Adaptive lattices on the unit sphere. Application to remote sensing

The aim of the study is to structure multivariate data in cylindrical polar representation, in order to segment them by composing the Rⁿ luminance and hue partitions according to that of the saturation. Remote sensing applications are in mind. This goal requires to equip with a total ordering the unite sphere in Rⁿ, where the hue is distributed. The proposed orders, and the subsequent lattices, do not depend on absolute coordinates, but are adapted either to the whole image under study, or to the neighbourhoods necessary for processing the image. They are obtained by choosing a few maxima on the distribution of the spherical coordinates of the image, and by taking as new variables the infima of the distances to these various maxima. The case of four dimensions, usual when dealing with the principal components in remote sensing, is treated in detail. A comparison of the results in various dimensions is performed on a multispectral aerial image of the university of Pavia.