Recently, I worked with Ingrid Daubechies and her postdoc Shira Golovin on animal tooth data sets during my visit at Duke University. I used two approaches to construct smooth (C^1) tooth surfaces. One is based on bivariate splines to fit a given tooth surface data (points and triangulation on the left of the following images) (local patches with a blending function). The MATLAB code is based on Tsung-Wei Hu's dissertation, 2023. The method is an extension of the smooth space curve construction which is published in 2025. See the article here. The other approach is to use trivariate splines which interpolate the function value 1 at the given tooth data points only and use the isosurface method to produce a smooth tooth surface. With these two tools, I am able to generate smooth tooth surfaces (to be shown below), compute the tangent bundle (the tangent vector space around each tooth surface) of tooth surfaces, and etc.. For bivariate and trivariate splines, see the monograph. Let me use the following images to show some computational results based on the first approach. All the data sets are from Dr. Tingran Gao who is a former student of Ingrid Daubechies. The tooth triangulations on the left of all images are based on the data sets from Dr. Gao. The surfaces on the right of all the images below are produced by MATLAB code developed by Dr. Tsung-Wei Hu who is a former student of mine. Each tooth surface needs 2 hours to be generated as the memory of my computer is not large enough. The following is a set of examples based on the first approach. The computational results based on the second approach can be found in these two webpages. One is Tooth Surface Construction by 3D Splines and the other one is Damaged Tooth Repairs.
