The Minimum Spanning Tree of Maximum Entropy

In computer vision, we have the problem of creating graphs out of unstructured point-sets, i.e. the data graph. A common approach for this problem consists of building a triangulation which might not always lead to the best solution. Small changes in the location of the points might generate graphs with unstable configurations and the topology of the graph could change significantly. After building the data-graph, one could apply Graph Matching techniques to register the original point-sets. In this paper, we propose a data graph technique based on the Minimum Spanning Tree of Maximum Entropty (MSTME). We aim at a data graph construction which could be more stable than the Delaunay triangulation with respect to small variations in the neighborhood of points. Our technique aims at creating data graphs which could help the point-set registration process. We propose an algorithm with a single free parameter that weighs the importance between the total weight cost and the entropy of the current spanning tree. We compare our algorithm on a number of different databases with the Delaunay triangulation.

Invariant Representative Cocycles of Cohomology Generators using Irregular Graph Pyramids

Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide more refined algebraic invariants to a topological space than does homology. It assigns `quantities' to the chains used in homology to characterize holes of any dimension. Graph pyramids can be used to describe subdivisions of the same object at multiple levels of detail. This paper presents cohomology in the context of structural pattern recognition and introduces an algorithm to efficiently compute representative cocycles (the basic elements of cohomology) in 2D using a graph pyramid. An extension to obtain scanning and rotation invariant cocycles is given.

Contains and Inside relationships within combinatorial Pyramids

Irregular pyramids are made of a stack of successively reduced graphs embedded in the plane. Such pyramids are used within the segmentation framework to encode a hierarchy of partitions. The different graph models used within the irregular pyramid framework encode different types of relationships between regions. This paper compares different graph models used within the irregular pyramid framework according to a set of relationships between regions. We also define a new algorithm based on a pyramid of combinatorial maps which allows to determine if one region contains the other using only local calculus.