Characterization of the joint (among wavebands) probability density function (pdf) of hyperspectral imaging (HSI) data is crucial for several applications, including the design of constant false alarm rate (CFAR) detectors and statistical classifiers. HSI data are vector (or equivalently multivariate) data in a vector space with dimension equal to the number of spectral bands. As a result, the scalar statistics utilized by many detection and classification algorithms depend upon the joint pdf of the data and the vector-to--scalar mapping defining the specific algorithm. For reasons of analytical tractability, the multivariate Gaussian assumption has dominated the development and evaluation of algorithms for detection and classification in HSI data, although it is widely recognized that it does not always provide an accurate model for the data. The purpose ofthis paper is to provide a detailed investigation ofthejoint and marginal distributional properties of HSI data. To this end, we assess how well the multivanate Gaussian pdf describes HSI data using univariate techniques for evaluating marginal normality, and techniques that use unidimensional views (projections) of multivanate data. We show that the class of elliptically contoured distributions, which includes the multivariate normal distribution as a special case, provides a better characterization of the data. Finally, it is demonstrated that the class of univariate stable random variables provides a better model for the heavy-tailed output distribution of the well known matched filter target detection algorithm.
In this paper, we explore the use of Elliptically Contoured Distributions (ECDs) to model the statistical variability of hyperspectral imaging (HSI) data. ECDs have the elliptical symmetry of the multivariate Gaussian distribution and therefore share most of its properties. However, the presence of additional parameters, allows to control the behavior of their tails to match the distribution of the data more accurately than the normal distribution. More specifically, the purpose of our paper is two fold. First, we provide a brief introduction to ECDs and their key properties. Second, we introduce the multivariate EC t-distribution and investigate its capability to accurately describe the joint statistics of HSI data from the HYDICE sensor.
Developing proper models for Hyperspectral imaging (HSI) data allows for useful and reliable algorithms for data exploitation. These models provide the foundation for development and evaluation of detection, classification, clustering, and estimation algorithms. To date, most algorithms have modeled real data as multivariate normal, however it is well known that real data often exhibits non-normal behavior. In this paper, Elliptically Contoured Distributions (ECDs) are used to model the statistical variability of HSI data. Non-homogeneous data sets can be modeled as a finite mixture of more than one ECD, with different means and parameters for each component. A larger family of distributions, the family of ECDs includes the multivariate normal distribution and exhibits most of its properties. ECDs are uniquely defined by their multivariate mean, covariance and the distribution of its Mahalanobis distance metric. This metric lets multivariate data be identified using a univariate statistic and can be adjusted to more closely match the longer tailed distributions of real data. One ECD member of focus is the multivariate t-distribution, which provides longer tailed distributions than the normal, and has an F-distributed Mahalanobis distance statistic. This work will focus on modeling these univariate statistics, using the Exceedance metric, a quantitative goodness-of-fit metric developed specifically to improve the accuracy of the model to match the long probabilistic tails of the data. This metric will be shown to be effective in modeling the univariate Mahalanobis distance distributions of hyperspectral data from the HYDICE sensor as either an F-distribution or as a weighted mixture of F-distributions. This implies that hyperspectral data has a multivariate t-distribution. Proper modeling of Hyperspectral data leads to the ability to generate synthetic data with the same statistical distribution as real world data.
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