Commute time distance transformation applied to spectral imagery and its utilization in material clustering

Spectral image analysis problems often begin by applying a transformation that generates an alternative representation of the spectral data with the intention of exposing hidden features not discernable in the original space. We introduce and demonstrate a transformation based on a Markov-chain model of a random walk on a graph via application to spectral image clustering. The random walk is quantified by a measure known as the average commute time distance (CTD), which is the average length that a random walker takes, when starting at one node, to transition to another and return to the starting node. This distance metric has the important characteristic of increasing when the number of paths between two nodes decreases and/or the lengths of those paths increase. Once a similarity graph is built on the spectral data, a transformation based on an eigendecomposition of the graph Laplacian matrix is applied that embeds the nodes of the graph into a Euclidean space with the separation between nodes equal to the square-root of the average commute time distance. This is referred to as the Commute Time Distance transformation. As an example of the utility of this data transformation, results are shown for standard clustering algorithms applied to hyperspectral data sets. 1 Introduction Imaging spectrometer data can be represented as a three-dimensional structure that encompasses both spatially and spectrally sampled data of a given scene. In an image with d spectral bands, the observed spectral radiance data, or estimated surface reflectance data, can be modeled as a scattering of points in a d-dimensional Euclidean space, such that each of the coordinate axes corresponds uniquely to a single spectral band. 1 In other words, a particular pixel of the spectral image is plotted as a vector with coordinates corresponding to the brightness values of the pixel in the spectral bands. The vector space representation permits the application of linear and nonlinear transformations that result in alternative representations of the data. These alternative representations are often referred to as feature spaces. These derived feature spaces do not add any information, but rather redistribute the original information into a more useful form. 2 The original spectral space representation of the data is not necessarily the best feature space to extract information for spectral image applications, such as classification. One reason is that adjacent spectral bands are often highly correlated and, therefore, provide redundant information. One of the most common transformations applied …