The purpose of this article is to introduce a method for computing the homology groups of cellular complexes composed of cubes. We will pay attention to issues of storage and efficiency in performing computations on large complexes which will be required in applications to the computation of the Conley index. The algorithm used in the homology computations is based on a local reduction procedure, and we give a subquadratic estimate of its computational complexity. This estimate is rigorous in two dimensions, and we conjecture its validity in higher dimensions. 1 Introduction The computability of homology groups is well-known and is found in most standard textbooks, e.g. [18], and the classical algorithm is based on performing row and column operations on the boundary matrices as a whole and reducing them to the Smith Normal Form (SNF), which is known to exist for any integer matrix, [20]. The homology groups can then be immediately determined from this canonical form. However, explici...

Abstract. This paper presents new mathematical foundations for topologically correct surface reconstruction techniques that are applicable to 2-manifolds with boundary, where provable techniques previously had been limited to surfaces without boundary. This is done by an intermediate construction of the envelope (as defined herein) of the original surface. For any compact C 2 manifold M it is shown that its envelope is C 1,1 and this envelope can be reconstructed with topological guarantees. Then it is shown that there exists a piecewise linear (PL) subset of the reconstruction of the envelope that is ambient isotopic to M, whenever M is orientable. The emphasis of this paper is upon the mathematical proofs needed for these extensions, where more practical applications and examples are presented in a companion paper.

We present a fast enumeration algorithm for combinatorial 2- and 3-manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3-manifolds with 11 vertices. We further determine all equivelar maps on the non-orientable surface of genus 4 as well as all equivelar triangulations of the orientable surface of genus 3 and the non-orientable surfaces of genus 5 and 6. 1

In Geometry in Action: Cartography and geographic Information Systems David Eppstein [6] lists several important problems of computational geometry for cartography and GIS. This paper considers two of them: (1) matching/correlating similar features from different geospatial databases (the conflation problem), and (2) handling approximate and inconsistent data. These problems are of great practical importance for end users-- defense and intelligence analysts, geologists, geographers, ecologists and others. An adequate mathematical formulation and solution of these problems is still an open question due their complexity. This paper analyzes relations between these problems and topics in computational topology and geometry. This analysis concludes that a fundamentally new mathematical approach is needed. The paper develops such new approach based on the general concept of an abstract algebraic system. Such a system can uniformly express all major algebraic constructs such as groups, fields, algebras and models. We also show the benefit of developing a fundamentally new approach-- algebraic invariants for geospatial data analysis and correlation using this concept.

Given a nonsingular compact 2-manifold ¦ without boundary, we present methods for establishing a family of surfaces which can approximate ¦ so that each approximant is ambient isotopic to ¦. The methods presented here offer broad theoretical guidance for a rich class of ambient isotopic approximations, for applications in graphics, animation and surface reconstruction. They are also used to establish sufficient conditions for an interval solid to be ambient isotopic to the solid it is approximating. Furthermore, the normals of the approximant are compared to the normals of the original surface, as these approximating normals play prominent roles in many graphics algorithms. The methods are based on global theoretical considerations and are compared to existing local methods. Practical implications of these methods are also presented. For the global case, a differential surface analysis is performed to find a positive number § so that the offsets ¦©¨�����§� � of ¦ at distances �� § are nonsingular. In doing so, a normal tubular neighborhood, ¦���§� � , of ¦ is constructed. Then, each approximant of ¦ lies inside ¦���§� �. Comparisons between these global and local constraints are given. Key words: Ambient isotopy; computational topology; surface reconstruction; interval solids; offsets and deformations; reverse engineering Preprint submitted to Elsevier Science 30 July 2003 1

This paper presents our recent results in the field of surface representation based on topological coding. In particular, we investigate a possible way to adapt to discrete surface models some theoretical concepts as Morse theory and Reeb graphs which bases on differential topology. Starting from a triangulated surface, our aim is to code the relationship among critical points of the height function associated to the mesh. We named Extended Reeb Graph (ERG) the graph representation which can handle also degenerate critical points. The ERG gives an effective representation of the surface shape available for classification, simplification and restoring purposes.

In this note, we segment and topologically classify brain vessel data obtained from magnetic resonance angiography (MRA). The segmentation is done adaptively and the classification by means of cubical homology, i.e. the computation of homology groups. In this way the number of connected components (measured byH0), the tunnels (given byH1) and the voids (given byH2) are determined, resulting in a topological characterization of the blood vessels. 1.

Morse theory has been considered a powerful tool in its applications to computational topology, computer graphics and geometric modeling. It was originally formulated for smooth manifolds. Recently, Robin Forman formulated a version of this theory for discrete structures such as cell complexes. It opens up several categories of interesting objects (particularly meshes) to applications of Morse theory. Once a

Given a set of points and a set of prototypes representing them, how to create a graph of the prototypes whose topology accounts for that of the points? This problem had not yet been explored in the framework of statistical learning theory. In this work, we propose a generative model based on the Delaunay graph of the prototypes and the Expectation-Maximization algorithm to learn the parameters. This work is a first step towards the construction of a topological model of a set of points grounded on statistics. 1

Deformable objects play an important role in many applications, such as animation and simulation. Effective computation with deformable surfaces can be achieved through the use of dynamic meshes. In this paper, we introduce a framework for constructing and maintaining a timevarying adapted mesh structure that conforms to the underlying deformable surface. The adaptation function employs error metrics based on stochastic sampling. Our scheme combines normal and tangential geometric correction with refinement and simplification resolution control. Furthermore, it applies to both parametric and implicit surface descriptions. As the result, we obtain a simple and efficient general scheme that can be used for a wide range of computations. 1.