Using n-dimensional volumes for mathematical applications in spectral image analysis

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Title: Using n-dimensional volumes for mathematical applications in spectral image analysis
Author: Ziemann, Amanda
Abstract: The ability to detect an object or activity -- such as a military vehicle, construction area, campsite, or vehicle tracks -- is highly important to both military and civilian applications. Sensors that process multi and hyperspectral images provide a medium for performing such tasks. Hyperspectral imaging is a technique for collecting and processing imagery at a large number of visible and non-visible wavelengths. Different materials exhibit different trends in their spectra, which can be used to analyze the image. For an image collected at n different wavelengths, the spectrum of each pixel can be mathematically represented as an n-element vector. The algorithm established in this work, the Simplex Volume Estimation algorithm (SVE), focuses specifically on change detection and large area search. In hyperspectral image analysis, a set of pixels constitutes a data cloud, with each pixel corresponding to a vector endpoint in Euclidean space. The SVE algorithm takes a geometrical approach to image analysis based on the linear mixture model, which describes each pixel in an image collected at n spectral bands as a linear combination of n+1 pure-material component spectra (known as endmembers). Iterative endmember identification is used to construct a 'volume function,' where the Gram matrix is used to calculate the hypervolume of the data at each iteration as the endmembers are considered in Euclidean spaces of increasing dimensionality. Linear algebraic theory substantiates that the volume function accurately characterizes the inherent dimensionality of a set of data, and supports that the volume function provides a tool for identifying the subspace in which the magnitude of the spread of the data is the greatest. A metric is extracted from the volume function, and is used to quantify the relative complexity within a single image or the change in complexity across multiple images. The SVE algorithm was applied to hyperspectral images for the tasks of change detection and large area search, and the results from these applications will demonstrate the feasibility of this method as a cueing tool for analysts.
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Date: 2010-10-29

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