Dual-Primal Mesh Optimization for Polygonized Implicit Surfaces With Sharp Features

Abstract

A new method for improving polygonizations of implicit surfaces with sharp features is proposed. The method is based on the observation that, given an implicit surface with sharp features, a triangle mesh whose triangles are tangent to the implicit surface at certain inner triangle points gives a better approximation of the implicit surface than the standard Marching Cubes mesh [Lorensen, W.E., and Cline, H.E., 1987, Computer Graphics (Proceedings of SIGGRAPH ’87), 21(3), pp. 163–169] (in our experiments we use VTK Marching Cubes [Schroeder, W., Martin, K., and Lorensen, W., 1998, The Visualization Toolkit: An Object-Oriented Approach to 3-D Graphics, Prentice Hall]). First, given an initial triangle mesh, its dual mesh composed of the triangle centroids is considered. Then the dual mesh is modified such that its vertices are placed on the implicit surface and the mesh dual to the modified dual mesh is considered. Finally the vertex positions of that “double dual” mesh are optimized by minimizing a quadratic energy measuring a deviation of the mesh normals from the implicit surface normals computed at the vertices of the modified dual mesh. In order to achieve an accurate approximation of fine surface features, these basic steps are combined with adaptive mesh subdivision and curvature-weighted vertex resampling. The proposed method outperforms approaches based on the mesh evolution paradigm in speed and accuracy.

Progressive encoding and compression of surfaces generated from point cloud data

“…Therefore, good normal reproduction is also important, especially in the presence of sharp features [34]. We have not investigated the quality of normal approximation using our method.…”

Surface Curvature Estimation for Edge Spinning Algorithm

“…as in [7]. In many cases, the computation of gradient of the function f is influenced by a major error that depends on modeling techniques used [8], [9], [10], [11], [13], [14]. Because of these reasons, in our approach, we have defined these restrictions for finding a new surface point p new :…”

“…There are two different definitions of implicit surfaces. The first one [2], [3] defines an implicit object as f(x) < 0 and the second one, F-rep [8], [10], [11], [12], defines it as f(x) ≥ 0. In our implementation, we use the F-rep definition of implicit objects.…”

“…Surface tracking approaches (also known as continuation methods) iteratively create a triangulation from a seed element by marching along the surface [1], [2], [5], [7], [9], [15]. Surface fitting techniques progressively adapt and deform an initial mesh to converge to the implicit surface, [10].…”

Topology-Adaptive Mesh Deformation for Surface Evolution, Morphing, and Multiview Reconstruction

“…The explicit surface is recovered by nding the 0-level set of the implicit function. A number of methods have been proposed to extract surfaces from volumetric data [32,36,38,44].…”