Inverse synthetic aperture radar (ISAR) imaging of sparse aperture is a challenging problem. Noting the ISAR image generally exhibits strong sparsity, compressive sensing (CS) and sparse signal recovery methods are widely applied for sparse aperture data. However, the existing sparse aperture ISAR imaging algorithms are either computationally heavy or require manual adjustment of too many parameters, which limits their applications in the real-time ISAR imaging system. This paper proposes a high precision and computationally efficient ISAR imaging algorithm for sparse aperture. A generalized CS model with log-sum minimization is first considered for ISAR imaging, and then the iterative log-sum thresholding (ILT) algorithm is utilized to solve the optimization problem, where no large-scale matrix inversion is performed for reducing the computational overhead. Specifically, to accurately estimate ISAR image sparsity and avoid manually tuning parameters, a novel adjustment strategy of the trade-off parameter λ is applied for the ILT algorithm. To analyze the influence of different measurement matrix on the imaging algorithm, four sparse sampling patterns, including centralized sampling (CES), gap missing sampling (GMS), equally space sampling (ESS), and random missing sampling (RMS), are analyzed based on the mutual incoherence property (MIP). Experiments based on both simulated and measured data validate that the proposed algorithm can achieve well-focused ISAR images with a few seconds and is very efficient to implement.

Iterative thresholding-type algorithm, as one of the typical methods of compressed sensing (CS) theory, is widely used in sparse recovery field, because of its simple computational process. However, the estimation accuracy and convergence speed achieved by this type of algorithm with a nonconvex regularization, e.g., iterative half thresholding (IHalfT) algorithm, are not satisfactory, which limits its practical application. To improve the performance, a modified algorithm is proposed in this paper. Firstly, a novel non-negative expression is introduced in the algorithm to reduce the gap between the relaxation function and the objective function, which can bring tens of dB estimation accuracy improvement, and the convergence of the modified algorithm is verified. Secondly, the fundamental reasons for the remarkable improvement of performance are discussed and analyzed through theoretical derivation. Thirdly, the applicable conditions are elaborated for the modified algorithm. Finally, extensive experimental results demonstrate the effectiveness of the modified iterative thresholding-type algorithm with nonconvex regularization. Keywords-compressed sensing (CS), iterative hard thresholding (IHT), iterative soft thresholding (IST), iterative half thresholding (IHalfT), fast iterative shrinkage-thresholding algorithm (FISTA), normalized iterative hard thresholding (NIHT), alternating direction method of multipliers (ADMM).

The log-sum regularization has been always drawing widespread attention in the field of sparse problem. However, it brings about a non-convex, non-smooth, and non-Lipschitz optimization problem that is difficult to tackle. To overcome the problem, an iterative threshold algorithm of log-sum regularization is proposed in this paper. Firstly, by deducing the derivative mathematical expression of log-sum function, a property theorem about solution for log-sum regularization is established. Secondly, based on the above theorem, the optimal setting rules of the compromising parameters are elaborated, and the iterative log-sum threshold algorithm is proposed. Thirdly, under the situation that the compromising parameters of log-sum regularization are relatively small, it can be proven that the proposed algorithm converges to a local minimizer of log-sum regularization. Finally, a series of simulations are implemented to examine performance of the algorithm, and the results exhibit that the proposed algorithm outperforms the state-of-the-art algorithms in terms of iterations and precision.

The log-sum minimization has been widely employed in signal and image processing. In this paper, in order to analyze its applicable conditions and achievable boundary for compressed sensing scenario, some theoretical derivations of the log-sum function are studied based on the mutual coherence (MC) of measurement matrix. Firstly, the existence condition of the global minimum is demonstrated for the log-sum minimization without noise perturbation. Secondly, for the iterative threshold log-sum (ITL) algorithm, the theoretical performance guarantee of recovering sparse signal disturbed by noise is proposed. Additionally, comprehensive experimental results verify the theoretical analysis for the log-sum one and display the advantages and disadvantages of four sparse estimation algorithms, i.e., the ITL algorithm, ℓ1/2 regularization, ℓ1 regularization and orthogonal matching pursuit (OMP).