Dirichlet energy of Delaunay meshes and intrinsic Delaunay triangulations. (September 2020) - Ye, Zipeng; Yi, Ran; Gong, Wenyong; He, Ying; Liu, Yong-Jin
Abstract: The Dirichlet energy of a smooth function measures how variable the function is. Due to its deep connection to the Laplace–Beltrami operator, Dirichlet energy plays an important role in digital geometry processing. Given a 2-manifold triangle mesh M with vertex set V, the generalized Rippa's theorem shows that the Dirichlet energy among all possible triangulations of V arrives at its minimum on the intrinsic Delaunay triangulation (IDT) of V . Recently, Delaunay meshes (DM) – a special type of triangle mesh whose IDT is the mesh itself – were proposed, which can be constructed by splitting mesh edges and refining the triangulation to ensure the Delaunay condition. This paper focuses on Dirichlet energy for functions defined on DMs. Given an arbitrary function f defined on the original mesh vertices V, we present a scheme to assign function values to the DM vertices V n e w ⊃ V by interpolating f . We prove that the Dirichlet energy on DM is no more than that on the IDT. Furthermore, among all possible functions defined on V n e w by interpolating f, our scheme attains the global minimum of Dirichlet energy on a given DM. Graphical abstract: Highlights: The relation of Dirichlet energies on DM and IDT is revealed in this paper, which is not reported in existing literature. We propose a holistic COT harmonic interpolation scheme which ensures the unique global minimum of Dirichlet energy on DM. We have proved that the Dirichlet energy of DM is not larger than that of IDT according to the proposed interpolation.