Independent Component Analysis as a tool for the dimensionality reduction and the representation of hyperspectral images M. Lennon 1 , G. Mercier 1 , M.C. Mouchot 1 , L. Hubert-Moy 2 1 Ecole Nationale Supérieure des Télécommunications de Bretagne - Département ITI Technopôle de Brest Iroise - BP 832 - 29285 Brest Cédex - France 2 Laboratoire Costel - Université de Rennes 2 - 6, avenue Gaston Berger - 35043 Rennes Cédex- France Tél : (33) (0)229001069 ; Fax : (33) (0)229001098 ; Email : marc.lennon@enst-bretagne.fr Abstract – Independent Component Analysis (ICA) is a multivariate data analysis process largely sudied these last years in the signal processing community for blind source separation. This paper proposes to show the interest of ICA as a tool for unsupervised analysis of hyperspectral images. The commonly used Principal Component Analysis (PCA) is the mean square optimal projection for gaussian data leading to uncorrelated components by using second order statistics. ICA rather uses higher order statistics and leads to independent components, a stronger statistical assumption revealing interesting features in the usually non gaussian hyperspectral data sets. I. INTRODUCTION Current hyperspectral imaging sensors provide images including a huge number of spectral bands, typically from a few to several hundred ones. These data can be represented in a vectorial space whose dimension is equal to this number of spectral bands. However, the intrinsic dimensionality of data is usually largely less important than the dimension of the support vectorial space. In order to analyse such data, it is also necessary to project the data in a lower dimension space according to the intrinsic dimensionality of the data. Original data should be compressed while preserving the maximum amount of information for processing or visualisation. The main reason is that pattern recognition algorithms have poor capabilities in large dimension spaces. For example, the estimation of the statistical properties of classes in a supervised classification process needs the number of training samples to exponentially increase when the number of dimensions of the data increase. Multidimensional segmentation process also needs a “reasonable” number of dimensions to be achievable, usually simply for execution time considerations. Moreover, the visualisation of such data asks for the creation of new channels allowing the data to be represented the most synthetic way while preserving the maximum amount of visual information. PCA is one of the techniques commonly used for the purpose of this dimensionality reduction step in the process of data analysis. PCA leads to optimal projections in the case of a single source of information corrupted with gaussian noise. From a geometrical point of vue, PCA is optimal when the multidimensional scatterplot of data reveals to be hyperelliptic. The distributions of hyperspectral data sets are usually not gaussian and interesting information about “pure endmembers” is concentrated on vertices in the scatterplot of data. Such information can then become indistinguishable with principal components projections. In addition, small size endmembers which do not contribute very much to the total amount of the variance of information may be lost with a principal component analysis. Independent Component Analysis [1] enables the data to be represented by statistically independent components unlike PCA which leads to uncorrelated components. The statistical independance takes into consideration higher order moments and is so a stronger statistical property than decorrelation (the second order statistic used in PCA). ICA has been largely studied these last years by reasearchers from the signal processing community in the field of blind source separation [2]. The technique is based on the optimisation of an independance criterion between the observed signals and leads to a linear non orthogonal projection of the data. In the case of blind source separation, the observed signals are unknown but assumed to be linear combinations of unknown non gaussian sources. With the assumption that the sources are statistically independent, ICA recovers sources from the observed signals. It has been demonstrated [3] that in the case of non gaussian sources, the optimisation of an independence criterion is equivalent to the maximisation of a non gaussianity criterion due to the central limit theorem. Consequently, ICA is able to compute projections which leads to the least gaussian projected data. In other words, probability density functions of projected data are the most skewed or multimodal. This feature is quite interesting for future classification of the data set. II. INDEPENDENT COMPONENT ANALYSIS The model of ICA if given by : x = A s (1) with : x : Vector of observed signals A : Scalar matrix of mixing coefficients s : Vector of source signals ICA requires 2 hypothesis : (H1) : The components s i of s are statistically independent. (H2) : The components s i of s have non gaussian distribution (It can be shown that no more than one component can be gaussian).