- Series
- School of Mathematics Colloquium
- Time
- Thursday, September 29, 2016 - 11:05am for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Gitta Kutyniok – Technical University of Berlin – kutyniok@math.tu-berlin.de – http://www.math.tu-berlin.de/fachgebiete_ag_modnumdiff/angewandtefunktionalanalysis/v_menue/mitarbeiter/kutyniok/v_menue/home/parameter/en/
- Organizer
- Christopher Heil

Modern imaging data are often composed of several geometrically
distinct constituents. For instance, neurobiological images could
consist of a superposition of spines (pointlike objects) and
dendrites (curvelike objects) of a neuron. A neurobiologist might
then seek to extract both components to analyze their structure
separately for the study of Alzheimer specific characteristics.
However, this task seems impossible, since there are two unknowns
for every datum.
Compressed sensing is a novel research area, which was introduced in
2006, and since then has already become a key concept in various
areas of applied mathematics, computer science, and electrical
engineering. It surprisingly predicts that high-dimensional signals,
which allow a sparse representation by a suitable basis or, more
generally, a frame, can be recovered from what was previously
considered highly incomplete linear measurements, by using efficient
algorithms.
Utilizing the methodology of Compressed Sensing, the geometric
separation problem can indeed be solved both numerically and
theoretically. For the separation of point- and curvelike objects,
we choose a deliberately overcomplete representation system made of
wavelets (suited to pointlike structures) and shearlets (suited to
curvelike structures). The decomposition principle is to minimize
the $\ell_1$ norm of the representation coefficients. Our theoretical
results, which are based on microlocal analysis considerations, show
that at all sufficiently fine scales, nearly-perfect separation is
indeed achieved.
This project was done in collaboration with David Donoho (Stanford
University) and Wang-Q Lim (TU Berlin).