Contour interpolation with bounded dihedral angles

ACM Transactions on Graphics, 2009

We present a practical approach to isotropic tetrahedral meshing of 3D domains bounded by piecewise smooth surfaces. Building upon recent theoretical and practical advances, our algorithm interleaves Delaunay refinement and mesh optimization to generate quality meshes that satisfy a set of user-defined criteria. This interleaving is shown to be more conservative in number of Steiner point insertions than refinement alone, and to produce higher quality meshes than optimization alone. A careful treatment of boundaries and their features is presented, offering a versatile framework for designing smoothly graded tetrahedral meshes.

Computer Methods in Applied Mechanics and Engineering, 1995

1, Introduction A triangulation of a set of points in three-dimensional space is a decomposition of the convex hull of the points into non-overlapping tetKihedK% Tic Delauwy triangulation, in which the circumsphere of every tetrahedron in the triangulation contains no point in its interior, IS of particular interest to mathematicians as well as to engineers. It has been used extensively in the design of efficient algorithms and in many direct applications such as interpolation [I], contouring (21 and mesh generation [3-51. Since the Delaunay triangulation has some optimal properties in two dimensions and efficient global and incremental algorithms exist for their construction, they have been user' in finite element mesh generation to produce well-proportioned elements [h. 71. In a recent paper hv Rajan [g], some new optimality results for Delaunay triangulation in higher dimensions are discussed. It is suggested that ttc Delaunay triangulation is the most compact one in the sense that: (1) the min-containment sphere is the smallest. (2) the circumspheres of the tetrahedra incident on an interior point is closest to the point, and (3) the weighted average of the square of edge length, 15 the smallest. Algorithms for the construction of Delaunay triangulations in n-dimensional space for n 3 2 are given by Bowyer [9], Watson [ 101 and Avis and Bhattcharya [ 1 I]. The incremental algorithms of Bowyer and Watson is especially popular in the large scale three-dimensional finite element generation [ 12, 131. The popularity of the method may be accounted for by the speed wtth which a large number of points can be ~-* Corresponding author 004%7825/95/%OY 50 0 1995 Elxwr Sc~cnce SSDi 0045.7825(95)00854-3 S.A. All rlghtc reserved

Graphical Models, 2000

In this paper we consider the problem of reconstructing triangular surfaces from given contours. An algorithm solving this problem has to decide which contours of two successive slices should be connected by the surface (branching problem), and, given that, which vertices of the assigned contours should be connected for the triangular mesh (correspondence problem). We present a new approach that solves both tasks in an elegant way. The main idea is to employ discrete distance fields enhanced with correspondence information. This allows us not only to connect vertices from successive slices in a reasonable way but also to solve the branching problem by creating intermediate contours where adjacent contours differ too much. Last but not least we show how the 2D-distance fields used in the reconstruction step can be converted to a 3D-distance field that can be advantageously exploited for distance calculations during a subsequent simplification step.

The Journal of Visualization and Computer Animation, 1997

Metamorphosis, or morphing, is the gradual transformation of one shape into another. It generally consists of two subproblems: the correspondence problem and the interpolation problem. This paper presents a solution to the interpolation problem of transforming one polyhedral model into another. It is an extension of the intrinsic shape interpolation scheme (T. W. Sederberg, P. Gao, G. Wang and H. Mu, '2-D shape blending: an intrinsic solution to the vertex path problem, SIGGRAPH '93, pp. 15-18.) for 2D polygons. Rather than considering a polyhedron as a set of independent points or faces, our solution treats a polyhedron as a graph representing the interrelations between faces. Intrinsic shape parameters, such as dihedral angles and edge lengths that interrelate the vertices and faces in the two graphs, are used for interpolation. This approach produces more satisfactory results than the linear or cubic curve paths would, and is translation and rotation invariant.

Communications of the ACM, 1977

In many scientific and technical endeavors, a threedimensional solid must be reconstructed from serial sections, either to aid in the comprehension of the object's structure or to facilitate its automatic manipulation and analysis. This paper presents a general solution to the problem of constructing a surface over a set of cross-sectional contours. This surface, to be composed of triangular tiles, is constructed by separately determining an optimal surface between each pair of consecutive contours. Determining such a surface is reduced to the problem of finding certain minimum cost cycles in a directed toroidal graph. A new fast algorithm for finding such cycles is utilized. Also developed is a closed-form expression, in terms of the number of contour points, for an upper bound on the number of operations required to execute the algorithm. An illustrated example which involves the construction of a minimum area surface describing a human head is included.

