Chong-Yung Chi

A Convex Analysis-Based Minimum-Volume Enclosing Simplex Algorithm for Hyperspectral Unmixing

Hyperspectral unmixing aims at identifying the hidden spectral signatures (or endmembers) and their corresponding proportions (or abundances) from an observed hyperspectral scene. Many existing hyperspectral unmixing algorithms were developed under a commonly used assumption that pure pixels exist. However, the pure-pixel assumption may be seriously violated for highly mixed data. Based on intuitive grounds, Craig reported an unmixing criterion without requiring the pure-pixel assumption, which estimates the endmembers by vertices of a minimum-volume simplex enclosing all the observed pixels. In this paper, we incorporate convex analysis and Craigs criterion to develop a minimum-volume enclosing simplex (MVES) formulation for hyperspectral unmixing. A cyclic minimization algorithm for approximating the MVES problem is developed using linear programs (LPs), which can be practically implemented by readily available LP solvers. We also provide a non-heuristic guarantee of our MVES problem formulation, where the existence of pure pixels is proved to be a sufficient condition for MVES to perfectly identify the true endmembers. Some Monte Carlo simulations and real data experiments are presented to demonstrate the efficacy of the proposed MVES algorithm over several existing hyperspectral unmixing methods.

QoS-Based Transmit Beamforming in the Presence of Eavesdroppers: An Optimized Artificial-Noise-Aided Approach

Secure transmission techniques have been receiving growing attention in recent years, as a viable, powerful alternative to blocking eavesdropping attempts in an open wireless medium. This paper proposes a secret transmit beamforming approach using a quality-of-service (QoS)-based perspective. Specifically, we establish design formulations that: i) constrain the maximum allowable signal-to-interference-and-noise ratios (SINRs) of the eavesdroppers, and that ii) provide the intended receiver with a satisfactory SINR through either a guaranteed SINR constraint or SINR maximization. The proposed designs incorporate a relatively new idea called artificial noise (AN), where a suitable amount of AN is added in the transmitted signal to confuse the eavesdroppers. Our designs advocate joint optimization of the transmit weights and AN spatial distribution in accordance with the channel state information (CSI) of the intended receiver and eavesdroppers. Our formulated design problems are shown to be NP-hard in general. We deal with this difficulty by using semidefinite relaxation (SDR), an approximation technique based on convex optimization. Interestingly, we prove that SDR can exactly solve the design problems for a practically representative class of problem instances; e.g., when the intended receivers instantaneous CSI is known. Extensions to the colluding-eavesdropper scenario and the multi-intended-receiver scenario are also examined. Extensive simulation results illustrate that the proposed AN-aided designs can yield significant power savings or SINR enhancement compared to some other methods.

A Signal Processing Perspective on Hyperspectral Unmixing: Insights from Remote Sensing

Blind hyperspectral unmixing (HU), also known as unsupervised HU, is one of the most prominent research topics in signal processing (SP) for hyperspectral remote sensing [1], [2]. Blind HU aims at identifying materials present in a captured scene, as well as their compositions, by using high spectral resolution of hyperspectral images. It is a blind source separation (BSS) problem from a SP viewpoint. Research on this topic started in the 1990s in geoscience and remote sensing [3]-[7], enabled by technological advances in hyperspectral sensing at the time. In recent years, blind HU has attracted much interest from other fields such as SP, machine learning, and optimization, and the subsequent cross-disciplinary research activities have made blind HU a vibrant topic. The resulting impact is not just on remote sensing - blind HU has provided a unique problem scenario that inspired researchers from different fields to devise novel blind SP methods. In fact, one may say that blind HU has established a new branch of BSS approaches not seen in classical BSS studies. In particular, the convex geometry concepts - discovered by early remote sensing researchers through empirical observations [3]-[7] and refined by later research - are elegant and very different from statistical independence-based BSS approaches established in the SP field. Moreover, the latest research on blind HU is rapidly adopting advanced techniques, such as those in sparse SP and optimization. The present development of blind HU seems to be converging to a point where the lines between remote sensing-originated ideas and advanced SP and optimization concepts are no longer clear, and insights from both sides would be used to establish better methods.

