A Minimization Approach for Constructing Generalized Barycentric Coordinates and Its Computation

Abstract

We are interested in constructing more generalized barycentric coordinates (GBC) over arbitrary polygon in the 2D setting. We propose a constrained minimization over the class of infinitely differentiable functions subject to the GBC constraints of preserving linear functions and the non-negativity condition. It includes the harmonic GBC, biharmonic GBC, maximum entropy GBC, local barycentric coordinates as special cases. We mainly show that the constrained minimization has a unique solution when the minimizing functional is strictly convex. Next we use a $C^r$ smoothness spline function space $S^r_d(\triangle )$ with $r\ge 2$ over a triangulation $\triangle$ of a polygon of interest in $\mathbb {R}^2$ to approximate the minimizer. One advantage of using smooth splines is that derivaties, e.g. the mean curvature and/or Gaussian curvature of spline GBC functions can be calculated. As the minimization restricted to the spline space $S^r_d$ certainly has a unique minimizer, we use the standard projected gradient descent (PGD) method to approximate the spline minimizer. To find the projection of each iteration, we shall explain an alternating projection algorithm (APA). A convergence of the APA and the convergence of the PGD with the APA will be presented. As an example of this approach, a new kind of biharmonic GBC functions which preserve the nonnegativity is constructed. Finally, we have implemented the PGD method based on bivariate splines of arbitrary degree d and arbitrary smoothness r over arbitrary triangulation as long as $d>>r$. The surfaces of many new GBC’s will be shown. Some standard GBC applications will be demonstrated.