# Flow-based Delaunay Surface Reconstruction in Two Dimensions

The process starts with a Delaunay triangulation of the convex hull. In two dimensions, the Delaunay triangulation of a set of points is such that the circumcircle of any element (circle passing through the three vertices of the element) is empty.

Here's an example of the Delaunay triangulation of the convex hull of a point cloud (note that I did this by hand, so I don't actually guarantee it's a Delaunay triangulation but it does look like one):

We basically want to remove elements from the boundary in until we get this (the two-dimensional model in mesh form):

Looking very carefully at the Delaunay triangulation of the convex hull, for a given boundary edge and its unique connected element, the circumcenter (center of the circumcircle) is either on the outside side of the boundary edge or the inside. Any time the circumcenter is on the outside side of the boundary edge, it's an element we probably don't want because it's very likely to be an element that's outside whatever the point cloud is supposed to model. Note that if the vertex opposite the boundary edge is inside the smallest circle passing through the edge's end vertices (we will refer to that circle as the circumcircle of the boundary edge), the circumcenter is on the outside side of the boundary edge and vice versa. This means that we have two ways to tell if an element should be removed. In practice, the latter interpretation of the test is preferred.

The two equivalent ways to tell if an element connected to a boundary edge should be removed:

Here's the pseudo code for the carving or sculpting algorithm that starts from the Delaunay triangulation of the convex hull and ends with the proper (or not so proper) mesh of the two-dimensional model:

Of course, it can't be that easy, all the time. In some cases, the algorithm above may terminate prematurely. Here's an example:

When this happens, the element must be deleted manually or some heuristics based on element size must be used. The algorithm above then continues where it left off. This is clearly the Achille's heal of the method since any decision making based on size can be dicey when sampling is not dense enough.

# Flow-based Delaunay Surface Reconstruction in Three Dimensions

In three dimensions, things are little bit more complicated but the methodology is exactly the same as in two dimensions. Instead of triangles, the Delaunay triangulation of the convex hull of the point cloud is made up of tetrahedra. The test to see whether an element connected to a boundary face should be removed from the mesh involves either i) computing the circumsphere of the tetrahedron and determining if the circumcenter is on the outside side of the boundary face or ii) computing the circumsphere of the boundary face (the smallest sphere passing through its three bounding vertices) and determining if the vertex opposite the boundary face is inside. Unlike the two dimensional case, there is a bit of a problem when the element connected to a boundary face is bounded by another boundary face. In that case, you can remove the element if both tests (on both boundary faces) tell you that you should remove the element or if just one test (on one of the two boundary faces) tells you that you should remove the element. The former is obviously more aggressive than the later at carving elements away.

Here's the pseudo code for the carving or sculpting algorithm that starts from the Delaunay triangulation of the convex hull and ends with the proper mesh of the three-dimensional model:

In reality, it's a bit more complicated than that since you also have to deal with boundary faces that become disconnected (dangling boundary faces), which may have to be carved away as well (using a methodology similar to the two dimensional case). This may come as no surprise that when all is said and done, you may have to manually delete some elements (or use some potentially dangerous heuristics based on size) and restart in order to get the mesh you really want. The big issue is how much of this manual deleting has to be done. In two dimensions, it may not be too big of a deal but in three dimensions, it may be a problem. Unless there's a "good" automated way to perform those manual deletions, I don't see how a Delaunay based surface reconstruction like this one can be that useful.

What this methodology is lacking is a global view of the problem especially when you have to do manual deletion to "unstuck" the algorithm. It's very hard to tell if an element should be removed (is outside) when all the information at your disposal is very local. This in contrast with another Delaunay-based explicit method called Eigencrust ("Spectral Surface Reconstruction from Noisy Point Clouds" by Ravikrishna Kolluri, Jonathan Richard Shewchuk, and James F. O’Brienwhich) which partitions a graph that connects the elements to find the surface in one global operation (a costly proposition though, clock wise).

Delaunay-based methods like this one are referred to as explicit because there's no surface approximation involved, in other words, the surface passes through points that are actually in the cloud (To be pedantic, the surface is interpolated, not approximated.) This is unlike implicit methods like Poisson or "marching cubes" that approximate the object's surface. Note that those implicit methods need a good normal extrapolation scheme to be effective whereas an explicit method doesn't. If one is able to get decent, consistent with each other normals, a Poisson surface reconstruction ("Poisson Surface Reconstruction" by Michael Kazhdan, Matthew Bolitho and Hugues Hoppe) is extremely hard to beat.

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