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Table Of Integrals Series And Products

Robust principal component analysis (RPCA) via decomposition into low-rank plus sparse matrices offers a powerful framework for a large variety of applications such as image processing, video processing, and 3-D computer vision. Indeed, most of the time these applications require to detect sparse outliers from the observed imagery data that can be approximated by a low-rank matrix. Moreover, most of the time experiments show that RPCA with additional spatial and/or temporal constraints often outperforms the state-of-the-art algorithms in these applications. Thus, the aim of this paper is to survey the applications of RPCA in computer vision. In the first part of this paper, we review representative image processing applications as follows: 1) low-level imaging such as image recovery and denoising, image composition, image colorization, image alignment and rectification, multifocus image, and face recognition; 2) medical imaging such as dynamic magnetic resonance imaging (MRI) for acceleration of data acquisition, background suppression, and learning of interframe motion fields; and 3) imaging for 3-D computer vision with additional depth information such as in structure from motion (SfM) and 3-D motion recovery. In the second part, we present the applications of RPCA in video processing which utilize additional spatial and temporal information compared to image processing. Specifically, we investigate video denoising and restoration, hyperspectral video, and background/foreground separation. Finally, we provide perspectives on possible future research directions and algorithmic frameworks that are suitable for these applications.

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On the Applications of Robust PCA in Image and Video Processing

This letter considers the coherent integration problem for a maneuvering target, involving range migration (RM) and Doppler frequency migration (DFM) within one coherent pulse interval. A new coherent integration method, known as Radon-Lv’s distribution (RLVD), is proposed. It can not only eliminate the RM effect via jointly searching in the target’s motion parameters space, but also remove the DFM and achieve the coherent integration via Lv’s distribution (LVD). Finally, several simulations are provided to demonstrate the effectiveness. The results show that for detection ability, the proposed method is superior to the moving target detection (MTD), Radon-Fourier transform (RFT), and Radon-fractional Fourier transform (RFRFT) under low signal-to-noise-ratio (SNR) environment.

Coherent Integration for Maneuvering Target Detection Based on Radon-Lv’s Distribution

In this two-part paper, we consider the problem of adaptive multidimensional/multichannel signal detection in Gaussian noise with unknown covariance matrix. The test data (primary data) is assumed as a collection of sample vectors, arranged as the columns of a rectangular data array. The rows and columns of the signal matrix are both assumed to lie in known subspaces, but with unknown coordinates. Due to this feature of the signal structure, we name this kind of signal as the double subspace signal. Part I of this paper focuses on the adaptive detection in homogeneous environments, while Part II deals with the adaptive detection in partially homogeneous environments. Precisely, in this part, we derive the generalized likelihood ratio test (GLRT), Rao test, Wald test, as well as their two-step variations, in homogeneous environments. Three types of spectral norm tests (SNTs) are also introduced. All these detectors are shown to possess the constant false alarm rate (CFAR) property. Moreover, we discuss the differences between them and show how they work. Another contribution is that we investigate various special cases of these detectors. Remarkably, some of them are well-known existing detectors, while some others are still new. At the stage of performance evaluation, conducted by Monte Carlo simulations, both matched and mismatched signals are dealt with. For each case, more than one scenario is considered.

Adaptive Double Subspace Signal Detection in Gaussian Background—Part I: Homogeneous Environments

In the airborne or spaceborne radar applications, prolonging the coherent integration time is one of the effective methods to improve the radar detection ability of a weak maneuvering target, whereas the coherent integration performance may degrade due to the complex range migration (RM) and Doppler frequency migration (DFM) effects. In this paper, detection and motion parameter estimation for a weak maneuvering target with the third-order RM and DFM are considered. Firstly, Keystone transform is applied to compensate the linear range walk. Then, the matched filtering processing is performed in the range-frequency and azimuth-time domain to eliminate the residual coupling effects between range and azimuth. Finally, a well-focused image of a moving target is obtained, and three motion parameters, i.e., velocity, acceleration, and acceleration rate, are effectively estimated. In addition, as for a fast-moving target with Doppler ambiguity, two cases, i.e., target azimuth spectrum within a pulse repetition frequency (PRF) and spanning over neighboring PRF bands, are analyzed. Compared with the generalized Radon Fourier transform (GRFT), the proposed method can acquire a close integration performance but with lower computational complexity since the parameter searching dimension is reduced. Simulated processing results are provided to validate the effectiveness of the proposed method.

Long-Time Coherent Integration for Weak Maneuvering Target Detection and High-Order Motion Parameter Estimation Based on Keystone Transform

Weak-target detection and imaging are the challenging problems of airborne or spaceborne early warning radar. The envelope of a high-speed weak target after range compression spreads over range during the long observation period. To finely refocus a high-speed weak maneuvering target, motion parameters should be accurately obtained for compensating the envelope. This letter proposes a new imaging approach for high-speed maneuvering targets without a priori knowledge of their motion parameters. In this method, the azimuth compression function is constructed in a range and azimuth 2-D frequency domain, which can eliminate the coupling effect between range and azimuth. Theoretical analysis confirms that the methodology can precisely focus targets. Simulation results show that the proposed algorithm improves the performance for detecting and imaging high-speed maneuvering targets.

