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Emerging Hyper-Spectral imaging technology allows the acquisition of data 'cubes' which simultaneously have high- resolution spatial and spectral components. There is a wealth of information in this data and effective techniques for extracting and processing this information are vital. Previous work by ERIM on man-made object detection has demonstrated that there is a huge amount of discriminatory information in hyperspectral images. This work used the hypothesis that the spectral characteristics of natural backgrounds can be described by a multivariate Gaussian model. The Mahalanobis distance (derived from the covariance matrix) between the background and other objects in the spectral data is the key discriminant. Other work (by DERA and Pilkington Optronics Ltd) has confirmed these findings, but indicates that in order to obtain the lowest possible false alarm probability, a way of including higher order statistics is necessary. There are many ways in which this could be done ranging from neural networks to classical density estimation approaches. In this paper we report on a new method for extending the Gaussian approach to more complex spectral signatures. By using ideas from the theory of Support Vector Machines we are able to map the spectral data into a higher dimensional space. The co- ordinates of this space are derived from all possible multiplicative combinations of the original spectral line intensities, up to a given order d -- which is the main parameter of the method. The data in this higher dimensional space are then analyzed using a multivariate Gaussian approach. Thus when d equals 1 we recover the ERIM model -- in this case the mapping is the identity. In order for such an approach to be at all tractable we must solve the 'combinatorial explosion' problem implicit in this mapping for large numbers of spectral lines in the signature data. In order to do this we note that in the final analysis of this approach it is only the inner (dot) products between vectors in the higher dimensional space that need to be computed. This can be done by efficient computations in the original data space. Thus the computational complexity of the problem is determined by the amount of data -- rather than the dimensionality of the mapping. The novel combination of non- linear mapping and high dimensional multivariate Gaussian analysis, only possible by using techniques from SVM theory, allows the practical application to hyperspectral imagery. We note that this approach also generates the non-linear Principal Components of the data, which have applications in their own right. In this paper we give a mathematical derivation of the method from first principles. The method is illustrated on a synthetic data set where complete control over the true statistics is possible. Results on this data show that the method is very powerful. It naturally extends the Gaussian approach to a variety of more complex probability distributions, including multi-modal and other manifestly non- Gaussian examples. Having shown the potential of this approach it is then applied to real hyperspectral trials data. The relative improvement in performance over the Gaussian approach is demonstrated for the real data.
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