In many sparse reconstruction problems, M observations are used to estimate K components in an N dimensional basis, where N > M ≫ K. The exact basis vectors, however, are not known a priori and must be chosen from an M × N matrix. Such under-determined problems can be solved using an ℓ2 optimization with an ℓ1 penalty on the sparsity of the solution. There are practical applications in which multiple measurements can be grouped together, so that K × P data must be estimated from M × P observations, where the ℓ1 sparsity penalty is taken with respect to the vector formed using the ℓ2 norms of the rows of the data matrix. In this paper we develop a computationally efficient block partitioned ho-motopy method for reconstructing K × P data from M × P observations using a grouped sparsity constraint, and compare its performance to other block reconstruction algorithms.
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