<title>What is the visual information loss in a spatial-point-pattern statistical characterization?</title>

Abstract

Spatial point pattern recognition is a frequent step, sometimes the last one, in a general pattern recognition process. Some techniques have been devised to this purpose, generally based on graphs. From statistical geometry considerations we demonstrate the optimal graph representation to be the minimal spanning tree one. The minimal spanning tree (MST) is a graph which provides several ways to analyze the topography (spatial relationships) of objects sets: global degree of order (the so-called m-(sigma) diagram), hierarchical classification (single linkage cluster analysis), non- hierarchical pattern recognition (by graph theory or anisotropy diagrams). The statistical geometry derivation, based on the maximum entropy principle, leads as well to estimate the allowed compression rate of information by using this graph. Anyway the rightist test of an information compression quality is to compare the original pattern to the retrieved one. We have thus investigated various ways to reconstruct those patterns from informations derived, with various compression levels, from the MST. Among them one of the most promising (figure) is the simulated annealing technique with parameters related to the statistical geometry of the graph. Starting from the hypothesis that the analysis of the spatial patterns of objects may lead to display and determine the interactions and control processes between the objects which have induced those patterns, the MST is well suited to analyze these interactions simultaneously at the local and global levels. The method has been applied to the analysis of physical as well as biological systems.