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SUMMARY:Constrained image restoration problems - Gabriele Steidl (Universi
 ty of Kaiserslautern)
DTSTART:20130228T130000Z
DTEND:20130228T140000Z
UID:TALK39398@talks.cam.ac.uk
CONTACT:Carola-Bibiane Schoenlieb
DESCRIPTION:We are interested in solving various image restoration problem
 s\nby constraint convex models.\n\nIn particular\, we deal with the minimi
 zation of seminorms $\\|L \\cdot\\|$ on $\\R^n$ under\nthe constraint of a
  bounded $I$-divergence $D(b\,H \\cdot)$.\nThe $I$-divergence is also know
 n as Kullback-Leibler divergence and appears\nin many models in imaging sc
 ience\, in particular when dealing with Poisson data.\nTypically\, $H$ rep
 resents here\, e.g.\, a linear blur operator and $L$ is some discrete deri
 vative operator.\nOur preference for the constrained approach over the cor
 responding penalized version\nis based on the fact that the $I$-divergence
  of data corrupted\, e.g.\, by Poisson noise\nor multiplicative Gamma nois
 e can be estimated by statistical methods.\nOur minimization technique res
 ts upon relations between constrained and penalized convex problems\nand r
 esembles the idea of Morozov's discrepancy principle.\nMore precisely\, we
  propose first-order primal-dual algorithms\nwhich reduce the problem to t
 he\nsolution of certain proximal minimization problems in each iteration s
 tep.\nThe most interesting of these proximal minimization problems\nis an 
 $I$-divergence constrained least squares problem.\nWe solve this problem b
 y connecting it to the corresponding $I$-divergence penalized least square
 s problem\nwith an appropriately chosen regularization parameter.\nTherefo
 re\, our algorithm produces not only a sequence of vectors\nwhich converge
 s to a minimizer of the constrained problem\nbut also a sequence of parame
 ters which convergences to a regularization parameter\nso that the penaliz
 ed problem has the same solution as our constrained one.\nFinally\, we dea
 l with Anscombe constrained problems via epigraphical projections.
LOCATION:MR 13\, CMS
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