Laplacian Framework for Interactive Mesh Editing

Recent works in geometric modeling show the advantage of local differential coordinates in various surface processing applications. In this paper we review recent methods that advocate surface representation via differential coordinates as a basis to interactive mesh editing. One of the main challenges in editing a mesh is to retain the visual appearance of the surface after applying various modifications. The differential coordinates capture the local geometric details and therefore are a natural surface representation for editing applications. The coordinates are obtained by applying a linear operator to the mesh geometry. Given suitable deformation constraints, the mesh geometry is reconstructed from the differential representation by solving a sparse linear system. The differential coordinates are not rotation-invariant and thus their rotation must be explicitly handled in order to retain the correct orientation of the surface details. We review two methods for computing the local rotations: the first estimates them heuristically using a deformation which only preserves the underlying smooth surface, and the second estimates the rotations implicitly through a variational representation of the problem. We show that the linear reconstruction system can be solved fast enough to guarantee interactive response time thanks to a precomputed factorization of the coefficient matrix. We demonstrate that this approach enables to edit complex meshes while retaining the shape of the details in their natural orientation