbf{\Large{\textsc{STANFORD UNIVERSITY}}}\\[5pt]
  \textbf{\Large{\textsc{DEPARTMENT OF STATISTICS}}}\\[5pt]
  \Large{\textsc{DEPARTMENTAL SEMINAR}}
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  4:15 p.m., Tuesday, May 6, 2008\\
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  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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  \textsl{Gitta Kutyniok} \\
  Department of Statistics\\
  Stanford University
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  \subsection*{$\ell_1$-Minimization and the Geometric Separation Problem}
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\noindent

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 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 sparsifies the pointlike structures
(wavelets) and the other sparsifies the curvelike structures
(curvelets or shearlets). The decomposition principle is to minimize
the $\ell_1$ 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 $\ell_1$-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).


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