This dissertation deals with the topic of non-negative blind source separation (nBSS), a widely-applicable technique in many real-world applications, such as multichannel biomedical image analysis and hyper-spectral image analysis. Fundamentally, unlike the skills involved in relevant existing frameworks, such as non-negative extension of independent component analysis (ICA) and non-negative matrix factorization (NMF), we exploit convex geometry to develop two nBSS frameworks without source statistical independence/uncorrelatedness assumption.

The first framework called convex analysis of mixtures of non-negative sources (CAMNS) makes use of an insightful and practical model assumption (called source local dominance) to connect nBSS and convex geometry. It leads to a deterministic, convex analysis based nBSS criterion that boils down nBSS problem to the problem of finding all the extreme points of an observation-constructed polyhedral set (or an extreme point enumeration problem). We derive two linear programming based methods for efficiently locating the extreme points. One is analytically based and provides
some appealing theoretical guarantees, while the other is heuristic but provides better robustness when model assumptions are not perfectly satisfied. Simulation results
for several data sets are presented to demonstrate the efficacy of the CAMNS-based methods over several existing benchmark nBSS methods. In addition, experimental results with real biomedical images are presented to evaluate the high practical applicability of CAMNS.

In hyperspectral remote sensing, unmixing a data cube into the spectral signatures (or endmenbers) and their corresponding mixing proportion (or abundance fractions)
plays a crucial role in analyzing the mineralogical composition of a solid surface. Such an unmixing problem nature has a lot in common with nBSS problem. The
second framework describes a new convex analysis and optimization perspective to hyperspectral unmixing. By the notion of convex analysis, we formulate two optimization
problems for hyperspectral unmixing, which have intuitive ideas (or beliefs without any rigorous analysis and proof) that “the endmembers are determined by vertices of the maximum volume simplex within all the observed pixels” proposed by Winter in late 1990, and that “the endmembers are determined by the vertices of a minimum volume simplex enclosing all the observed pixels” proposed by Craig in mid
1990, respectively. We show the relation between the two formulated optimization problems, by proving that both of their optimal solutions yield the true endmembers when the abundance fractions (sources) are locally dominant. We also illustrate how the two problems can be efficiently solved by alternating linear programming. Monte Carlo simulation results for several data sets are presented to validate our analytical results, and demonstrate the efficacy of the proposed algorithms. The experimental results of our nBSS method for real hyperspectral image data collected by airbone visible/infrared imaging spectrometer flight over the Cuprite mining site, Nevada, in 1997, show a high agreement with the reported ground truth. We believe that the proposed two nBSS frameworks in this dissertation have provided new dimensions to the nBSS research area, and will expectantly serve as important signal processing tools not only for biomedical image analysis and hyperspectral image analysis but also for other potential applications, such as analytical chemistry, deconvolution of genomic signals, and superresolution image reconstruction, where the sources are non-negative in nature.

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