Outage Constrained Robust Transmit Optimization for Multiuser MISO Downlinks: Tractable Approximations by Conic Optimization

In this paper, we study a probabilistically robust transmit optimization problem under imperfect channel state information (CSI) at the transmitter and under the multiuser multiple-input single-output (MISO) downlink scenario. The main issue is to keep the probability of each users achievable rate outage as caused by CSI uncertainties below a given threshold. As is well known, such rate outage constraints present a significant analytical and computational challenge. Indeed, they do not admit simple closed-form expressions and are unlikely to be efficiently computable in general. Assuming Gaussian CSI uncertainties, we first review a traditional robust optimization-based method for approximating the rate outage constraints, and then develop two novel approximation methods using probabilistic techniques. Interestingly, these three methods can be viewed as implementing different tractable analytic upper bounds on the tail probability of a complex Gaussian quadratic form, and they provide convex restrictions, or safe tractable approximations, of the original rate outage constraints. In particular, a feasible solution from any one of these methods will automatically satisfy the rate outage constraints, and all three methods involve convex conic programs that can be solved efficiently using off-the-shelf solvers. We then proceed to study the performance-complexity tradeoffs of these methods through computational complexity and comparative approximation performance analyses. Finally, simulation results are provided to benchmark the three convex restriction methods against the state of the art in the literature. The results show that all three methods offer significantly improved solution quality and much lower complexity.

DOA Estimation of Quasi-Stationary Signals With Less Sensors Than Sources and Unknown Spatial Noise Covariance: A Khatri–Rao Subspace Approach

In real-world applications such as those for speech and audio, there are signals that are nonstationary but can be modeled as being stationary within local time frames. Such signals are generally called quasi-stationary or locally stationary signals. This paper considers the problem of direction-of-arrival (DOA) estimation of quasi-stationary signals. Specifically, in our problem formulation we assume: i) sensor array of uniform linear structure; ii) mutually uncorrelated wide-sense quasi-stationary source signals; and iii) wide-sense stationary noise process with unknown, possibly nonwhite, spatial covariance. Under the assumptions above and by judiciously examining the structures of local second-order statistics (SOSs), we develop a Khatri-Rao (KR) subspace approach that has two notable advantages. First, through an identifiability analysis, it is proven that this KR subspace approach can operate even when the number of sensors is about half of the number of sources. The idea behind is to make use of a ¿virtual¿ array structure provided inherently in the local SOS model, of which the degree of freedom is about twice of that of the physical array. Second, the KR formulation naturally provides a simple yet effective way of eliminating the unknown spatial noise covariance from the signal SOSs. Extensive simulation results are provided to demonstrate the effectiveness of the KR subspace approach under various situations.

Distributed Robust Multicell Coordinated Beamforming With Imperfect CSI: An ADMM Approach

Multicell coordinated beamforming (MCBF), where multiple base stations (BSs) collaborate with each other in the beamforming design for mitigating the intercell interference (ICI), has been a subject drawing great attention recently. Most MCBF designs assume perfect channel state information (CSI) of mobile stations (MSs); however CSI errors are inevitable at the BSs in practice. Assuming elliptically bounded CSI errors, this paper studies the robust MCBF design problem that minimizes the weighted sum power of BSs subject to worst-case signal-to-interference-plus-noise ratio (SINR) constraints on the MSs. Our goal is to devise a distributed optimization method to obtain the worst-case robust beamforming solutions in a decentralized fashion with only local CSI used at each BS and limited backhaul information exchange between BSs. However, the considered problem is difficult to handle even in the centralized form. We first propose an efficient approximation method for solving the nonconvex centralized problem, using semidefinite relaxation (SDR), an approximation technique based on convex optimization. Then a distributed robust MCBF algorithm is further proposed, using a distributed convex optimization technique known as alternating direction method of multipliers (ADMM). We analytically show the convergence of the proposed distributed robust MCBF algorithm to the optimal centralized solution. We also extend the worst-case robust beamforming design as well as its decentralized implementation method to a fully coordinated scenario. Simulation results are presented to examine the effectiveness of the proposed SDR method and the distributed robust MCBF algorithm.

