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A Bregman iterative approach to linear inverse problems with compound regularizers

Afonso, M. ; Bioucas-Dias, J. ; Figueiredo, M. A. T.

A Bregman iterative approach to linear inverse problems with compound regularizers, Proc Workshop on Signal Processing with Adaptive Sparse Structured Representations - SPARS’09, Saint-Malo, France, Vol. , pp. - , April, 2009.

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Abstract
Some imaging inverse problems may require the solution to simultaneously exhibit properties that are not enforceable by a single regularizer. One way to attain this goal is to use a linear combinations of regularizers, thus encouraging the solution to simultaneously exhibit the characteristics enforced by each individual regularizer. In this paper, we address the optimization
problem resulting from this type of compound regularization using the split Bregman iterative method. The resulting algorithm only requires the ability to efficiently compute the denoising operator associated to each involved regularizer. Convergence is guaranteed by the theory behind the Bregman iterative approach to solving constrained optimization problems. In experiments with images that are simultaneously sparse and piece-wise
smooth, the proposed algorithm successfully solves the
deconvolution problem with a compound regularizer that is the linear combination of the l1 and total variation (TV) regularizers. The lowest MSE obtained with the (l1+TV) regularizer is lower than that obtained with TV or l1 alone, for any value of the corresponding regularization parameters.