Delaunay Triangulation & Application Course Presentation by Wei-Chao Chen Oct. 29, 1998 2/23/2019

Outline Applications of Delaunay Triangulation Data-Dependent Triangulation Constrained Delaunay Triangulation Mesh Generation using Delaunay Refinement 3-D Delaunay Triangulation Rhythmes, Sonia Delaunay 2/23/2019

Applications Geoscientific Modelling: Collect data about spatial objects and domains. Categorize features in soils or ocean. Unique triangulation, adaptive user control. Lattuada et al. (http://www.iah.bbsrc.ac.uk/phd/gisruk95.html) 2/23/2019

Applications Volume visualization for 3D scalar function Given a set of 3D points with scalar value, visualize the dataset, one or more level with transparency. Requires intensive computation power. Delaunay Triangulation is used to interpolate the scalar function between given vertices (Delaunay Interpolation) Ray-Tracing image refinement Use Delaunay Triangulation, interpolate by using convolution and integral. 2/23/2019

Applications Medical Imaging, Feature Extraction Input: A medical image such as CT or MRI. Process: Find vertices on the edges. Output: The medial axis and feature surface by combining Voronoi diagram and Delaunay triangulation. Robinson. (http://noodle.med.yale.edu/~robinson/) 2/23/2019

Data-Dependent Triangulation For applications like height fields and terrain, we want to have triangulation to approximate the true surface condition. Taking the height into account, we may get a better triangulation than merely doing the Delaunay Triangulation in 2-D. Dyn et al.: Minimizing the size of normal derivative discontinuities across edges of the triangulation will provide dramatic improvement. 2/23/2019

Data-Dependent Triangulation Quak, Schumaker: Given a set of vertices in 3D, find the triangulation that is closest to the actual surface (assuming cubic spline fitting). Specify an energy function and flip edge recursively to obtain the minimal energy. The “actual surface” is globally C1 across vertices. The vertices lies on the “actual surface”. Delaunay Triangulation is actually a triangulation for certain energy function. 2/23/2019

Data-Dependent Triangulation Algorithm 1: Given any triangulation of vertices, calculate the Bernstein-Bézier representation, compute the energy of the associated surface over each triangles. Sort the energies computed on the previous step in decreasing order. From the triangle with biggest energy, swap the edges to reduce the energy. Algorithm 2: Use conventional Delaunay Triangulation and replace angle-optimal swap test into energy-optimal swap test. 2/23/2019

Data-Dependent Triangulation Results: Both algorithms yield similar results. Smooth surfaces: Only limited reduction in energy function is observed. Non-smooth surfaces: Substantial decrease in energy is found. However, this does not imply the resulting surface is visually more pleasing. 2/23/2019

Constrained Delaunay Triangulation Given a set of constraining edges, find an optimal triangulation including the constraining edges. User can specify the edges that must appear in the triangulation. Standard Delaunay Triangulation always triangulate the convex hull. Constraining edges are given in the form of planar straight line graph (PSLG). There are no intersection among these edges. We can avoid this limitation by converting intersections into vertices. 2/23/2019

Constrained Delaunay Triangulation Conforming Delaunay Triangulation: Given a set of vertices {vi}, find intersections {vj} between PSLG and Delaunay Triangulation of {vi}. Do the Delaunay Triangulation on vertices ({vi}{vj}) Drawback: Too many vertices! O(m2n) 2/23/2019 Schewchuk. (http://www.cs.cmu.edu/~jrs/jrspapers.html)

Constrained Delaunay Triangulation Given a set of vertices {vi}, triangulate using Delaunay Triangulation. Delete all the triangles that overlap constraining edges. Retriangulate both sides of these edges. 2/23/2019 Schewchuk. (http://www.cs.cmu.edu/~jrs/jrspapers.html)

Constrained Delaunay Triangulation Triangulation by splitting constraining edges: Do the Delaunay Triangulation for {vi}. Find constraining edges not in the triangulation. If the diametral circle of the edge contains some points (encroached), split this edge by adding a vertice in the center of the segment. 2/23/2019 Schewchuk. (http://www.cs.cmu.edu/~jrs/jrspapers.html)

