A tool for the nonparametric characterization of the geometry of spectra in hyperspace

Abstract

There is a relative paucity of studies that seek to nonparametrically describe or categorize hyperspectral image (HSI) data in the most natural context within which those data are best represented: n-dimensional (n-D) hyperspace, where n is the number of bands in an HSI data cube. Statistical techniques-and specifically those based on second-order statistics-are predominant and generally do not provide (nor are they utilized to provide or 'leave behind') an understanding of the distribution of spectra in hyperspace for the purpose of discovering patterns, comparing data sets, developing a deeper understanding of HSI data, etc. With the terabytes of HSI data in archives today representing a huge diversity of geographical settings, it seems natural to enquire as to whether there are universal patterns among HSI data sets. If such patterns exist, new techniques may be developed to exploit them, e.g., to enhance information extraction speed and accuracy. Since an understanding of the nature of HSI data has rarely gone beyond cursory, visual assessments of geographical setting; i.e., assessments of whether the data are of desert, forest, urban, littoral, etc., environments, it is perhaps worthwhile to develop methods to probe n-D hyperspace with the intent to identify and describe universal data commonalities, differences, classifications, and patterns. Building upon earlier work (Resmini, 2003 [1], 2006 [2]), an algorithm was created to describe the volume (and label the individual volume elements) of hyperspace occupied by an HSI data set. The algorithm essentially discretizes hyperspace assigning addresses to boxes in which data points (i.e., spectra) reside. The algorithm has been applied to visible/near-infrared to shortwave infrared (VNIR/SWIR) and longwave infrared (LWIR) HSI data cubes. Implemented in C code, the algorithm may also be used to calculate the fractal dimension of any array of points in any n-D hyperspace because it is essentially an implementation of box counting. The algorithm is described as are results of its application to actual HSI data sets. The fractal dimension of HSI data derived from the algorithm are also presented and compared to principal componentsbased estimates of data dimensionality. Application of the algorithm for developing a deeper understanding of the nature of HSI data as viewed as points in hyperspace is discussed as are practical applications for 'traditional' HSI exploitation activities such as anomaly and target detection.