Spectral unmixing and anomaly detection for hyperspectral images
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Least Angle Regression (LARS) solves the basis pursuit optimization problem, as a sparse signal recovery algorithm, for all relevant regularization parameter values simultaneously. However, despite this efficiency of LARS, it has not been applied to the spectral unmixing problem yet as large multichannel data could be very challenging in practice, due to the need for generation and storage of extremely large arrays (~ 1010 bytes for a relatively small spectral unmixing problem). In this thesis, we extend the standard LARS algorithm, using Kronecker products, to make it practical, i.e., without the need to construct or process very large arrays, for efficient recovery of sparse signals from large multichannel data. This thesis also presents a new recursive Kronecker LARS (K-LARS) algorithm based on a homotopy formulation, similar to recursive methods, to update the basis pursuit optimization problem. Instead of completely solving a new basis pursuit problem that is slightly different from the previous known problem solution, we use it as the starting point to solve the new problem in a more computationally efficient, thereby faster way. Afterward, we apply our new Kronecker LARS (K-LARS) algorithm and our new Kronecker homotopy algorithm to successfully unmix both synthetic and AVIRIS hyperspectral images. We compare our results to ones obtained using an earlier basis pursuit-based spectral unmixing algorithm, Generalized Morphological Component Analysis (GMCA), that uses a thresholding-based proximal optimization method. We show that the results are similar, albeit our results were obtained without any trial and error, or arbitrary choices, in specifying the needed regularization parameter. The spectral unmixing problem requires a sparse endmembers constraint in an appropriate basis known as a dictionary. In this thesis, we explored and compared the use of some standard, e.g., wavelets, Discrete Cosine Transform (DCT), and custom online learned dictionaries to promote sparsity of the unknown endmembers when solving the spectral unmixing problem. We show that because of their large number of vanishing moments, a Coiflet dictionary would be optimum to sparsely represent endmembers, who are mostly smooth functions with a small number of peaks. Finally, we present a high-spatial-resolution, i.e., using a small number of spectral pixels as background, anomaly detection algorithm that models this background using a lasso-penalized maximum likelihood estimation method.