### Mathematics Colloquium

# Gitta Kutyniok

### (Universität Osnabrück)

## "Compressed sensing and data separation"

** Date: ** |
Mon, November 15, 2010 |

** Time: ** |
17:15 |

** Place: ** |
Research II Lecture Hall |

**Abstract:** During the last two years, sparsity has become a key concept in various
areas of mathematics, computer science, and electrical engineering.
Sparsity methodologies explore the fundamental fact that many types of
data/signals can be represented or at least approximated by only a few
non-vanishing coefficients when choosing a suitable basis or, more
generally, a frame. If signals can be sparsely approximated, they can in
general be recovered from very few measurements using *l*_{1}
minimization techniques, which is typically coined Compressed Sensing.

One application of this novel methodology is the separation of data,
which are composed of two (or more) geometrically distinct constituents
-- for instance, spines (pointlike structures) and dendrites (curvelike
structures) in neurobiological imaging. Although it seems impossible to
extract those components -- as there are two unknowns for every datum --
suggestive empirical results using Compressed Sensing have already been
obtained.

In this talk we will first give an introduction into the concept of
sparse approximation and Compressed Sensing. Then we will develop a very
general theoretical approach to the problem of data separation based on
these methodologies. Finally, we will apply our results to the situation
of separation of pointlike and curvelike structures, for instance, in
neurobiological imaging, where a deliberately overcomplete
representation made of wavelets (suited to pointlike
structures) and shearlets (suited to curvelike structures) will be
chosen. Our theoretical results, which are based on applied harmonic
analysis and microlocal analysis considerations, show that at all
sufficiently fine scales, nearly-perfect
separation is indeed achieved.