Tree-based Algorithms for Compressed Sensing with Sparse-Tree Prior

Recent studies have shown that sparse representation can be used effectively as a prior in linear inverse problems. However, in many multiscale bases (e.g. wavelets), signals of interest (e.g. piecewisesmooth signals) not only have few significant coefficients, but also those significant coefficients are well-organized in trees. We propose to exploit this prior, named sparse-tree, for linear inverse problems with limited numbers of measurements. Toward this end, we present two efficient and effective algorithms named Tree-based Orthogonal Matching Pursuit (TOMP) and Tree-based Majorization-Minimization (TMM). Numerical results show that tree-based algorithms provide significantly better reconstruction quality compared to methods relying only on the sparse prior. Index Terms compressed sensing, linear inverse problems, sparse representations, sparse-tree prior, tree structures, wavelets, greedy algorithms, tree-based orthogonal matching pursuit, tree-based majorizationminimization.