Efficient Delaunay Mesh Generation from Sampled Scalar Functions

Chandrajit Bajaj

2008, Proceedings of the 16th International Meshing Roundtable

visibility

…

description

18 pages

link

1 file

Abstract

Many modern research areas face the challenge of meshing level sets of sampled scalar functions. While many algorithms focus on ensuring geometric qualities of the output mesh, recent attention has been paid to building topologically accurate Delaunay conforming meshes of any level set from such volumetric data.

Figures (7)

Fig. 1. Various stages of our algorithm. (a) A rectilinear grid with sample values of
an unknown function at the grid points. Within cells, the function is approximated
with a trilinear interpolant. For the purpose of visualization only, we collect a set
of points (green) on the surface and display them. A narrow region of the surface
is magnified below. (b) Another view of the data (right) and the same view of the
mesh generated by Marching Cubes [22]. Note that the mesh is disconnected in
the thin region. (c) The mesh generated by the restricted Delaunay triangulation
of only edge and grid points. Blue facets have a grid point as at least one of their
vertices. This point set is still not sufficient to produce a Delaunay-conforming mesh.
(d) The mesh generated by addition of sample points of X’. The topology is now
recovered (Property I). (e,f) Geometrical refinement for progressively smaller value
of e. (g) Even in the magnified portion of the thin region, the triangles approximate
the geometry nicely (Property II). (h) All the points involved in construction of the
mesh including grid points (blue), edge points (green), and new points added by
the algorithm (red). Observe that in order to recover the topology and reduce the
geometric error in the approximation, many surface sample points are added to the
point set. — Fig. 1. Various stages of our algorithm. (a) A rectilinear grid with sample values of an unknown function at the grid points. Within cells, the function is approximated with a trilinear interpolant. For the purpose of visualization only, we collect a set of points (green) on the surface and display them. A narrow region of the surface is magnified below. (b) Another view of the data (right) and the same view of the mesh generated by Marching Cubes [22]. Note that the mesh is disconnected in the thin region. (c) The mesh generated by the restricted Delaunay triangulation of only edge and grid points. Blue facets have a grid point as at least one of their vertices. This point set is still not sufficient to produce a Delaunay-conforming mesh. (d) The mesh generated by addition of sample points of X’. The topology is now recovered (Property I). (e,f) Geometrical refinement for progressively smaller value of e. (g) Even in the magnified portion of the thin region, the triangles approximate the geometry nicely (Property II). (h) All the points involved in construction of the mesh including grid points (blue), edge points (green), and new points added by the algorithm (red). Observe that in order to recover the topology and reduce the geometric error in the approximation, many surface sample points are added to the point set.

Fig. 5. (a) A picture proof of Lemma 1. The Voronoi cell of the edge point e must lie
in the shaded region bounded by the solid lines (green). By symmetry, this restricts
the Voronoi cell to a two pixel range. (b) A picture proof of Theorem 2 in the general
2D case. Here, each pixel has dimensions s1 by s2 with [si/s2] = 3. By symmetry,
the Voronoi cell of e is restricted to a six pixel range. (c) A Voronoi cell of an edge
point in 3D, visibly contained within four voxels. — Fig. 5. (a) A picture proof of Lemma 1. The Voronoi cell of the edge point e must lie in the shaded region bounded by the solid lines (green). By symmetry, this restricts the Voronoi cell to a two pixel range. (b) A picture proof of Theorem 2 in the general 2D case. Here, each pixel has dimensions s1 by s2 with [si/s2] = 3. By symmetry, the Voronoi cell of e is restricted to a six pixel range. (c) A Voronoi cell of an edge point in 3D, visibly contained within four voxels.

Fig. 6. Proof of Theorem 3 in the 2D case. We show that » lies entirely inside
the shaded yellow region, while the Voronoi edges formed between two grid points
necessarily lie outside of it. — Fig. 6. Proof of Theorem 3 in the 2D case. We show that » lies entirely inside the shaded yellow region, while the Voronoi edges formed between two grid points necessarily lie outside of it.

