Meshes obtained from laser scanner data often contain topological noise due to inaccuracies in the scanning and merging process. This topological noise complicates subsequent operations such as remeshing, parameterization and smoothing. We introduce an approach that removes unnecessary nontrivial topology from meshes. Using a local wave front traversal, we discover the local topologies of the mesh and identify features such as small tunnels. We then identify non-separating cuts along which we cut and seal the mesh, reducing the genus and thus the topological complexity of the mesh.

Associated to any simplicial complex \Delta on n vertices is a square-free monomial ideal I \Delta in the polynomial ring A = k[x 1 ; : : : ; xn ], and its quotient k[\Delta] = A=I \Delta known as the Stanley-Reisner ring. This note considers a simplicial complex which is in a sense a canonical Alexander dual to \Delta, previously considered in [Ba, BrHe]. Using Alexander duality and a result of Hochster computing the Betti numbers dim k Tor i (k[\Delta]; k), it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of faces in \Delta . As corollaries, we prove that I \Delta has a linear resolution as A-module if and only if \Delta is Cohen-Macaulay over k, and show how to compute the Betti numbers dim k Tor i (k[\Delta]; k) in some cases where \Delta is well-behaved (shellable, Cohen-Macaulay, or Buchsbaum). Some other applications of the notion of shellability are also discussed.

Abstract. We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allows us to address the various interactions between forms and vector fields (such as Lie derivatives) which are important in applications. Previous attempts at discrete exterior calculus have addressed only differential forms. We also introduce the notion of a circumcentric dual of a simplicial complex. The importance of dual complexes in this field has been well understood, but previous researchers have used barycentric subdivision or barycentric duals. We show that the use of circumcentric duals is crucial in arriving at a theory of discrete

This paper describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the Laboratory's artificial intelligence research is provided in part by the Advanced Research Projects Agency of the Department of Defense under Office of Naval Research contract N00014-89- J-3202, and in part by the National Science Foundation grant MIP-9001651. The author is also supported by a G.Y. Chu Fellowship

Abstract. This paper solves the problem of computing conformal structures of general 2manifolds represented as triangular meshes. We approximate the De Rham cohomology by simplicial cohomology and represent the Laplace-Beltrami operator, the Hodge star operator by linear systems. A basis of holomorphic one-forms is constructed explicitly. We then obtain a period matrix by integrating holomorphic differentials along a homology basis. We also study the global conformal mappings between genus zero surfaces and spheres, and between general surfaces and planes. Our method of computing conformal structures can be applied to tackle fundamental problems in computer aid geometry design and computer graphics, such as geometry classification and identification, and surface global parametrization.

We describe two methods for estimating the size and depth of decision trees where a linear test is performed at each node. Both methods are applied to the question of deciding, by a linear decision tree, whether given n real numbers, some k of them are equal. We show that the minimum depth of a linear decision tree for this problem is Θ(nlog(n/k)). The upper bound is easy; the lower bound can be established for k = O(n 1/4−ε) by a volume argument; for the whole range, however, our proof is more complicated and it involves the use of some topology as well as the theory of Möbius functions.

The authors of this article believe there is or should be a research area appropriately referred to as computational topology. Its agenda includes the identification and formalization of topological questions in computer applications and the study of algorithms for topological problems. It is hoped this article can contribute to the creation of a computational branch of topology with a unifying influence on computing and computer applications. Keywords. Survey; topology, geometry, algorithms, computer applications. INTRODUCTION The title of this article combines computation with topology, suggesting a general research activity that studies the computational aspects of problems with topological flavor. What we have in mind is distinctly different from studying the topology of computing or the computer animation of topology. Computational studies of topological questions can be found in the mathematics and the computer science literature, but no concerted effort is apparent. The auth...

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We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric

In this paper, we present a complete computational pipeline for extracting a compact shape descriptor for curve point cloud data (PCD). Our shape descriptor, called a barcode, is based on a blend of techniques from differential geometry and algebraic topology. We also provide a metric over the space of barcodes, enabling fast comparison of PCDs for shape recognition and clustering. To demonstrate the feasibility of our approach, we implement our pipeline and provide experimental evidence in shape classification and parametrization.