I am able to use a diffsion map to convert a set of points on a tooth surface into a point set on the surface of unit ball. Once we have a set of points on the surface of a ball, we are able to find a triangulation by using Delaunay triangulation method. A detail is described in a paper. A reference will be given once the status is known. Let me present some examples. Mainly I used the data sets from Dr. Tingran Gao, a former PhD. of Dr. Ingrid Daubechies and convert each of them into a bowl shaped data set which is then mapped into the surface of a unit ball for triangulation. Our main result is to show that there exists a set of diffusion map coordinates which form a bowl shaped data set. In addition, we use a lifting technique to find a triangulation from deformed point cloud by our diffusive map. In order to show that our surface triangulation works, I used a discrete harmonic map to deform the surface triangulation to a planar triangulation over the unit disk. In this way, we can see that the planar triangulation is a good looking triangulation. Finally, I have tried a few examples from morphosouce.org. Each of point clouds contains 128,787 points in R^3. Such the data is so large, I randomly take one fifth of the points and generate its surface triangulation. See examples in the end of this page. Example 1. Consider a tooth point cloud of a tooth from a kind of monkey called Alouatta. Example 2. Consider a tooth point cloud of a tooth from a kind of monkey called Alouatta. Example 3. Consider a tooth point cloud of a tooth from a kind of money called Alouatta. Example 4. Consider a tooth point cloud of a tooth from a kind of monkey called Alouatta. Example 5. Consider a tooth point cloud of a tooth from a kind of monkey called Alouatta. Example 6. Consider a tooth point cloud of a tooth from a kind of monkey called Alouatta. Example 7. Consider a tooth point cloud of a tooth from a kind of monkey called Alouatta. More data sets and their triangulations will be uploaded soon. Let me present more examples together with the images from our diffusion maps. Example 8. Consider a tooth point cloud of a tooth from a kind of monkey called Ateles. . Example 9. Next consider another tooth point cloud from a kind of monkey called Ateles. . We have experimented all 50 teeth point clouds from Dr. Tingran Gao. All results are very good. See below for yourself. One way to show that our surface triangulations are good is to deform a surface triangulation to a planar triangulation by a conformal map. We used a discrete version of harmonic map which preserves all angles of surface triangulation approximately. Here are a few examples. Finally, we present a few examples based on practical situations. The point clouds have a very large number of points, about 128,787. The method works very well. See examples below.