Delaunay meshing is a popular technique for mesh generation. Usually , the mesh has to be refined so that certain fidelity and quality criteria are met. Delaunay refinement is achieved by dynamically inserting and removing points in/from a Delaunay triangulation. In this paper, we present a robust parallel algorithm for computing Delaunay triangulations in three dimensions. Our triangulator offers fully dynamic parallel insertions and removals of points and is thus suitable for mesh refinement. As far as we know, ours is the first method that parallelizes point removals, an operation that significantly slows refinement down. Our shared memory implementation makes use of a custom memory manager and lightweight locks which greatly reduce the communication and synchronization cost. We also employ a contention policy which is able to accelerate the execution times even in the presence of high number of rollbacks. Evaluation on synthetic and real data shows the effectiveness of our method on widely used multi-core SMPs.

1992

We show how to triangulate a three dimensional polyhedral region with holes. Our triangulation is optimal in the following two senses. First, our triangulation achieves the best possible aspect ratio up to a constant. Second, for any other triangulation of the same region into m triangles with bounded aspect ratio, our triangulation has size n = O(m). Such a triangulation is desired as an initial mesh for a finite element mesh refinement algorithm. Previous three dimensional triangulation schemes either worked only on a restricted class of input, or did not guarantee well-shaped tetrahedral, or were not able to bound the output size. We build on some of the ideas presented in previous work by Bern, Eppstein, and Gilbert, who have shown how to triangulate a two dimensional polyhedral region with holes, with similar quality and optimality bounds.

SIAM Journal on Scientific Computing, 2010

Traditional refinement algorithms insert a Steiner point from a few possible choices at each step. Our algorithm, on the contrary, defines regions from where a Steiner point can be selected and thus inserts a Steiner point among an infinite number of choices. Our algorithm significantly extends existing generalized algorithms by increasing the number and the size of these regions. The lower bound for newly created angles can be arbitrarily close to 30 degrees. Both termination and good grading are guaranteed. It is the first Delaunay refinement algorithm with a 30 degree angle bound and with grading guarantees. Experimental evaluation of our algorithm corroborates the theory. ). 1 2 selection region is a one-dimensional region called selection interval.

2006

For 2D or 3D meshes that represent a continuous function to the reals, the contours|or isosurfaces|of a speci ed value are an important way to visualize it. To nd such contours, a seed set can be used for the starting points from which the traversal of the contours can start. This paper gives the rst methods to obtain seed sets that are provably small in size. They are based on a variant of the contour tree (or topographic change tree). We give a new, simple algorithm to compute such a tree in regular and irregular meshes that requires O(n log n) time in 2D for meshes with n elements, and in O(n 2 ) time in higher dimensions. The additional storage overhead is proportial to the maximum size of any contour (linear in the worst case, but typically less). Given the contour tree, a minimum size seed set can be computed in polynomial time and storage. Since in practice at most linear storage is allowed, we develop a simple approximation algorithm giving a seed set of size at most twice the size of the minimum. It requires O(n log 2 n) time in 2D and O(n 2 ) time otherwise, and requires linear storage. We also give experimental results, showing the size of the seed sets and supporting the claim that sublinear storage is used.

2000

A tool for constructing a "good" 3D triangulation of a given set of vertices in 3D is developed and studied. The constructed triangulation is "optimal" in the sense that it locally minimizes a cost function which measures a certain discrete curvature over the resulting triangle mesh. The algorithm for obtaining the optimal triangulation is that of swapping edges sequentially, such that the cost function is reduced maximally by each swap. In this paper three easy-to-compute cost functions are derived using a simple model for defining discrete curvatures of triangle meshes. The results obtained by the different cost functions are compared. Operating on data sampled from simple 3D models, we compare the approximation error of the resulting optimal triangle meshes to the sampled model in various norms. The conclusion is that all three cost functions lead to similar results, and none of them can be said to be superior to the others. The triangle meshes generated by our algorithm, when serving as initial triangle meshes for the butterfly subdivision scheme, are found to improve significantly the limit butterfly-surfaces compared to arbitrary initial triangulations of the given sets of vertices. Based upon this observation, we believe that any algorithm operating on triangle meshes such as subdivision, finite element solution of PDE, or mesh simplification, can obtain better results if applied to a "good" triangle mesh with small discrete curvatures. Thus our algorithm can serve for modelling surfaces from sampled data as well as for initialization of other triangle mesh based algorithms.