A New Method for Radar High-Speed Maneuvering Weak Target Detection and Imaging

High-resolution synthetic aperture radar (SAR) imagery of a wide area of surveillance is a difficult large-data problem. In the past few years, researchers have applied compressive sensing (CS) to SAR, as it exploits redundancy in signals. To further extend the sparse problem from the vector to the matrix, a new theory called matrix completion (MC) has attracted much attention, which can complete a matrix from a small set of corrupted entries based on the assumption that the matrix is essentially of low rank. Inspired by this technique, a novel SAR imaging algorithm is proposed in this letter to deal with the undersampled data. After representing the data of a range cell as a matrix, the phase is compensated to keep the matrix holding the property of low rank. Subsequently, MC can be utilized to recover the full-aperture data in the new constructed matrix. Since the data are completely unsampled in the corresponding azimuth cells, the proposed method has effectively conquered the restriction of previous applications that each received channel must have a small number of samples. The final results in both simulation and real-data experiments show that the targets can be well focused even in the scenario of discarding a large percentage of the received pulses. Moreover, when compared with CS, the method is not required to design the complicated measurement matrix.

SAR Imaging With Undersampled Data via Matrix Completion

This paper addresses the problem of detecting a signal in partially homogeneous and homogeneous environments. In partially homogeneous environments, i.e., both the test data and training data share the same noise covariance matrix structure up to an unknown scaling factor, a persymmetric adaptive coherence estimator (Per-ACE) detector is proposed. By exploiting the persymmetric structure of the covariance matrix, the Per-ACE can reduce training data requirements. Furthermore, the expressions for the probabilities of false alarm and detection are derived along with the distribution of the loss factor β. In homogeneous environments, a persymmetric adaptive matched filter (Per-AMF) detector has been presented. However, its probability of detection has not been obtained yet. Thus, we derive the expression for the probability of detection. For both the Per-ACE and Per-AMF, numerical results of these proposed expressions are confirmed with those of Monte Carlo trials. In addition, simulation results show that the proposed Per-ACE outperforms the conventional ACE in training-limited scenarios.

Persymmetric Adaptive Detectors in Homogeneous and Partially Homogeneous Environments

Clutter suppression and ground moving target indication are challenging tasks in multichannel synthetic aperture radar (SAR) systems. In recent years, robust principal component analysis (RPCA) has attracted much attention for its good performance in distinguishing the different parts from a set of correlative database. Therefore, we propose a fast RPCA-based detection method for multichannel SAR under a strong clutter background in this letter even with channel unbalance or platform motion error. Subsequently, as the existing space-time adaptive processing (STAP) method would fail when the training samples are contaminated by the moving target, we apply the RPCA-based method in the range-Doppler domain to improve the performance of STAP. Since the regions of targets can be detected via RPCA, the remaining samples, which can be regarded as only clutter, are used to estimate the covariance matrix for further processing. The final experiments based on real measured data set show its good performance under the strong clutter background. Although the RPCA-based result differs from that of the STAP method, they can work cooperatively to get a more robust detection performance.

Strong Clutter Suppression via RPCA in Multichannel SAR/GMTI System

Multiple-input multiple-output synthetic aperture radar (MIMO SAR) system has drawn considerable attention because of its extra degrees of freedom for high resolution and wide swath compared with the traditional multichannel SAR system. But how to extract the matched signal without the unmatched interferences is the foremost task for MIMO SAR system. In this paper, by using the orthogonal frequency division multiplexing chirp signals as the transmitted signals, it is demonstrated that the robust principal component analysis (RPCA) method can be successfully employed for ground moving target indication (GMTI) with no need for separating the matched signal and unmatched interferences. It is because the unmatched interference is proven to have low-rank property and noise-level magnitude, which can be separated apart from the matched signal with the RPCA method. However, the traditional RPCA methods may be restricted by the high computational burden due to the complex decompositions and multiple iterations. Hence, a fast interferometry RPCA method is proposed specially for GMTI mode, which takes full advantage of the characteristics of along-track interferometry SAR system. It can improve the probability of detection under low signal-to-clutter-and-noise ratio. Additionally, it will dramatically shorten the computational time. Furthermore, the proposed method can also estimate the radial velocities of the moving targets simultaneously. The results by applying the proposed method into a set of real SAR data are consistent with the analysis presented in this paper.

Dong Yang

Papers

A New Method for Radar High-Speed Maneuvering Weak Target Detection and Imaging

SAR Imaging With Undersampled Data via Matrix Completion

Persymmetric Adaptive Detectors in Homogeneous and Partially Homogeneous Environments

Strong Clutter Suppression via RPCA in Multichannel SAR/GMTI System

GMTI and Parameter Estimation for MIMO SAR System via Fast Interferometry RPCA Method