A Multiple Hypothesis Testing Approach to Low-Complexity Subspace Unmixing

Subspace-based signal processing traditionally focuses on problems involving a few subspaces. Recently, a number of problems in different application areas have emerged that involve a significantly larger number of subspaces relative to the ambient dimension. It becomes imperative in such settings to first identify a smaller set of active subspaces that contribute to the observation before further processing can be carried out. This problem of identification of a small set of active subspaces among a huge collection of subspaces from a single (noisy) observation in the ambient space is termed subspace unmixing. This paper formally poses the subspace unmixing problem under the parsimonious subspace-sum (PS3) model, discusses connections of the PS3 model to problems in wireless communications, hyperspectral imaging, high-dimensional statistics and compressed sensing, and proposes a low-complexity algorithm, termed marginal subspace detection (MSD), for subspace unmixing. The MSD algorithm turns the subspace unmixing problem for the PS3 model into a multiple hypothesis testing (MHT) problem and its analysis in the paper helps control the family-wise error rate of this MHT problem at any level $\alpha \in [0,1]$ under two random signal generation models. Some other highlights of the analysis of the MSD algorithm include: (i) it is applicable to an arbitrary collection of subspaces on the Grassmann manifold; (ii) it relies on properties of the collection of subspaces that are computable in polynomial time; and ($iii$) it allows for linear scaling of the number of active subspaces as a function of the ambient dimension. Finally, numerical results are presented in the paper to better understand the performance of the MSD algorithm.Comment: Submitted for journal publication; 33 pages, 14 figure

Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches

Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution enables material identification via spectroscopic analysis, which facilitates countless applications that require identifying materials in scenarios unsuitable for classical spectroscopic analysis. Due to low spatial resolution of HSCs, microscopic material mixing, and multiple scattering, spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus, accurate estimation requires unmixing. Pixels are assumed to be mixtures of a few materials, called endmembers. Unmixing involves estimating all or some of: the number of endmembers, their spectral signatures, and their abundances at each pixel. Unmixing is a challenging, ill-posed inverse problem because of model inaccuracies, observation noise, environmental conditions, endmember variability, and data set size. Researchers have devised and investigated many models searching for robust, stable, tractable, and accurate unmixing algorithms. This paper presents an overview of unmixing methods from the time of Keshava and Mustard's unmixing tutorial [1] to the present. Mixing models are first discussed. Signal-subspace, geometrical, statistical, sparsity-based, and spatial-contextual unmixing algorithms are described. Mathematical problems and potential solutions are described. Algorithm characteristics are illustrated experimentally.Comment: This work has been accepted for publication in IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensin

Linear models, signal detection, and the Grassmann manifold

2014 Fall.Standard approaches to linear signal detection, reconstruction, and model identification problems, such as matched subspace detectors (MF, MDD, MSD, and ACE) and anomaly detectors (RX) are derived in the ambient measurement space using statistical methods (GLRT, regression). While the motivating arguments are statistical in nature, geometric interpretations of the test statistics are sometimes developed after the fact. Given a standard linear model, many of these statistics are invariant under orthogonal transformations, have a constant false alarm rate (CFAR), and some are uniformly most powerful invariant (UMPI). These properties combined with the simplicity of the tests have led to their widespread use. In this dissertation, we present a framework for applying real-valued functions on the Grassmann manifold in the context of these same signal processing problems. Specifically, we consider linear subspace models which, given assumptions on the broadband noise, correspond to Schubert varieties on the Grassmann manifold. Beginning with increasing (decreasing) or Schur-convex (-concave) functions of principal angles between pairs of points, of which the geodesic and chordal distances (or probability distribution functions) are examples, we derive the associated point-to-Schubert variety functions and present signal detection and reconstruction algorithms based upon this framework. As a demonstration of the framework in action, we implement an end-to-end system utilizing our framework and algorithms. We present results of this system processing real hyperspectral images

Algorithms for feature selection and pattern recognition on Grassmann manifolds

Includes bibliographical references.2015 Summer.This dissertation presents three distinct application-driven research projects united by ideas and topics from geometric data analysis, optimization, computational topology, and machine learning. We first consider hyperspectral band selection problem solved by using sparse support vector machines (SSVMs). A supervised embedded approach is proposed using the property of SSVMs to exhibit a model structure that includes a clearly identifiable gap between zero and non-zero feature vector weights that permits important bands to be definitively selected in conjunction with the classification problem. An SSVM is trained using bootstrap aggregating to obtain a sample of SSVM models to reduce variability in the band selection process. This preliminary sample approach for band selection is followed by a secondary band selection which involves retraining the SSVM to further reduce the set of bands retained. We propose and compare three adaptations of the SSVM band selection algorithm for the multiclass problem. We illustrate the performance of these methods on two benchmark hyperspectral data sets. Second, we propose an approach for capturing the signal variability in data using the framework of the Grassmann manifold (Grassmannian). Labeled points from each class are sampled and used to form abstract points on the Grassmannian. The resulting points have representations as orthonormal matrices and as such do not reside in Euclidean space in the usual sense. There are a variety of metrics which allow us to determine distance matrices that can be used to realize the Grassmannian as an embedding in Euclidean space. Multidimensional scaling (MDS) determines a low dimensional Euclidean embedding of the manifold, preserving or approximating the Grassmannian geometry based on the distance measure. We illustrate that we can achieve an isometric embedding of the Grassmann manifold using the chordal metric while this is not the case with other distances. However, non-isometric embeddings generated by using the smallest principal angle pseudometric on the Grassmannian lead to the best classification results: we observe that as the dimension of the Grassmannian grows, the accuracy of the classification grows to 100% in binary classification experiments. To build a classification model, we use SSVMs to perform simultaneous dimension selection. The resulting classifier selects a subset of dimensions of the embedding without loss in classification performance. Lastly, we present an application of persistent homology to the detection of chemical plumes in hyperspectral movies. The pixels of the raw hyperspectral data cubes are mapped to the geometric framework of the Grassmann manifold where they are analyzed, contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows the time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmannian. This motivates the search for topological structure, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the manifold. The proposed framework affords the processing of large data sets, such as the hyperspectral movies explored in this investigation, while retaining valuable discriminative information. For a particular choice of a distance metric on the Grassmannian, it is possible to generate topological signals that capture changes in the scene after a chemical release

Fuzzy spectral and spatial feature integration for classification of nonferrous materials in hyperspectral data

Hyperspectral data allows the construction of more elaborate models to sample the properties of the nonferrous materials than the standard RGB color representation. In this paper, the nonferrous waste materials are studied as they cannot be sorted by classical procedures due to their color, weight and shape similarities. The experimental results presented in this paper reveal that factors such as the various levels of oxidization of the waste materials and the slight differences in their chemical composition preclude the use of the spectral features in a simplistic manner for robust material classification. To address these problems, the proposed FUSSER (fuzzy spectral and spatial classifier) algorithm detailed in this paper merges the spectral and spatial features to obtain a combined feature vector that is able to better sample the properties of the nonferrous materials than the single pixel spectral features when applied to the construction of multivariate Gaussian distributions. This approach allows the implementation of statistical region merging techniques in order to increase the performance of the classification process. To achieve an efficient implementation, the dimensionality of the hyperspectral data is reduced by constructing bio-inspired spectral fuzzy sets that minimize the amount of redundant information contained in adjacent hyperspectral bands. The experimental results indicate that the proposed algorithm increased the overall classification rate from 44% using RGB data up to 98% when the spectral-spatial features are used for nonferrous material classification