Fastest available Delaunay triangulation algorithm for GPU - geometry

Which is in your opinion the fastest available Delaunay triangulation algorithm for GPU? Or more general, in parallel

2D Delaunay triangulation
GPU-DT is the fastest 2D Delaunay implementation for the GPU.
It constructs a digital Voronoi diagram in 2D using the GPU Parallel Banding Algorithm. Next it fixes and dualizes this to obtain a 2D triangulation. Finally, it performs edge-flipping in parallel on the GPU to obtain the 2D Delaunay triangulation.
3D Delaunay triangulation
gStar4D is a fast and robust implementation of 3D Delaunay for the GPU.
Similar to GPU-DT, this algorithm constructs the 3D digital Voronoi diagram first. However, in 3D this cannot be dualized to a triangulation due to topological and geometrical problems. Instead, gStar4D uses the neighborhood information from this diagram to create stars lifted to 4D and performs star splaying on them efficiently on the GPU. By extracting the lower hull from this, the 3D Delaunay triangulation is obtained.
A faster alternative is gDel3D, which is a hybrid GPU-CPU algorithm.
It performs parallel insertion and flipping on the GPU. The result is close to Delaunay. It then fixes this result using a conservative star splaying method on the CPU.
All these methods are robust, so they can handle any kind of degenerate input.

Be careful with GPU's: Delaunay Triangulations require orientation tests. These do not work reliably with floating point arithmetic, and it might be hard to cope with that
problem using a GPU. Also the memory management is crucial.
You might want to try http://www.geom.at/fade2d/html/ which is among the fastest robust
single threaded implementations.

Related

Interpolation of Polyhedron

Given a polyhedron defined by a matrix of 3-Dimensional vertices and its faces(delaunay triangles), I want to be able to create a smooth 3-D object.
Is there any software that has built a built in function that would allow me to do this?
If not, I have found a paper that seems to describe what I want, but I am unable to fully understand the math. http://graphics.berkeley.edu/papers/Turk-MIS-2002-10/Turk-MIS-2002-10.pdf.
Here is an examples of what I am looking for.
Rabbit
One solution for "smoothing" geometry, if we state the problem a bit more formally, is to perform mean curvature flow on your mesh. Here are some search terms - "curve-shortening flow", "mean curvature flow", "willmore flow", "conformal curvature flow" ...
Image source: Keenan Crane. Context and permission
"Smoothness of a surface or curve is very hard to define. (For an empirical test on what people perceive as smooth see http://www.levien.com/phd/thesis.pdf#page=23).
If you only care about perceived smoothness, for example, smoother appearance while rendering in high resolution etc., an easier approach would be Catmull-Clark subdivision scheme.
The geometric intuition is quite simple. In the case of a 2D curve, in every instance, every point on a curve moves according to some function of the curvature at that point. If we let the curve or surface move like this for some time, it will start smoothing out areas with high curvature more and more, eventually becoming a circle (or a sphere in 3d) and then collapse to a point. So for smoothing usually we have to preserve areas or volumes.
One way to define it is in terms of some energy, and our goal is to minimise this energy on the mesh. For example willmore flow minimises the willmore energy. Sometimes this process is called fairing.
I am not aware of a prepackaged library or tool, that's freely available and open source for curvature flow.
Algorithms
2D only
K.Mikula, D.Sevcovic, "Tangentially stabilized Lagrangian algorithm for elastic curve evolution driven by intrinsic Laplacian of curvature",
pdf
2D and 3D
https://www.youtube.com/watch?v=Jhqlmcms04M.
Keenan Crane's page has more information on this and more examples too.
http://www.cs.cmu.edu/~kmcrane/Projects/ConformalWillmoreFlow/
2D and 3D (level set method)
https://math.berkeley.edu/~sethian/2006/level_set.html

voronoi diagram and delaunay triangulation understanding

Can someone help me to understand the purpose of delaunay triangulation? what is it use for?
Also I got to know that delaunay triangulation is the dual graph of voronoi diagram but not really fully understand it. Thanks so much
The Delaunay triangulation is a "good" triangulation in that it finds triangles that are not too skewed. It is unique and guarantees that the circumcircle of any triangle is empty. It is relatively fast to build, requiring no more than O(N Log(N)) operations.
It is used where triangulations are used, for instance for interpolation on irregular 2D point sets.
It is said to be the dual of the Voronoi diagram because vertexes (sites) are in one-to-one correspondence with the faces (regions) and vice versa. And when two Voronoi regions share an edge, the corresponding sites are linked by a triangle side.

intersection of two triangle meshes

Currently I am looking for an efficient algorithm to compute the intersection of two triangle meshes. I have searched over the internet, but haven't found valuable materials. The book Real-Time Collision Detection is a helpful book but is too complex for my task. I also found the post:Triangle to triangle collision detection in 3D. However I hope to find a detailed description about the algorithm.
Regards
Jogging
Well it depends on meshes size, testing each triangle in each mesh against the other is only valid in small meshes since it has n^2 complexity.
To work around that most algorithms use
Spatial portioning
first to subdivide the space into smaller ones and then tackles each one separately.
For spatial portioning most algorithms use
OcTrees
or BSPTrees however if you don't need to complicate things you can just subdivide the space into n boxes then check triangle triangle intersection in each box

Generating coordinates for abstract triangulation

I have a piece of abstract triangulation, made entirely out of equilateral triangles, that describes a curved 2d space. As such, some vertices have for example 7 equilateral triangles attached to them. Now I want to draw this as a terrain.
This has to be done in 3d, so I expect a lot of saddle nodes and some cone-like structures. I am currently trying to find a nice algorithm that does this for me, but as of yet I have come out empty handed. In principle you could 'just' solve a large set of quadratic equations that fixes all the distances, but this is unfeasible. I would be content with an algorithm that gives a best approximation.
Any advice?

Triangulated Irregular Networks from qhull

I wanted to create TINs from 3D points (about 7 million in every file) using qhull.
can anyone suggest a place where i could probably see how to do this? thanks!
I've never used QHull since it is hard to integrate as a library into an existing project. Try out Triangle; it is specialized for 2D and is very easy to use (it comes with an example of how to call it from other C code).
I could recommend you a software package called Streaming Computation of Delaunay Triangulations. On a normal computer it can compute
Delaunay triangulations for large,
well-distributed data sets in 2D and
3D which can be greatly accelerated by
exploiting the natural spatial
coherence in a stream of points.
In terms of performance:
We compute a billion-triangle terrain
representation for the Neuse River
system from 11.2 GB of LIDAR data in
48 minutes using only 70 MB of memory
on a laptop.
Here is teaser image on how it works:
You can check out this video explaining their method/software.
Wiki says,
A TIN comprises a triangular network
of vertices, known as mass points,
with associated coordinates in three
dimensions connected by edges to form
a triangular tessellation.
Three-dimensional visualizations are
readily created by rendering of the
triangular facets. In regions where
there is little variation in surface
height, the points may be widely
spaced whereas in areas of more
intense variation in height the point
density is increased.
A TIN is typically based on a Delaunay
triangulation but its utility will be
limited by the selection of input data
points: well-chosen points will be
located so as to capture significant
changes in surface form, such as
topographical summits, breaks of
slope, ridges, valley floors, pits and
cols.
MATLAB can generate 3-D Delaunay tesselation and n-D Delaunay tesselation using Qhull.
3-dimensional Delaunay tessellation - tetramesh is used to plot the tetrahedrons that form the corresponding simplex
(source: mathworks.com)

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