Topological correction of hypertextured implicit surfaces for ray casting

Hypertextures are a useful modelling tool in that they can add three-dimensional detail to the surface of otherwise smooth objects. Hypertextures can be rendered as implicit surfaces, resulting in objects with a complex but well defined boundary. However, representing a hypertexture as an implicit surface often results in many small parts being detached from the main surface, turning an object into a disconnected set. Depending on the context, this can detract from the realism in a scene where one usually does not expect a solid object to have clouds of smaller objects floating around it. We present a topology correction technique, integrated in a ray casting algorithm for hypertextured implicit surfaces, that detects and removes all the surface components that have become disconnected from the main surface. Our method works with implicit surfaces that are C2 continuous and uses Morse theory to find the critical points of the surface. The method follows the separatrix lines joining the critical points to isolate disconnected components

Reliable detection and separation of components for solid objects defined with scalar fields

The detection of the number of disjoint components is a well-known procedure for surface objects. However, this problem has not been solved for solid models defined with scalar fields in the so-called implicit form. In this paper, we present a technique which allows for detection of the number of disjoint components with a predefined tolerance for an object defined with a single scalar function. The core of the technique is a reliable continuation of the spatial enumeration based on the interval methods. We also present several methods for separation of components using set-theoretic operations for further handling these components individually in a solid modelling system dealing with objects defined with scalar fields

Ambient Isotopic Meshing of Implicit Algebraic Surface with Singularities

A complete method is proposed to compute a certified, or ambient isotopic, meshing for an implicit algebraic surface with singularities. By certified, we mean a meshing with correct topology and any given geometric precision. We propose a symbolic-numeric method to compute a certified meshing for the surface inside a box containing singularities and use a modified Plantinga-Vegter marching cube method to compute a certified meshing for the surface inside a box without singularities. Nontrivial examples are given to show the effectiveness of the algorithm. To our knowledge, this is the first method to compute a certified meshing for surfaces with singularities.Comment: 34 pages, 17 Postscript figure

Quantitative morphometric methods in diatom research

Morphometric methods have been used in diatom research for decades. We present a review of the history of usage of morphometric methods of outline shape analysis, pattern recognition, and landmarkbased analysis. In addition, we present how morphometric methods are important in diatom taxonomy and classifi cation and what connections exist between morphometric methods and biologically meaningful results. Next, we present some details about calculating shape descriptors and using them in analysis of shape variation, the issues to be aware of, and what such results mean when defi ning shape groups as species groups. Finally, we provide a glimpse of the future in using morphometric methods in diatom research.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/117649/1/bibl_diatom_62_strelnikova.pdfDescription of bibl_diatom_62_strelnikova.pdf : Main articl

Computational Topology Methods for Shape Modelling Applications

This thesis deals with computational topology, a recent branch of research that involves both mathematics and computer science, and tackles the problem of discretizing the Morse theory to functions defined on a triangle mesh. The application context of Morse theory in general, and Reeb graphs in particular, deals with the analysis of geometric shapes and the extraction of skeletal structures that synthetically represents shape, preserving the topological properties and the main morphological characteristics. Regarding Computer Graphics, shapes, that is a one-, two- or higher- dimensional connected, compact space having a visual appearance, are typically approximated by digital models. Since topology focuses on the qualitative properties of spaces, such as the connectedness and how many and what type of holes it has, topology is the best tool to describe the shape of a mathematical model at a high level of abstraction. Geometry, conversely, is mainly related to the quantitative characteristics of a shape. Thus, the combination of topology and geometry creates a new generation of tools that provide a computational description of the most representative features of the shape along with their relationship. Extracting qualitative information, that is the information related to semantic of the shape and its morphological structure, from discrete models is a central goal in shape modeling. In this thesis a conceptual model is proposed which represents a given surface based on topological coding that defines a sketch of the surface, discarding irrelevant details and classifying its topological type. The approach is based on Morse theory and Reeb graphs, which provide a very useful shape abstraction method for the analysis and structuring of the information contained in the geometry of the discrete shape model. To fully develop the method, both theoretical and computational aspects have been considered, related to the definition and the extension of the Reeb graph to the discrete domain. For the definition and automatic construction of the conceptual model, a new method has been developed that analyzes and characterizes a triangle mesh with respect to the behavior of a real and at least continuous function defined on the mesh. The proposed solution handles also degenerate critical points, such as non-isolated critical points. To do that, the surface model is characterized using a contour-based strategy, recognizing critical areas instead of critical points and coding the evolution of the contour levels in a graph-like structure, named Extended Reeb Graph, (ERG), which is a high-level abstract model suitable for representing and manipulating piece-wise linear surfaces. The descriptive power of the (ERG) has been also augmented with the introduction of geometric information together with the topological ones, and it has been also studied the relation between the extracted topological and morphological features with respect to the real characteristics of the surface, giving and evaluation of the dimension of the discarded details. Finally, the effectiveness of our description framework has been evaluated in several application contexts

Morse Theory for Implicit Surface Modeling

. Morse theory describes the relationship between a function's critical points and the homotopy type of the function's domain. The theorems of Morse theory were developed specifically for functions on a manifold. This work adapts these theorems for use with parameterized families of implicit surfaces in computer graphics. The result is a theoretical basis for the determination of the global topology of an implicit surface, and supports the interactive modeling of implicit surfaces by direct manipulation of a topologically-correct triangulated representation. 1 Introduction Implicit surfaces provide a powerful and versatile shape model in computer graphics by representing geometry as the zero-set of a function over threespace, although displaying such surfaces requires a search through space. The display of an implicit surface is hastened by maintaining a triangulation that can be quickly rendered on modern graphics workstations. However, when the implicit surface changes topological t..