470 research outputs found
Dimension-independent simplification and refinement of Morse complexes
No full textAscending and descending Morse complexes, determined by a scalar field f defined over a manifold M, induce a subdivision of M into regions associated with critical points of f, and compactly represent the topology of M. We define two simplification operators on Morse complexes, which work in arbitrary dimensions, and we define their inverse refinement operators. We describe how simplification and refinement operators affect Morse complexes on M, and we show that these operators form a complete set of atomic operators to create and update Morse complexes on M. Thus, any operator that modifies Morse complexes on M can be expressed as a suitable sequence of the atomic simplification and refinement operators we have defined. The simplification and refinement operators also provide a suitable basis for the construction of a multi-resolution representation of Morse complexes
Spatial queries on a hierarchical terrain model
No full textIn this paper we consider the problem of defining and answering spatial queries on hierarchical terrain models that provide a multiresolution representation. In particular, we focus our attention on interference queries in which the query object is a spatial entity not belonging to the model. We propose algorithms for efficiently answering such queries on a triangle-based hierarchical model. -Author
Extracting contour lines from a hierarchical surface model
No full textThe Hierarchical Triangulated Irregular Network (HTIN) is a structure for representing 21⁄2‐dimensional surfaces at different levels of detail through piecewise‐linear approximations based on triangulations of the surface domain. In this paper, we present two algorithms that allow extracting a representation of the surface and contour lines at a given level of detail, directly from the HTIN. © 1993 Eurographics Associatio
Building Morphological Representations for 2D and 3D Scalar Fields
No full textAscending and descending Morse complexes, defined by the critical points and integral lines of a scalar field f defined on a manifold domain D, induce a subdivision of D into regions of uniform gradient flow, and thus provide a compact description of the morphology of f on D. We propose a dimension-independent representation for the ascending and descending Morse complexes, and we describe a data structure which assumes a discrete representation of the field as a simplicial mesh, that we call the incidence-based data structure. We present algorithms for building such data structure for 2D and 3D scalar fields, which make use of a watershed approach to compute the cells of the Morse decompositions
Efficient computation of simplicial homology through acyclic matching
No full textWe consider the problem of efficiently computing homology with Z coefficients as well as homology generators for simplicial complexes of arbitrary dimension. We analyze, compare and discuss the equivalence of different methods based on combining reductions, co reductions and discrete Morse theory. We show that the combination of these methods produces theoretically sound approaches which are mutually equivalent. One of these methods has been implemented for simplicial complexes by using a compact data structure for representing the complex and a compact encoding of the discrete Morse gradient. We present experimental results and discuss further developments
Multiresolution models for topographic surface description
No full textMultiresolution terrain models describe a topographic surface at various levels of resolution. Besides providing a data compression mechanism for dense topographic data, such models enable us to analyze and visualize surfaces at a variable resolution. This paper provides a critical survey of multiresolution terrain models. Formal definitions of hierarchical and pyramidal models are presented. Multiresolution models proposed in the literature (namely, surface quadtree, restricted quadtree, quaternary triangulation, ternary triangulation, adaptive hierarchical triangulation, hierarchical Delaunay triangulation, and Delaunay pyramid) are described and discussed within such frameworks. Construction algorithms for all such models are given, together with an analysis of their time and space complexities
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