Sparse Representation Using Multidimensional Mixed-Norm Penalty With Application to Sound Field Decomposition

A sparse representation method for multidimensional signals is proposed. In generally used group-sparse representation algorithms, the sparsity is imposed only on a single dimension and the signals in the other dimensions are solved in the least-square-error sense. However, multidimensional signals can be sparse in multiple dimensions. For example, in acoustic array processing, in addition to the sparsity of the spatial distribution of acoustic sources, acoustic source signals will also be sparse in the time-frequency domain. We define a multidimensional mixed-norm penalty, which enables the sparsity to be controlled in each dimension. The majorization-minimization approach is applied to derive the optimization algorithm. The proposed algorithm has the advantages of a wide range for the sparsity-controlling parameters, a small cost of adjusting the balancing parameters, and a low computational cost compared with current methods. Numerical experiments indicate that the proposed method is also effective for application to sound field decomposition.