In this page, I explain how to use trivariate splines for constructing a smooth interpolatory spline for a given data set. The computational method was explained in my paper. Mainly, we look for a trivariate spline $S$ which interpolates $1$ at the given data points, interpolates $0$ on the auxiliary points on the bounding box, and interpolates $2$ on the points near the center of the given data set. The spline S is also a minimization of the energy function and satisfies C^1 smoothness condition. In general, this requires the spline is of degree 9. Due to the equally spaced points for a tetrahedral partition of the bounding box of the given data set, the spline space over such a tetrahedralization is not empty when d >=6 and r=1. We can find a spline satisfying the above constraints. These are mathematical heuristic for our computation. We have already implemented trivariate splines 20 years ago. See our paper [Awanou, Lai, Wenston,2006]. In addition, my former Ph.D. students Clayton Mersmann and Yidong Xu had also implemented trivariate splines for various computational problems, e.g. numerical solutions of PDEs and data fitting problems. I simply used their MATLAB codes to do the tooth surface construction. The tooth data sets are from Dr. Tingran Gao, a Ph.D. graduated from Duke University, 2015 under Dr. Ingrid Daubechies' supervision. In addition to this approach, I have also used another approach developed by Dr. Tsung-Wei Hu who is a former Ph.D. student of mine to construct tooth surfaces. See some constructions in the webpage. The following are some examples. Example 1. Let me first show ten tooth surfaces. They are generated by using trivariate splines of degree 8 and C^1. Example 2. In order to see the smooth surfaces better, let me use trivariate spline functions of degree 10 and C^2 smoothness. I used MATLAB transparency method to show the surfaces. The following surfaces are the surface by using splines of C^1 smooth and degree 9. The other one is based on trivariate splines of C^2 smoothness and degree 9. We can see that the last one is the best as it is very clean smooth tooth surface. I finally see how to generate very nice surfaces.
