Morphological image distances for hyperspectral dimensionality exploration using Kernel-PCA and ISOMAP

The application of nonlinear manifold learning for hyperspectral image analysis has been widely studied in last years. One of the main ingredients of these data reduction techniques is the distance used to compare the spectral band images. By means of this distance the pairwise similarity matrix is built and then, the matrix is used to explore the intrinsic dimensionality of the hyperspectral image. There are two main families of image distances which have been considered in previous works: i) the distance between the pixels using Minkowski metrics, such as the Euclidean distance or the L
₁
distance; ii) the distances between the image histograms, such as the Kullback-Leibler Divergence or the chi-squared distance. The aim of this paper is to propose two new families of spatial image distances for spectral band comparison. Both are based on notions from mathematical morphology, a nonlinear image processing methodology based on the application of lattice theory to spatial structures. The first distance is based on the formulation using morphological dilations of Hausdorff distance for gray-scale images. The second distance is more original and it is founded in the leveling operator. Levelings are geodesic filters which modify, without blurring the contours, one of the images according to the other image. The application of these morphological distances for hyperspectral dimensionality exploration is illustrated with two powerful nonlinear data analysis techniques: Kernel-PCA and ISOMAP. Using standard image examples, their performance is studied in comparison with other image distances such as Euclidean distance and KL-divergence.