Laplacian guided editing, synthesis, and simulation

Abstract

The Laplacian has been playing a central role in numerous scientific and engineering problems. It has also become popular in computer graphics. This talk presents a series of our work that exploits the Laplacian in mesh editing, texture synthesis and flow simulation. First, a review is given on mesh editing using differential coordinates and the Poisson equation, which involves the Laplacian. The distinctive feature of this approach is that it modifies the original geometry implicitly through gradient field manipulation. This approach can produce desired results for both global and local editing operations, such as deformation, object merging, and denoising. This technique is computationally involved since it requires solving a large sparse linear system. To overcome this difficulty, an efficient multigrid algorithm specifically tailored for geometry processing has been developed. This multigrid algorithm is capable of interactively processing meshes with hundreds of thousands of vertices. In our latest work, Laplacian-based editing has been generalized to deforming mesh sequences, and efficient user interaction techniques have also been designed. Second, this talk presents a Laplacian-based method for surface texture synthesis and mixing from multiple sources. Eliminating seams among texture patches is important during texture synthesis. In our technique, it is solved by performing Laplacian texture reconstruction, which retains the high frequency details but computes new consistent low frequency components. Third, a method for inviscid flow simulation over manifold surfaces is presented. This method enforces incompressibility on closed surfaces by solving a discrete Poisson equation. Different from previous work, it performs simulations directly on triangle meshes and thus eliminates parametrization distortions. © 2007 IEEE.