24 research outputs found
Analysis and Parameterization of Triangulated Surfaces
Get PDFThis dissertation deals with the analysis and parameterization of surfaces represented by triangle meshes, that is, piecewise linear surfaces which enable a simple representation of 3D models commonly used in mathematics and computer science. Providing equivalent and high-level representations of a 3D triangle mesh M is of basic importance for approaching different computational problems and applications in the research fields of Computational Geometry, Computer Graphics, Geometry Processing, and Shape Modeling. The aim of the thesis is to show how high-level representations of a given surface M can be used to find other high-level or equivalent descriptions of M and vice versa. Furthermore, this analysis is related to the study of local and global properties of triangle meshes depending on the information that we want to capture and needed by the application context. The local analysis of an arbitrary triangle mesh M is based on a multi-scale segmentation of M together with the induced local parameterization, where we replace the common hypothesis of decomposing M into a family of disc-like patches (i.e., 0-genus and one boundary component) with a feature-based segmentation of M into regions of 0-genus without constraining the number of boundary components of each patch. This choice and extension is motivated by the necessity of identifying surface patches with features, of reducing the parameterization distortion, and of better supporting standard applications of the parameterization such as remeshing or more generally surface approximation, texture mapping, and compression. The global analysis, characterization, and abstraction of M take into account its topological and geometric aspects represented by the combinatorial structure of M (i.e., the mesh connectivity) with the associated embedding in R^3. Duality and dual Laplacian smoothing are the first characterizations of M presented with the final aim of a better understanding of the relations between mesh connectivity and geometry, as discussed by several works in this research area, and extended in the thesis to the case of 3D parameterization. The global analysis of M has been also approached by defining a real function on M which induces a Reeb graph invariant with respect to affine transformations and best suited for applications such as shape matching and comparison. Morse theory and the Reeb graph were also used for supporting a new and simple method for solving the global parameterization problem, that is, the search of a cut graph of an arbitrary triangle mesh M. The main characteristics of the proposed approach with respect to previous work are its capability of defining a family of cut graphs, instead of just one cut, of bordered and closed surfaces which are treated with a unique approach. Furthermore, each cut graph is smooth and the way it is built is based on the cutting procedure of 0-genus surfaces that was used for the local parameterization of M. As discussed in the thesis, defining a family of cut graphs provides a great flexibility and effective simplifications of the analysis, modeling, and visualization of (time-depending) scalar and vector fields; in fact, the global parameterization of M enables to reduce th
Tailor: understanding 3D shapes using curvature
Get PDFTools for the automatic decomposition of a surface into shape features will facilitate the editing, matching, texturing, morphing, compression, and simplification of 3D shapes. Different features, such as flats, limbs, tips, pits, and various blending shapes that transition between them may be characterized in terms of local curvature and other differential properties of the surface or in terms of a global skeletal organization of the volume it encloses. Unfortunately, both solutions are extremely sensitive to small perturbations in the surface smoothness and to quantization effects when they operate on triangulated surfaces. Thus, we propose a multi-resolution approach, which not only estimates the curvature of a vertex over neighborhoods of variable size, but also takes into account the topology of the surface in that neighborhood. Our approach is based on blowing a spherical bubble at each vertex and studying how the intersection of that bubble with the surface evolves. For example, for a thin limb, that intersection will start simply connected and will rapidly split into two components. For a point on the tip of a limb, that intersection will usually simply remain connected, but the ratio of its length to the radius of the bubble will be decreasing. For a point on a blend, that ratio will exceed 2p. We describe an efficient approach for computing these characteristics for a sampled set of bubble radii and for using them to identify features, based on easily formulated f i lters, that may capture the needs of a particular application
A multi-resolutive extraction of geometric descriptors for virtual shapes and humans
Get PDFTools for the automatic decomposition of a surface into shape features will facilitate and optimize the classification, matching, texturing, morphing, and simplification of 3D shapes. Different features, such as flats, limbs, tips, pits, and various blending shapes between them may be characterized in terms of local curvature and other differential properties of the surface, or in terms of a global skeletal organization of the volume it encloses. However, both solutions are extremely sensitive to small perturbations in the surface smoothness and to quantization effects when they operate on triangulated surfaces. The paper presents a shape characterization based on a multi-resolutive curvature computation where the vertices of a given triangle mesh are classified according to their curvature and shape behavior in neighborhoods of increasing size, and whose final goal is to segment 3D models into main bodies and tubular parts, and to code the tube/body connectivity with their geometric parameters. Last, we propose to apply the morphological analysis for the automatically extraction of the semantic of human body models for their representation, retrieval and applications to animation. We prove the efficacy of our tool in automatically extracting morphological shape parameters and locating feature points on the human body identifying fingertips, nose,armpits, ankles, umbilicus, and so on
Multi-Segmentation and Annotation of 3D Surface Meshes
Get PDFThe ShapeAnnotator is a graphical tool designed to assist an expert user in the task of annotating a surface mesh with concepts belonging to a domain of expertise. The user is first required to identify meaningful surface features; Each such feature can then be "labeled" with a concept, and the surface mesh, along with the features identified and their labels, can be saved
A Minimal Contouring Approach to the Computation of the Reeb Graph
No full textGiven a manifold surface M and a continuous scalar function f:M-->R, the Reeb graph of (M,f) is a widely-used high-level descriptor of M and its usefulness has been demonstrated for a variety of applications, which range from shape parameterization and abstraction to deformation and comparison. In this context, we propose a novel contouring algorithm for the construction of a discrete Reeb graph with a minimal number of nodes, which correspond to the critical points of f (i.e., minima, maxima, saddle points) and its level-sets passing through the saddle points. In this way, we do not need to sample, sweep, or increasingly sort the f-values. Since most of the computation uses only local information on the mesh connectivity, equipped with the f-values at the surface vertices, the proposed approach is insensitive to noise and requires a small memory footprint and temporary data structures. Furthermore, we maintain the parametric nature of the Reeb graph with respect to the input scalar function and we efficiently extract the Reeb graph of time-varying maps. Indicating with n and s the number of vertices of M and saddle points of f, the overall computational cost O(sn) is competitive with respect to the O(nlog n) cost of previous work. This cost becomes optimal if M is highly-sampled or s<log n, as it happens for Laplacian eigenfunctions, harmonic maps and 1-forms