A Simplex Volume Maximization Framework for Hyperspectral Endmember Extraction

In the late 1990s, Winter proposed an endmember extraction belief that has much impact on endmember extraction techniques in hyperspectral remote sensing. The idea is to find a maximum-volume simplex whose vertices are drawn from the pixel vectors. Winters belief has stimulated much interest, resulting in many different variations of pixel search algorithms, widely known as N-FINDR, being proposed. In this paper, we take a continuous optimization perspective to revisit Winters belief, where the aim is to provide an alternative framework of formulating and understanding Winters belief in a systematic manner. We first prove that, fundamentally, the existence of pure pixels is not only sufficient for the Winter problem to perfectly identify the ground-truth endmembers but also necessary. Then, under the umbrella of the Winter problem, we derive two methods using two different optimization strategies. One is by alternating optimization. The resulting algorithm turns out to be an N-FINDR variant, but, with the proposed formulation, we can pin down some of its convergence characteristics. Another is by successive optimization; interestingly, the resulting algorithm is found to exhibit some similarity to vertex component analysis. Hence, the framework provides linkage and alternative interpretations to these existing algorithms. Furthermore, we propose a robust worst case generalization of the Winter problem for accounting for perturbed pixel effects in the noisy scenario. An algorithm combining alternating optimization and projected subgradients is devised to deal with the problem. We use both simulations and real data experiments to demonstrate the viability and merits of the proposed algorithms.

A Convex Analysis Framework for Blind Separation of Non-Negative Sources

This paper presents a new framework for blind source separation (BSS) of non-negative source signals. The proposed framework, referred herein to as convex analysis of mixtures of non-negative sources (CAMNS), is deterministic requiring no source independence assumption, the entrenched premise in many existing (usually statistical) BSS frameworks. The development is based on a special assumption called local dominance. It is a good assumption for source signals exhibiting sparsity or high contrast, and thus is considered realistic to many real-world problems such as multichannel biomedical imaging. Under local dominance and several standard assumptions, we apply convex analysis to establish a new BSS criterion, which states that the source signals can be perfectly identified (in a blind fashion) by finding the extreme points of an observation-constructed polyhedral set. Methods for fulfilling the CAMNS criterion are also derived, using either linear programming or simplex geometry. Simulation results on several data sets are presented to demonstrate the efficacy of the proposed method over several other reported BSS methods.

A comparative study of several wind estimation algorithms for spaceborne scatterometers

The authors compare the performance of seven wind-estimation algorithms, including the weighted least squares in the log domain, maximum-likelihood (ML), least squares, weighted least squares, adjustable weighted least squares, L1 norm, and least wind speed squares algorithms, for wind retrieval. For each algorithm, they present performance simulation results for the NASA scatterometer system planned to be launched in the 1990s. A relative performance merit based on the root-mean-square value of wind vector error is devised. It is found that performances of all algorithms are quite comparable. However, the results do indicate that the ML algorithm performs best for the 50-km wind resolution cell case and the L1 norm algorithm performs best for the 25-km wind resolution cell case. >

Batch processing algorithms for blind equalization using higher-order statistics

Statistical signal processing has been one of the key technologies in the development of wireless communication systems, especially for broadband multiuser communication systems which severely suffer from intersymbol interference (ISI) and multiple access interference (MAI). This article reviews batch processing algorithms for blind equalization using higher-order statistics for mitigation of the ISI induced by single-input, single-output channels as well as of both the ISI and MAI induced by multiple-input, multiple-output channels. In particular, this article reviews the typical inverse filter criteria (IFC) based algorithm, super-exponential algorithm, and constant modulus algorithm along with their relations, performance, and improvements. Several advanced applications of these algorithms are illustrated, including blind channel estimation, simultaneous estimation of multiple time delays, signal-to-noise ratio (SNR) boost by blind maximum ratio combining, blind beamforming for source separation in multipath, and multiuser detection for direct sequence/code division multiple access (DS/CDMA) systems in multipath.