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SUMMARY:SURE-based Parameter Selection for Sparse Regularization of Invers
 e Problems - Gabriel Peyré (CNRS and University Paris-Dauphine\, FRANCE)
DTSTART:20130307T150000Z
DTEND:20130307T160000Z
UID:TALK42971@talks.cam.ac.uk
CONTACT:Carola-Bibiane Schoenlieb
DESCRIPTION: In this talk\, I will reviews both theoretical and numerical 
 aspects of parameter selection for inverse problems regularization. We foc
 us our attention to a set of methods built on top of the Generalized Stein
  Unbiased Risk Estimator (GSURE) [1]. GSURE allows one to estimate without
  bias the squared error on the orthogonal of the kernel of the imaging ope
 rator. One can thus automatically set the value of some parameters of the 
 method by minimizing the GSURE. Computing the GSURE necessitates the estim
 ation of the generalized degree of freedom of the method. We prove in [2] 
 a formula that gives an unbiased estimator of this degree of freedom for s
 parse l1 analysis regularization. This includes analysis-type translation 
 invariant wavelet sparsity and total variation. This theoretical analysis 
 provides a better understanding of the behavior of the methods\, but is di
 fficult to compute numerically for large scale imaging problems. Indeed\, 
 convex optimization solvers only provide an approximate solution\, which d
 oes not lead to a stable estimation of the number of degrees of freedom. W
 e addressed this issue in [3] by proposing a novel algorithm that computes
  a unbiased and stable estimator of the risk associated to each iterate of
  a large class of convex optimization methods. The algorithms that I will 
 present can be implemented and tested via the "Numerical Tours" plateform 
 that is available online from www.numerical-tours.com. This is a joint wor
 k with Samuel Vaiter\, Charles Deledalle\, Jalal Fadili and Charles Dossal
 \n\nBibliography:\n\n[1] Y. C. Eldar\, “Generalized sure for exponential
  families: Applications to regularization\,” IEEE Transactions on Signal
  Processing\, vol. 57\, pp. 471– 481\, 2009.\n\n[2] S. Vaiter\, C. Deled
 alle\, G. Peyre\, J. Fadili\, and C. Dossal\, “Local be- havior of spars
 e analysis regularization: Applications to risk estimation\,” Technical 
 report\, Preprint Hal-00687751\, 2012.\n\n[3] C. Deledalle\, S. Vaiter\, G
 . Peyre\, J. Fadili\, and C. Dossal\, “Proximal splitting derivatives fo
 r risk estimation\,” Proc. NCMIP’12\, 2012.
LOCATION:MR 14\, CMS
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