Phase retrieval has become a very active area of research. We will classify when phase retrieval by Parseval frames passes to the Naimark complement and when phase retrieval by projections passes to the orthogonal complements. We introduce a new concept we call norm retrieval and show that this is what is necessary for passing phase retrieval to complements. This leads to a detailed study of norm retrieval and its relationship to phase retrieval. One fundamental result: a frame {ϕ i } M i=1 yields phase retrieval if and only if {T ϕ i } M i=1 yields norm retrieval for every invertible operator T .1991 Mathematics Subject Classification. Primary 32C15 .
This paper considers compressed sensing of different size block-sparse signals, i.e. signals with nonzero elements occurring in blocks with different lengths. A new sufficient condition for mixed l2/l1-optimization algorithm is derived to successfully recover k-sparse signals. We show that if the signal possesses k-block sparse structure, then via mixed l2/l1-optimization algorithm, a better reconstruction results can be achieved in comparison with the conventional l1-optimization algorithm and fixed-size mixed l2/l1-optimization algorithm. The significance of the results presented in this paper lies in the fact that making explicit use of different block-sparsity can yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the structure in the signal.Index Terms-Compressed sensing, block-sparsity, mixed l2/l1-optimization algorithm.
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