Application of compressed sensing for image compression based on optimized Toeplitz sensing matrices

In compressed sensing, the Toeplitz sensing matrices are generated by randomly drawn entries and further optimizes them with suitable optimization methods. However, during an optimization process, state-of-the-art optimization methods tend to lose control over the structure of measurement matrices. In this paper, we proposed the novel approach for optimization of Toeplitz sensing matrices based on evolutionary algorithms such as Genetic Algorithm (GA), Simulated Annealing (SA), and Particle Swarm Optimization (PSO) for compression of an image signal. Furthermore, we investigated the performance of Basis Pursuit (BP) and Orthogonal Matching Pursuit (OMP) algorithms for the reconstruction of the images. The proposed optimized Toeplitz sensing matrices based on evolutionary algorithms such as GA, SA, and PSO exhibit a significant reduction in the mutual coherence (μ) and thus improved the recovery performance of 2D images compared to state-of-the-art non-optimized Toeplitz sensing matrices. The result reveals that the optimized Toeplitz sensing matrices with Basis Pursuit (BP) achieved more accurate results with a robust and uniform reconstruction guarantee compared to the OMP algorithm. However, BP shows the slow reconstruction performance of the image signal. On the other hand, an optimized Toeplitz sensing matrix with OMP shows a fast reconstruction guarantee, but at the cost of a reduction in the PSNR. Furthermore, the proposed approach retains the structure of Toeplitz sensing matrices and improves the image recovery performance of compressed sensing. Finally, the experimental results validate the effectiveness of the proposed method based on evolutionary algorithms for image compression.

Tight bounds on the mutual coherence of sensing matrices for Wigner D-functions on regular grids

Many practical sampling patterns for function approximation on the rotation group utilizes regular samples on the parameter axes. In this paper, we analyze the mutual coherence for sensing matrices that correspond to a class of regular patterns to angular momentum analysis in quantum mechanics and provide simple lower bounds for it. The products of Wigner d-functions, which appear in coherence analysis, arise in angular momentum analysis in quantum mechanics. We first represent the product as a linear combination of a single Wigner d-function and angular momentum coefficients, otherwise known as the Wigner 3j symbols. Using combinatorial identities, we show that under certain conditions on the bandwidth and number of samples, the inner product of the columns of the sensing matrix at zero orders, which is equal to the inner product of two Legendre polynomials, dominates the mutual coherence term and fixes a lower bound for it. In other words, for a class of regular sampling patterns, we provide a lower bound for the inner product of the columns of the sensing matrix that can be analytically computed. We verify numerically our theoretical results and show that the lower bound for the mutual coherence is larger than Welch bound. Besides, we provide algorithms that can achieve the lower bound for spherical harmonics.