Lobbes: An Algorithm for Sparse-Spike Deconvolution

This letter proposes an algorithm for solving the sparse-spike deconvolution problem, named Lobbes (Lasso-based binary search for parameter selection). It improves the fast iterative shrinkage and threshold algorithm for Toeplitz-sparse matrix factorization by performing three steps to find a suitable regularization parameter: 1) a normalization procedure over the input data; 2) a binary search step based on the least absolute shrinkage and selection operator; and 3) the elimination of consecutive peaks similar to non-maximum suppression. Such parameter allows us to find a solution with a specified sparsity. We compare our results against the original algorithm and with the known sparse-inducing greedy approach of orthogonal matching pursuit. Relative to state-of-the-art, results demonstrate that Lobbes generates better results: better signal-to-noise ratio of the reconstructed signal and better result for reflectivity peaks. We also derive a new way to measure the quality of the deconvolution.