Abstract Adaptive Fairing of Surface Meshes by Geometric Diffusion

In triangulated surface meshes, there are often very noticeable size variances (the vertices are distributed unevenly). The presented noise of such surface meshes is therefore composite of vast frequencies. In this paper, we solve a diffusion partial differential equation numerically for noise removal of arbitrary triangular manifolds using an adaptive time discretization. The proposed approach is simple and is easy to incorporate into any uniform timestep diffusion implementation with significant improvements over evolution results with the uniform timesteps. As an additional alternative to the adaptive discretization in the time direction, we also provide an approach for the choice of an adaptive diffusion tensor in the diffusion equation

Smooth surface constructions via a higher order level set method

We present a general framework for a higher-order spline level-set (HLS) method and apply this to smooth surface constructions. Starting from a first order energy functional, we derive the general level set formulation, and provide an efficient solution of a second order geometric partial differential equation using a C² spline basis. We also present a fast cubic spline interpolation algorithm based on convolution and the Z-transform, which exploits the local relationship of interpolatory cubic spline coefficients with respect to given function data values. We provide two demonstrative smooth surface construction examples of our HLS method. The first is the construction of a smooth surface model (an implicit solvation interface) of bio-molecules in solvent, given their individual atomic coordinates and solvated radii. The second is the smooth surface reconstruction from a cloud of points generated from a 3D surface scanner

Discrete Surface Modeling using Geometric Flows

We use various nonlinear geometric partial differential equations to efficiently solve several surface modeling problems, including surface blending, N-sided hole filling and free-form surface fitting. The nonlinear equations used include two second order flows (mean curvature flow and average mean curvature flow), one fourth order flow (surface diffusion flow) and a sixth order flow. These nonlinear equations are discretized based on discrete differential geometry operators. The proposed approach is simple, efficient and gives very desirable results, for a range of surface models, possibly having sharp creases and corners

Identifying flat and tubular regions of a shape by unstable manifolds

Figure 1: The steps of the algorithm are shown on an example dataset CLUB. Starting with an input set of points sampled from the surface (a), the medial axis in the interior of the shape is computed (b). The algorithm then detects the set of index 1 and index 2 saddle points lying on the interior medial axis and computes the unstable manifold of these saddle points (c). The unstable manifold of an index 1 saddle point is two dimensional (green) and the unstable manifold of an index 2 saddle point is one dimensional (red). The algorithm then collects the local maxima lying on the boundaries of these two types of unstable manifolds and tag them as falling into two different categories. The stable manifolds of these maxima are then used to map the 2-dimensional and 1-dimensional part of the medial axis back to the surface. The flat portion on the surface is colored cyan and the tubular region is colored golden (e). We present an algorithm to identify the flat and tubular regions of a three dimensional shape from its point sample. We consider the distance function to the input point cloud and the Morse structure induced by it on R 3. Specifically we focus on the index 1 and index 2 saddle points and their unstable manifolds. The unstable manifolds of index 2 saddles are one dimensional whereas those of index 1 saddles are two dimensional. Mapping these unstable manifolds back onto the surface, we get the tubular and flat regions. The computations are carried out on the Voronoi diagram of the input points by approximating the unstable manifolds with Voronoi faces. We demonstrate the performance of our algorithm on several point sampled objects

Automatic 3d mesh generation for a domain with multiple materials

Abstract: This paper describes an approach to construct unstructured tetrahedral and hexahedral meshes for a domain with multiple materials. In earlier works, we developed an octreebased isocontouring method to construct unstructured 3D meshes for a single-material domain. Based on this methodology, we introduce the notion of material change edge and use it to identify the interface between two or several materials. We then mesh each material region with conforming boundaries. Two kinds of surfaces, the boundary surface and the interface between two different material regions, are distinguished and meshed. Both material change edges and interior edges are analyzed to construct tetrahedral meshes, and interior grid points are analyzed for hexahedral mesh construction. Finally the edge-contraction and smoothing method is used to improve the quality of tetrahedral meshes, and a combination of pillowing, geometric flow and optimization techniques is used for hexahedral mesh quality improvement. The shrink set is defined automatically as the boundary of each material region. Several application results in different research fields are shown

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The XS Suite : Portable 3D Graphics with X

Computer Aided Geometric Design is a young and rapidly growing field that deals with the construction and manipulation of geometric objects. It relies heavily on high-performance computer graphics for the purpose of rendering realistic models. The SHASTRA system aims to provide a distributed, collaborative geometric design and scientific manipulation environment across a heterogeneous hardware platform. It is therefore necessary to achieve truly portable computer graphics. Since all of the subsystems under the SHASTRA umbrella require high-performance graphics, it is also very useful to provide a high-level, abstract interface. The XS suite was designed and built as a solution to these requirements, and is now used by all SHASTRA applications. XS is a powerful mechanism for engineering user interfaces for scientific manipulation applications on hardware graphics systems. It provides access to system-dependent graphics facilities in a uniform, system- independent manner. We describe o..