Joint random stepped frequency ISAR imaging and autofocusing based on 2D alternating direction method of multipliers

Abstract

Random stepped frequency (RSF) inverse synthetic aperture radar (ISAR) has attracted more attention due to its outstanding electronic counter-countermeasure (ECCM) performance by frequency agility. However, the two-dimensional (2D) sparse sampling echo, i.e. sparse sub-pulses and sparse aperture, increases the difficulty of 2D high-resolution ISAR imaging and autofocusing. Aiming to solve this problem, a novel 2D alternating direction method of multipliers (2D-ADMM) based imaging and autofocusing framework is proposed to achieve 2D high-resolution for 2D sparse RSF ISAR. In order to obtain the optimal value for phase correction, the phase error corresponding to each sub-pulse is estimated by solving an unconstrained optimization problem. To jointly achieve autofocusing during the process of ISAR image reconstruction, the phase error estimation is merged into the image reconstruction framework, and then two different strategies under the 2D ADMM framework are derived to iteratively solve this compound optimization problem in matrix form. Experiments based on both simulated and measured data validate the effectiveness of the proposed 2D joint methods, meanwhile, maintaining an acceptable imaging efficiency.

Introduction

Inverse synthetic aperture radar (ISAR) has the capacity to obtain two-dimensional (2D) images of noncooperative moving targets, which is widely used in modern military and civilian areas [1]. Generally, the high range resolution is achieved by transmitting wideband signal, which is a great challenge for electronic interference and radar hardware requirements. Random stepped frequency (RSF) waveform has attracted more interests in recent years due to its outstanding electronic counter-countermeasure (ECCM) performance [2]. Furthermore, RSF ISAR can achieve high-resolution images by using multiple narrow band sub-pulses, which reduces the instantaneous bandwidth dramatically. However, unlike traditional ISAR, this radar requires a long coherence processing interval (CPI) to transmit several bursts of sub-pulses, which is usually unachievable in practical applications. To address this problem, sparse RSF (SRSF) technique has been introduced into the ISAR imaging filed by transmitting less sub-pulses and bursts to achieve a short CPI. Furthermore, it is suitable for the modern multifunctional radar with the ability to observe multiple targets, which inevitably leads to the absence of pulses corresponding to the fixed target. Therefore, it has great significance to pay attention to the sparse RSF ISAR imaging.

In general, a 2D high-resolution ISAR imaging result can be easily reconstructed by the traditional 2D range-Doppler (RD) algorithm under the conditions of full data and no translational motion or phase error. Nevertheless, the frequency agility of the sub-pulses and the incomplete received data cause the traditional RD algorithm invalid. In recent years, compressive sensing (CS) [3], which can reconstruct a sparse signal with partial measurements, has been introduced in sparse ISAR imaging and is proven effective than many other methods [4,5]. Compared with traditional linear stepped frequency signal, the random jumping of the sub-pulse carrier frequency follows with the randomness requirements of the CS theory. Furthermore, random sampling can also be used to reduce echo data. Therefore, numerous CS based RSF ISAR imaging works have been done. One of the most common schemes is like the conventional RD methods, in which the high-resolution profiles corresponding to the range and cross-range directions are synthesized individually [6,7]. Therefore, many CS algorithms can be easily applied to RSF ISAR imaging, including orthogonal matching pursuit (OMP) [8], fast iterative shrinkage thresholding algorithm (FISTA) [9], and alternating direction method of multipliers (ADMM) [10]. However, this scheme damages the 2D coupling information of the received data, which inevitably influence the imaging performance. To address this problem, the 2D joint imaging scheme, that is achieved high-resolution profiles in both the range and cross-range simultaneously, has been proposed. In [11,12], they studied a 2D joint imaging method by converting the sparse data matrix into a 1D sparse reconstruction model. Nevertheless, vectorized optimization would face with the high computational complexity and a large amount of memory due to the big dictionary dimension. Furthermore, some 2D joint algorithms, i.e., 2D smoothed $l_0$ norm (2D-SL0) [13], 2D-FISTA [14], 2D-ADMM [15], have been developed to deal with the data matrix directly. Owing to the matrix form of the reconstruction model, not only greatly reduces the amount of computational complexity, but also the 2D coupling information can be protected. Considering the high efficiency and precision of the 2D-ADMM, we choose it to deal with the CS framework proposed in this paper.

Accurate motion compensation is the premise of high-resolution ISAR imaging. If the target motion is not accurately compensated, the residual motion will cause uncompensated phase errors, leading to defocusing of the imaging result. In addition, some other factors, such as system error and rotation error, further deteriorate the imaging quality. Therefore, phase autofocusing is a key step for high-resolution imaging. However, most autofocusing algorithms, which rely on the high coherence of the phase error between different pulses, are unsuitable for the incomplete data because of the coherence destroyed by the sparse aperture [16]. Some parametric algorithms have been adopted to the RSF ISAR, which model the target motion as a first or second order parameters approximation problem [17]. In [18], a modified eigenvector-based autofocus method, which can correct phase errors within sparse aperture (SA) measurements, is studied. Recently, embedding phase autofocusing into the ISAR imaging framework, that is iteratively completing phase compensation and ISAR imaging simultaneously, has been proved to have better performance than other algorithms [19]. Thereafter, minimum entropy minimum entropy [20, 21] and maximum contrast [22] based phase autofocusing have been integrated into the sparse ISAR imaging framework, which can be efficiently solved by Bayesian methods [23] and ADMM. Similar work has also been studied in [24] and [25]. However, although these algorithms can realize phase error correction and high-resolution imaging at the same time, they only consider the influence of phase error on cross-range processing. In fact, for RSF ISAR, the phase error exists in all received sub-pulses, which affects not only the cross-range processing, but also the range profile synthesizing. Therefore, Shao et al. [26] proposed a 2D joint ISAR imaging and motion compensation method for 2D sparse stepped-frequency signal. They constructed the target motion as a second-order approximation model and compensated it after estimating those parameters. However, it cannot eliminate the phase error induced by the complex motion or system imperfection. In [27], a phase compensation and image autofocusing method for RSF ISAR is studied. In fact, it can be regarded as a vectorized reconstruction model, which is difficult to be widely used in practical work because of the huge computational complexity and the demand for storage space.

Aiming to achieve the autofocusing and high-resolution ISAR imaging for the 2D sparse received data, we proposed a novel joint RSF ISAR imaging and autofocusing method in this paper. First, in order to make full use of the 2D coupling information, a vectorized sparse representation framework integrated with 2D phase error estimation is constructed for autofocusing and high-resolution ISAR imaging. Then, ADMM is utilized to optimize this sparsity constraint under-determined problem. Furthermore, the vectorized reconstruction problem has been reshaped to matrix form based on the 2D-ADMM framework, to reduce the computation and memory usage. In order to obtain the optimal value for phase correction, the phase error corresponding to each sub-pulse is estimated by solving an unconstrained optimization problem. Then, two easy-to-implement iterative algorithms are derived to solve this compound optimization problem efficiently to jointly achieve autofocusing and high-resolution ISAR imaging. Finally, both simulated and measured data experiments show that the proposed 2D joint methods have an outstanding performance than other imaging and autofocusing methods, meanwhile, maintaining an acceptable imaging efficiency.