Multi-frame representations in linear inverse problems with mixed multi-constraints

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 Torrésani F. Jaillet, B. Torrésani, Time–frequency jigsaw puzzle: Adaptive multi-window and multi-layered Gabor expansions, preprint, 2005] and Molla and Torrésani S. Molla, B. Torrésani, 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.