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SUMMARY:The Geometric Separation Problem: Imaging Science meets Compressed
  Sensing - Gitta Kutyniok (Technische Universität Berlin)
DTSTART:20120531T140000Z
DTEND:20120531T150000Z
UID:TALK36079@talks.cam.ac.uk
CONTACT:Dr Hansen
DESCRIPTION:Modern imaging data are often composed of several geometricall
 y\ndistinct constituents. For instance\, neurobiological images could\ncon
 sist of a superposition of spines (pointlike objects) and\ndendrites (curv
 elike objects) of a neuron. A neurobiologist might\nthen seek to extract b
 oth components to analyze their structure\nseparately for the study of Alz
 heimer specific characteristics.\nHowever\, this task seems impossible\, s
 ince there are two unknowns\nfor every datum.\n\nCompressed sensing is a n
 ovel research area\, which was introduced in\n2006\, and since then has al
 ready become a key concept in various\nareas of applied mathematics\, comp
 uter science\, and electrical\nengineering. It surprisingly predicts that 
 high-dimensional signals\,\nwhich allow a sparse representation by a suita
 ble basis or\, more\ngenerally\, a frame\, can be recovered from what was 
 previously\nconsidered highly incomplete linear measurements\, by using ef
 ficient\nalgorithms.\n\nUtilizing the methodology of Compressed Sensing\, 
 the geometric\nseparation problem can indeed be solved both numerically an
 d\ntheoretically. For the separation of point- and curvelike objects\,\nwe
  choose a deliberately overcomplete representation system made of\nwavelet
 s (suited to pointlike structures) and shearlets (suited to\ncurvelike str
 uctures). The decomposition principle is to minimize\nthe $\\ell_1$ norm o
 f the representation coefficients. Our theoretical\nresults\, which are ba
 sed on microlocal analysis considerations\, show\nthat at all sufficiently
  fine scales\, nearly-perfect separation is\nindeed achieved.\n\nThis proj
 ect was done in collaboration with David Donoho and Wang-Q\nLim.
LOCATION:MR19 (Potter Room\, Pavilion B)\, CMS
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