Discrete Smooth Interpolation: Constrained Discrete Fairing for Arbitrary Meshes

In this paper, it is shown how the Discrete Smooth Interpolation method (D.S.I.) may be used as a new framework for creating, fairing and editing triangulated surfaces. Given an arbitrary mesh with an arbitrary set of vertices fixed by the user, D.S.I. assigns coordinates to the other nodes of the mesh, enabling the fixed vertices to be interpolated smoothly. The squared discrete Laplacian criterion minimized by D.S.I. is an objective function similar to the bending energy of a thin-plate. This approach fulfills the requirements of subdivision methods, in that it provides arbitrary topology, simplicity, the possibility to define creases of variable sharpness, as well as the convergence of recursive subdivisions to a smooth surface. It does not suffer from the limitations inherent to more classic subdivision methods, such as the subdivision connectivity requirement. Moreover, D.S.I. offers a high degree of flexibility. It then becomes possible to define the surface zones to be smoothed in order of preference. Furthermore, constraints linearly combining the coordinates at the vertices of the surface may be taken into account in a least square sense. As a result, a surface can be fitted to an arbitrary set of points with, or without, specified normals. The method might also have important implications for multi-resolution editing and mesh compression. CR Categories: I.3.5 [Computational Geometry and Object Modeling]: curve, surface, solid and object representations