Sparse approximations in complex domain based on BM3D modeling

In this paper the concept of sparsity for complex-valued variables is introduced in the following three types: directly in complex domain and for two real-valued pairs phase/amplitude and real/imaginary parts of complex variables. The nonlocal block-matching technique is used for sparsity implementation and filter design for each type of sparsity. These filters are complex domain generalizations of the Block Matching 3D collaborative (BM3D) filter based on the high-order singular value decomposition (HOSVD) in order to generate group-wise adaptive analysis/synthesis transforms. Complex domain denoising is developed and studied as a test-problem for comparison of the designed filters as well as the different types of sparsity modeling.