| Hyperspectral remote sensing is a valuable new technology that has numerous commercial and scientific applications. For example, it has been used to study crop health, mineral and soil composition, and pollution levels. Hyperspectral imaging also has important military and intelligence applications such as the identification of man-made materials, and detection of chemical and biological plumes. The key mathematical challenges of hyperspectral imaging include image classification, anomaly detection, and target detection. Image classification is the process of grouping pixels into spectrally similar clusters. This thesis describes a new topological and network-theoretic approach for classifying pixels in hyperspectral image data.Pixels in hyperspectral image data sets are thought of as constituting a point cloud in a high dimensional topological space, and a network structure is imposed on the data by considering the spectral distance between pairs of pixels. We use the tools of persistent homology to argue that the resulting network effectively models the complex nonlinear structures in the data. We then perform data clustering by applying a network based community detection algorithm called the method of maximum modularity. The method of maximum modularity is an unsupervised, deterministic method for detecting communities in networks where neither the number of communities nor their sizes needs to be specified in advance. Examples of real hyperspectral images that have been classified using the method of maximum modularity are provided in order to demonstrate the feasibility of the approach. |
