Hierarchical Modeling of Multidimensional Data in Regularly Decomposed Spaces: Main Principles

TL;DR

This paper presents a hierarchical modeling approach for multidimensional data using regular decompositions and generalized trees, enabling advanced pattern recognition and data analysis in various fields.

Contribution

It introduces a multidimensional generalization of quaternary and octernary trees and develops algorithms for data modeling, transformation, and pattern recognition in high-dimensional spaces.

Findings

Effective data decomposition for disordered streams

Algorithms for pattern recognition without affine covering

Implementation of attribute calculus with Eigen trees

Abstract

The described works have been carried out in the framework of a mid-term study initiated by the Centre Electronique de l'Armement and led by ADERSA, a French company of research under contract. The aim was to study the techniques of regular dividing of numerical data sets so as to provide tools for problem solving enabling to model multidimensional numerical objects and to be used in computer-aided design and manufacturing, in robotics, in image analysis and synthesis, in pattern recognition, in decision making, in cartography and numerical data base management. These tools are relying on the principle of regular hierarchical decomposition and led to the implementation of a multidimensional generalization of quaternary and octernary trees: the trees of order 2**k or 2**k-trees mapped in binary trees. This first tome, dedicated to the hierarchical modeling of multidimensional numerical data, describes the principles used for building, transforming, analyzing and recognizing patterns on which is relying the development of the associated algorithms. The whole so developed algorithms are detailed in pseudo-code at the end of this document. The present publication especially describes: - a building method adapted disordered and overcrowded data streams ; - its extension in inductive limits ; - the computation of the homographic transformation of a tree ; - the attribute calculus based on generalized moments and the provision of Eigen trees ; - perception procedures of objects without any covering in affine geometry ; - several supervised and unsupervised pattern recognition methods.