Construction of unimodular tight frames for compressed sensing using majorization-minimization

Abstract

In this paper, we propose a method to construct uni-modular tight frames (UMTFs), which are tight frames with the additional constraint that every entry of the matrix has the same magnitude. UMTFs are useful in many applications, since multiplication of a UMTF by a vector can be implemented in polar coordinates using very low computational cost. Since normalized UMTFs are unit norm tight frames (UNTFs), and since a UNTF is a minimizer of the frame potential, we propose an algorithm to find UMTFs by minimizing the frame potential. We show that minimizing the frame potential is equivalent to minimizing the total coherence when the frame is unimodular. We use the majorization-minimization approach to propose a low complexity, iterative, fast-converging algorithm for minimizing the frame potential. We also extend our algorithm to the cases where the phase angles of the sensing matrix are required to belong to a given finite set of feasible angles, and to the case where the signal being sampled is sparse in an arbitrary, possibly non-canonical basis. We illustrate the utility of our proposed construction in the context of sparse signal recovery. Partial DFT matrices, obtained by randomly selected rows from the full DFT matrix, are UMTFs. However, they perform poorly when dealing with signals that admit a sparse representation in the wavelet, Fourier and discrete cosine transform domains. In such scenarios, we illustrate the superior performance of our construction compared to the partial DFT, complex Gaussian and Bernoulli random matrices through simulations. The proposed algorithm offers the same performance as the partial DFT matrix, and outperforms the complex Gaussian and Bernoulli random matrices, when the signal is sparse in the canonical basis.