Delaunay Refinement Delaunay Triangulation does not guarantee the actual minimum angle of triangles. We can refine the triangulation by adding more vertices. Specify the maximal acceptable aspect ratio for triangles. Add vertex for skinny triangles. Split edge when the triangle is too skinny. Guarantee the quality of generated triangle mesh. (Chew, Ruppert, Shewchuk) 2/23/2019

Delaunay Refinement Input: PSLG {ej} and vertices {vi} Steps: Schewchuk. (http://www.cs.cmu.edu/~jrs/jrspapers.html) Steps: 1. Obtain Delaunay Triangulation of vertices {vi}. Divide & conquer algorithm (Lee and Schachter) or Plane-sweep algorithm (Fortune) or Incremental insertion algorithm (Lawson). 2/23/2019

Delaunay Refinement Schewchuk. (http://www.cs.cmu.edu/~jrs/jrspapers.html) 2. Do the constrained Delaunay Triangulation on triangles generated at previous step. Schewchuk. (http://www.cs.cmu.edu/~jrs/jrspapers.html) 2/23/2019

Delaunay Refinement 3. Delete triangles outside the concavity of PSLG. User define the triangle to be deleted by assigning a point inside the Constrained Delaunay Triangulation, but outside the concavity. Run the point location algorithm, find the triangles, recursively delete until hits the boundary. Schewchuk. (http://www.cs.cmu.edu/~jrs/jrspapers.html) 2/23/2019

Delaunay Refinement 4. Correct “Bad Triangles” in the triangulation generated from previous step. Find triangles with large aspect ratio (skinny triangles). For each triangle, find the circle it inscribes, and add the center of the circle as a vertex of the mesh. Recursively flip edge for inserted vertex. If the circle encroaches upon any segment: Delete the vertex. Split the segments this circle encroaches upon. 2/23/2019 Schewchuk. (http://www.cs.cmu.edu/~jrs/jrspapers.html)

Delaunay Refinement Running of Delaunay Refinement algorithm. Highlighted segments are segments before split, or triangles before vertex insertion. 2/23/2019 Schewchuk. (http://www.cs.cmu.edu/~jrs/jrspapers.html)

3-D Delaunay Triangulation We can do triangulation for higher dimensions by using similar technique of 2-D Delaunay Triangulation. 3-D: Replace circles by spheres, triangles by tetrahedrons. 2/23/2019 Schewchuk. (http://www.cs.cmu.edu/~jrs/jrspapers.html)

3-D Delaunay Triangulation Input: Set of vertices, constraining edges & facets. Steps: For all constraining edges, make a diametral sphere. If some vertices encroach (inside) the sphere, split this edge by adding a vertex in the middle. Step 1. Schewchuk. (http://www.cs.cmu.edu/~jrs/jrspapers.html) 2/23/2019

3-D Delaunay Triangulation For all constraining facets, find its equatorial sphere. If the equatoral sphere are encroached by some vertices, add the center of the sphere as a new vertex. Step 2. Schewchuk. (http://www.cs.cmu.edu/~jrs/jrspapers.html) 2/23/2019

3-D Delaunay Triangulation Find all skinny tetrahedrons, find the spheres they inscribe. Add the center of the spheres as a vertex. “Skinny” criteria: The ratio of the sphere radius to shortest tetrahedron edge. Step 3. Schewchuk. (http://www.cs.cmu.edu/~jrs/jrspapers.html) 2/23/2019

References D Dobkin et al., Special Issue on Computational Geometry. Proceedings of the IEEE, Vol. 80, No.9, Sep. 1992. Applications for Delaunay Triangulation can be found on http://www.ics.uci.edu/~eppstein/gina/delaunay.html N. Max, P. Hanrahan et al., Area and Volumn Coherence for Efficient Visualization of 3D Scalar Functions, http://www.llnl.gov/graphics/docs/VolViz90.pdf E. Quak, L. Schumaker, Cublic Spline Fitting using Data Dependent Triangulations. Computer Aided Geometric Design, Vol.7, 1990, pp293-301 J. Ruppert. A Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh Generation. J. Algorithms, pp.548-585, May 1995 http://science.nas.nasa.gov/Pubs/TechReports/RNRreports/jruppert/RNR-94-002/RNR-94-002.html J. Shewchuk, Selected Papers on Triangulation, http://www.cs.cmu.edu/~jrs/jrspapers.html 2/23/2019