Abstract

One crucial issue in dealing with physical objects is to obtain a good representation of the objects. Topological features of object constitute the highest abstraction in object representation which is important for various pattern recognition, image analysis, medical image processing, and machine vision tasks. Though useful, the computation of topological features is not straightforward. Skeletonization is a topology preserving operation which allows one to compute the topology of objects, but it is very time-consuming. The Euler characteristic is a widely used invariant in computational topology. However, the Euler characteristic, either calculated on the boundary of the object or over the whole object, cannot correctly classify objects in R2 and R3 topologically. So far it is not clear either how we can compute the fundamental group of an arbitrary object.

Our investigation suggests that the intrinsic notion of homeomorphism in general topology is not powerful enough and an extrinsic one that preserves the embedding space is needed for computer applications. In this talk, following a brief introduction of digital topology, a field investigating the topological properties of digital objects in computer images, we describe a topological representation of digital objects through their structured boundaries. In Z3, for example, the boundary of an object is divided to surfaces, and the surfaces are structured from the external to internal surfaces based on their surrounding relationship. Each surface is then described by its topological characteristics such as the number of handles. This representation preserves the embedding space and enables an unambiguous topological classification of objects in Z2 and Z3. A key issue with our representation is that we need to compute topological properties of surfaces rather than that of the whole object. However, there is no theory available that can be applied to digital objects which are often non-manifolds. We propose a topological boundary invariant of 2 and 3-dimensional digital objects, called BIUP2 and BIUP3, that can be obtained through a special deformation: homogenous front propagation, BIUP standing for boundary/ surface invariant under propagation. With digital objects, BIUP2 and BIUP3 can be computed on contours in Z2 and surfaces in Z3, respectively, and they are invariant on both digital manifolds and non-manifolds. Our research shows that, in Z2 and Z3, the topology of digital objects can be fully determined by their boundary, given the structure of the boundary. Further investigation suggests that BIUP could be invariant in Z4, and that surfaces of digital objects in even and odd dimensional spaces may have different topological properties. Potential computer applications of our new invariants will be discussed. It would be interesting to explore, from a topology perspective, whether our results could be generalized to a broad category of objects.

Biography: Franck Xia received his Ph.D. degree in computer science from University Pierre-Marie Curie (Paris VI), Paris, France. Dr. Xia was with the University of Macao between 1993 and 1999. Currently, he is an associate professor of computer science at the University Missouri-Rolla, Missouri. His research interests range from digital topology, discrete differential geometry, computer vision, foundation of software engineering to philosophy of sciences.