ArulMurugan Ambikapathi

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.

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.

Chance-Constrained Robust Minimum-Volume Enclosing Simplex Algorithm for Hyperspectral Unmixing

Effective unmixing of hyperspectral data cube under a noisy scenario has been a challenging research problem in remote sensing arena. A branch of existing hyperspectral unmixing algorithms is based on Craigs criterion, which states that the vertices of the minimum-volume simplex enclosing the hyperspectral data should yield high fidelity estimates of the endmember signatures associated with the data cloud. Recently, we have developed a minimum-volume enclosing simplex (MVES) algorithm based on Craigs criterion and validated that the MVES algorithm is very useful to unmix highly mixed hyperspectral data. However, the presence of noise in the observations expands the actual data cloud, and as a consequence, the endmember estimates obtained by applying Craig-criterion-based algorithms to the noisy data may no longer be in close proximity to the true endmember signatures. In this paper, we propose a robust MVES (RMVES) algorithm that accounts for the noise effects in the observations by employing chance constraints. These chance constraints in turn control the volume of the resulting simplex. Under the Gaussian noise assumption, the chance-constrained MVES problem can be formulated into a deterministic nonlinear program. The problem can then be conveniently handled by alternating optimization, in which each subproblem involved is handled by using sequential quadratic programming solvers. The proposed RMVES is compared with several existing benchmark algorithms, including its predecessor, the MVES algorithm. Monte Carlo simulations and real hyperspectral data experiments are presented to demonstrate the efficacy of the proposed RMVES algorithm.

Hyperspectral Data Geometry-Based Estimation of Number of Endmembers Using p-Norm-Based Pure Pixel Identification Algorithm

Hyperspectral endmember extraction is a process to estimate endmember signatures from the hyperspectral observations, in an attempt to study the underlying mineral composition of a landscape. However, estimating the number of endmembers, which is usually assumed to be known a priori in most endmember estimation algorithms (EEAs), still remains a challenging task. In this paper, assuming hyperspectral linear mixing model, we propose a hyperspectral data geometry-based approach for estimating the number of endmembers by utilizing successive endmember estimation strategy of an EEA. The approach is fulfilled by two novel algorithms, namely geometry-based estimation of number of endmembers—convex hull (GENE-CH) algorithm and affine hull (GENE-AH) algorithm. The GENE-CH and GENE-AH algorithms are based on the fact that all the observed pixel vectors lie in the convex hull and affine hull of the endmember signatures, respectively. The proposed GENE algorithms estimate the number of endmembers by using the Neyman–Pearson hypothesis testing over the endmember estimates provided by a successive EEA until the estimate of the number of endmembers is obtained. Since the estimation accuracies of the proposed GENE algorithms depend on the performance of the EEA used, a reliable, reproducible, and successive EEA, called

Identifiability of the Simplex Volume Minimization Criterion for Blind Hyperspectral Unmixing: The No-Pure-Pixel Case

p

A robust alternating volume maximization algorithm for endmember extraction in hyperspectral images

-norm-based pure pixel identification (TRI-P) algorithm is then proposed. The performance of the proposed TRI-P algorithm, and the estimation accuracies of the GENE algorithms are demonstrated through Monte Carlo simulations. Finally, the proposed GENE and TRI-P algorithms are applied to real AVIRIS hyperspectral data obtained over the Cuprite mining site, Nevada, and some conclusions and future directions are provided.

Two effective and computationally efficient pure-pixel based algorithms for hyperspectral endmember extraction

In blind hyperspectral unmixing (HU), the pure-pixel assumption is well known to be powerful in enabling simple and effective blind HU solutions. However, the pure-pixel assumption is not always satisfied in an exact sense, especially for scenarios where pixels are heavily mixed. In the no-pure-pixel case, a good blind HU approach to consider is the minimum volume enclosing simplex (MVES). Empirical experience has suggested that MVES algorithms can perform well without pure pixels, although it was not totally clear why this is true from a theoretical viewpoint. This paper aims to address the latter issue. We develop an analysis framework wherein the perfect endmember identifiability of MVES is studied under the noiseless case. We prove that MVES is indeed robust against lack of pure pixels, as long as the pixels do not get too heavily mixed and too asymmetrically spread. The theoretical results are supported by numerical simulation results.

A robust minimum volume enclosing simplex algorithm for hyperspectral unmixing

Accurate estimation of endmember signatures and the associated abundances of a scene from its hyperspectral observations is at present, a challenging research area. Many of the existing hyper-spectral unmixing algorithms are based on Winters belief, which states that the vertices of the maximum volume simplex inside the data cloud (observations) will yield high fidelity estimates of the endmember signatures if pure-pixels exist. Based on Winters belief, we recently proposed a convex analysis based alternating volume maximization (AVMAX) algorithm. In this paper we develop a robust version of the AVMAX algorithm. Here, the presence of noise in the hyperspectral observations is taken into consideration with the original deterministic constraints suitably reformulated as probabilistic constraints. The subproblems involved are convex problems and they can be effectively solved using available convex optimization solvers. Monte Carlo simulations are presented to demonstrate the efficacy of the proposed RAVMAX algorithm over several existing pure-pixel based hyperspectral unmixing methods, including its predecessor, the AVMAX algorithm.

Convex geometry based outlier-insensitive estimation of number of endmembers in hyperspectral images

Endmember extraction is of prime importance in the process of hyperspectral unmixing so as to study the mineral composition of a landscape from its hyperspectral observations. Though, a whole bunch of pure-pixel based endmember extraction algorithms exists, the quest for a reliable, repeatable, and computationally efficient endmember extraction algorithm still prevails. In this work, we propose two pure-pixel based endmember extraction algorithms called simplex estimation by projection (SIMPLE-Pro) algorithm and p-norm based pure pixel identification (TRI-P) algorithm. The endmember identifiability of the proposed two algorithms is theoretically proved under the pure pixel assumption. Both algorithms never require any initializations and hence they are repeatable. Monte Carlo simulations are performed to demonstrate the superior efficacy and computational efficiency of the proposed two algorithms over some existing benchmark endmember extraction algorithms.

On the endmember identifiability of Craig's criterion for hyperspectral unmixing: A statistical analysis for three-source case

Hyperspectral unmixing is a process of extracting hidden spectral signatures (or endmembers) and the corresponding proportions (or abundances) of a scene, from its hyperspectral observations. Motivated by Craigs belief, we recently proposed an alternating linear programming based hyperspectral unmixing algorithm called minimum volume enclosing simplex (MVES) algorithm, which can yield good unmixing performance even for instances of highly mixed data. In this paper, we propose a robust MVES algorithm called RMVES algorithm, which involves probabilistic reformulation of the MVES algorithm, so as to account for the presence of noise in the observations. The problem formulation for RMVES algorithm is manifested as a chance constrained program, which can be suitably implemented using sequential quadratic programming (SQP) solvers in an alternating fashion. Monte Carlo simulations are presented to demonstrate the efficacy of the proposed RMVES algorithm over several existing benchmark hyperspectral unmixing methods, including the original MVES algorithm.