Topics in High-Dimensional Statistics and the Analysis of Large Hyperspectral Images.

Abstract

Advancement in imaging technology has made hyperspectral images gathered from remote sensing much more common. The high-dimensional nature of these large scale data coupled with wavelength and spatial dependency necessitates high-dimensional and efficient computation methods to address these issues while producing results that are concise and easy to understand. The thesis addresses these issues by examining high-dimensional methods in the context of hyperspectral image classification, unmixing and wavelength correlation estimation. Chapter 2 re-examines the sparse Bayesian learning (SBL) of linear models in a high-dimensional setting with sparse signal. The hard-thresholded version of the SBL estimator, under orthogonal design, achieves non-asymptotic error rate that is comparable to LASSO. We also establish in the chapter that with high-probability the estimator recovers the sparsity structure of the signal. The ability to recover sparsity structures in high dimensional settings is crucial for unmixing with high-dimensional libraries in the next chapter. In Chapter 3, the thesis investigates the application of SBL on the task of linear/bilinear unmixing and classification of hyperspectral images. The proposed model in this chapter uses latent Markov random fields to classify pixels and account for the spatial dependence between pixels. In the proposed model, the pixels belonging to the same group share the same mixture of pure endmembers. The task of unmixing and classification are performed simultaneously, but this method does not address wavelength dependence. Chapter 4 is a natural extension of the previous chapter that contains the framework to account for both spatial and wavelength dependence in the unmixing of hyperspectral images. The classification of the images are performed using approximate spectral clustering while the unmixing task is performed in tandem with sparse wavelength concentration matrix estimation.