Fig. 7. Performance of the meshing algorithm. The top row shows the geometry,
surface mesh, a closeup on the surface mesh and a cut-away of the volumetric tetra-
hedral mesh for 1CID. The second and the third rows show the same for 1MAG
and BONE, respectively.
As described in the problem statement in Section 1, only the parameter € is
needed to run the algorithm on a given data set. This input dictates the desired
upper bound on the Hausdorff distance between the surface approximation M
and the interpolated surface ’. Given this parameter, we reduce the amount
of computation needed by snapping some of the points of EF in the following
manner. If there exist points g € G and eo, e1, e2 € E such that the e; are on
grid edges incident to g and d(g,e;) < € for i = 0,1,2, we snap the three e;
points onto the common point g. Since all e; points are within e€ distance of
the grid point and 2’ passes through each e;, the closet point of 3’ to g cannot
be more than € away, which is within the user specified limit of tolerance. — Fig. 7. Performance of the meshing algorithm. The top row shows the geometry, surface mesh, a closeup on the surface mesh and a cut-away of the volumetric tetra- hedral mesh for 1CID. The second and the third rows show the same for 1MAG and BONE, respectively. As described in the problem statement in Section 1, only the parameter € is needed to run the algorithm on a given data set. This input dictates the desired upper bound on the Hausdorff distance between the surface approximation M and the interpolated surface ’. Given this parameter, we reduce the amount of computation needed by snapping some of the points of EF in the following manner. If there exist points g € G and eo, e1, e2 € E such that the e; are on grid edges incident to g and d(g,e;) < € for i = 0,1,2, we snap the three e; points onto the common point g. Since all e; points are within e€ distance of the grid point and 2’ passes through each e;, the closet point of 3’ to g cannot be more than € away, which is within the user specified limit of tolerance.

not produce sufficient samples for the extracted approximation to be embed-
ded in the Delaunay triangulation of the sampled set of points. On the other
hand, there are elegant approaches for meshing general implicit surfaces, yet
these algorithms suffer from the computational overhead of computing the
intersection of a ray and the level set. Typically, the level set of a sampled
scalar function is approximated via a number of piecewise interpolating func-
tions and therefore finding the intersection of a ray with the level set requires
a large number of checks. In light of this, we have presented an algorithm
which not only overcomes the sampling issue of traditional isosurfacing tech-
niques but also adopts a well-known provable algorithm [9] for generating a
good sample of the isosurface efficiently. We have also shown that for the
commonly used trilinear interpolant, many of the difficult cases that arise in
implementation of the algorithm can be avoided, thereby improving numerical
robustness.
Fig. 8. Data may be partitioned to run the algorithm in parallel. In this 2D example,
the voxels indicated by the P; are to be processed. One can send data from each P;
voxel to a separate processor, along with the collar of protective voxels around it
indicated by the same color shading. Two iterations are shown. — not produce sufficient samples for the extracted approximation to be embed- ded in the Delaunay triangulation of the sampled set of points. On the other hand, there are elegant approaches for meshing general implicit surfaces, yet these algorithms suffer from the computational overhead of computing the intersection of a ray and the level set. Typically, the level set of a sampled scalar function is approximated via a number of piecewise interpolating func- tions and therefore finding the intersection of a ray with the level set requires a large number of checks. In light of this, we have presented an algorithm which not only overcomes the sampling issue of traditional isosurfacing tech- niques but also adopts a well-known provable algorithm [9] for generating a good sample of the isosurface efficiently. We have also shown that for the commonly used trilinear interpolant, many of the difficult cases that arise in implementation of the algorithm can be avoided, thereby improving numerical robustness. Fig. 8. Data may be partitioned to run the algorithm in parallel. In this 2D example, the voxels indicated by the P; are to be processed. One can send data from each P; voxel to a separate processor, along with the collar of protective voxels around it indicated by the same color shading. Two iterations are shown.