Abstract
In this paper, a new class of Fourier-based matrices is studied for deterministic compressed sensing. Initially, a basic partial Fourier matrix is introduced by choosing the rows deterministically from the inverse discrete Fourier transform (DFT) matrix. By row/column rearrangement, the matrix is represented as a concatenation of DFT-based submatrices. Then, a full or a part of columns of the concatenated matrix is selected to build a new M × N deterministic compressed sensing matrix, where M = p r and N = L(M + 1) for prime p, and positive integers r and L ≤ M - 1. Theoretically, the sensing matrix forms a tight frame with small coherence. Moreover, the matrix theoretically guarantees unique recovery of sparse signals with uniformly distributed supports. From the structure of the sensing matrix, the fast Fourier transform (FFT) technique can be applied for efficient signal measurement and reconstruction. Experimental results demonstrate that the new deterministic sensing matrix shows empirically reliable recovery performance of sparse signals by the CoSaMP algorithm.
Highlights
Compressed sensing is a novel and emerging technology with a variety of applications in imaging, data compression, and communications
From the structure of our new sensing matrix, the fast Fourier transform (FFT) technique can be applied for efficient signal measurement and reconstruction
Experimental results demonstrate that the new deterministic compressed sensing matrix, together with the CoSaMP recovery algorithm [20], empirically guarantees sparse signal recovery with high reliability
Summary
Compressed sensing (or compressive sampling) is a novel and emerging technology with a variety of applications in imaging, data compression, and communications. The concatenated structure allows the new sensing matrix to offer the various admissible column numbers while keeping it as an incoherent tight frame and enables efficient processing for measurement and reconstruction in compressed sensing. We would like to stress that it is not a trivial task to obtain the concatenated structure from the basic partial Fourier matrix by row/column rearrangement. Our new sensing matrix theoretically guarantees unique recovery of sparse signals with uniformly distributed supports with high probability. Experimental results demonstrate that the new deterministic compressed sensing matrix, together with the CoSaMP recovery algorithm [20], empirically guarantees sparse signal recovery with high reliability.
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