The inertial relaxed algorithm with Armijo-type line search for solving multiple-sets split feasibility problem

The multiple-sets split feasibility problem is the generalization of split feasibility problem, which has been widely used in fuzzy image reconstruction and sparse signal processing systems. In this paper, we present an inertial relaxed algorithm to solve the multiple-sets split feasibility problem by using an alternating inertial step. The advantage of this algorithm is that the choice of stepsize is determined by Armijo-type line search, which avoids calculating the norms of operators. The weak convergence of the sequence obtained by our algorithm is proved under mild conditions. In addition, the numerical experiments are given to verify the convergence and validity of the algorithm.

On full spark frames via Cauchy matrices

Full spark frames have been widely applied in sparse signal processing, signal reconstruction with erasures and phase retrieval. Since testing whether a given frame is full spark is hard for NP under randomized polynomial-time reductions, hence the deterministic full spark (DFS) frames are particularly significant. However, the degree of freedom of choices of DFS frames is not enough in practical applications because the DFS frames are well known as Vandermonde frames and harmonic frames. In this paper, we focus on the deterministic constructions of full spark frames. We present a new and effective method to construct DFS frames by using Cauchy matrices. We also construct the DFS frames by using Cauchy-Vandermonde matrices. Finally, we show that full spark tight frames can be constructed from generalized Cauchy matrices.

Sparse SAR Imaging and Quantitative Evaluation Based on Nonconvex and TV Regularization

Sparse signal processing has been used in synthetic aperture radar (SAR) imaging due to the maturity of compressed sensing theory. As a typical sparse reconstruction method, L1 regularization generally causes bias effects as well as ignoring region-based features. Our team has proposed to linearly combine the nonconvex penalty and the total variation (TV)-norm penalty as a compound regularizer in the imaging model, called nonconvex and TV regularization, which can not only reduce the bias caused by L1 regularization but also enhance point-based and region-based features. In this paper, we use the variable splitting scheme and modify the alternating direction method of multipliers (ADMM), generating a novel algorithm to solve the above optimization problem. Moreover, we analyze the radiometric properties of sparse-signal-processing-based SAR imaging results and introduce three indexes suitable for sparse SAR imaging for quantitative evaluation. In experiments, we process the Gaofen-3 (GF-3) data utilizing the proposed method, and quantitatively evaluate the reconstructed SAR image quality. Experimental results and image quality analysis verify the effectiveness of the proposed method in improving the reconstruction accuracy and the radiometric resolution without sacrificing the spatial resolution.