A Sparse Regression Approach to Hyperspectral Unmixing

Spectral unmixing is an important problem in hyperspectral data exploitation. It amounts at characterizing the mixed spectral signatures collected by an imaging instrument in the form of a combination of pure spectral constituents (endmembers), weighted by their correspondent abundance fractions. Linear spectral unmixing is a popular technique in the literature which assumes linear interactions between the endmembers, thus simplifying the characterization of the mixtures and approaching the problem from a general perspective independent of the physical properties of the observed materials. However, linear spectral unmixing suffers from several shortcomings. First, it is unlikely to find completely pure spectral endmembers in the image data due to spatial resolution and mixture phenomena. Second, the linear mixture model does not naturally include spatial information, which is an important source of information (together with spectral information) to solve the unmixing problem. In this thesis, we propose a completely new approach for spectral unmixing which makes use of spectral libraries of materials collected on the ground or in a laboratory, thus circumventing the problems associated to image endmember extraction. Due to the increasing availability and dimensionality of spectral libraries, this problem calls for efficient sparse regularizers. The resulting approach is called sparse unmixing, which represents a unique contribution of this research work and which opens a new direction in the field of spectral unmixing, which is a very active research topic in the hyperspectral image analysis literature. Another important contribution of this work is the inclusion of spatial information in sparse unmixing, which is achieved in this work by means of a Total Variation (TV) regularizer. In order to take advantage of the arrangement of spectral signatures in the form of groups of materials in available spectral libraries, the thesis also explores the use of sparse unmixing for groups of materials, resorting to the Sparse Group Lasso (SGL). iii This technique focuses on obtaining sparse solutions for groups of signatures from the spectral library which correspond to the same type of materials, and not only with respect to the constituent endmembers of individual pixels. Finally, another innovatice contribution of this thesis is the joint consideration of groups of pixels and groups of materials, using the Collaborative Hierarchical Lasso (CHL). Combined, these innovative contributions establish the field of sparse unmixing in hyperspectral analysis and offer a thoughtful perspective on the possibility of using sparse regression techniques in this context, leading to solutions to problems that could not be solved before in the context of linear spectral unmixing. The effectiveness of all the proposed techniques is illustrated by providing exhaustive comparisons with state-of-the-art methods for spectral unmixing using both simulated and real hyperspectral data sets.