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This paper is concerned with linear inverse problems where the solution is assumed to have a sparse expansion with respect to several bases or frames. We were mainly motivated by the following two different approaches: (1) Jaillet and Torresani [F. Jaillet, B. Torresani, Time–frequency jigsaw puzzle: Adaptive multi-window and multi-layered Gabor expansions, preprint, 2005] and Molla and Torresani [S. Molla, B. Torresani, A hybrid audio scheme using hidden Markov models of waveforms, Appl. Comput. Harmon. Anal. (2005), in press] have suggested to represent audio signals by means of at least a wavelet for transient and a local cosine dictionary for tonal components. The suggested technology produces sparse representations of audio signals that are very efficient in audio coding. (2) Also quite recently, Daubechies et al. [I. Daubechies, M. Defrise, C. DeMol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math. 57 (2004) 1413–1541] have developed an iterative method for linear inverse problems that promote a sparse representation for the solution to be reconstructed. Here in this paper, we bring both ideas together and construct schemes for linear inverse problems where the solution might then have a sparse representation (we also allow smoothness constraints) with respect to several bases or frames. By a few numerical examples in the field of audio and image processing we show that the resulting method works quite nicely. |
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