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 graphical mesh editing, texture synthesis and flow simulation. First, we have recently introduced a new approach to mesh editing using the Poisson equation, which involves the Laplacian. The distinctive feature of this approach is that it modifies the original mesh geometry implicitly through gradient field manipulation. Our 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, we have developed a multigrid algorithm specifically tailored for geometry processing. This multigrid algorithm is capable of interactively processing meshes with hundreds of thousands of vertices. In our latest work, we have also generalized Laplacian-based editing to deforming mesh sequences and designed efficient user interaction techniques. Second, we introduce a Laplacian-based method for surface texture synthesis and mixing from multiple sources. Eliminating seams among texture patches is very important during texture synthesis. We solve this problem by performing Laplacian texture reconstruction, which retains the high frequency details but computes new consistent low frequency components. Third, we introduce a method for inviscid flow simulation over manifold surfaces, which is a challenging problem because of the difficulty of establishing plausible physical models. Our method enforces incompressibility on closed surfaces by utilizing a discrete vector field decomposition algorithm. Different from previous work, our method performs simulations directly on triangle meshesand thus eliminates parametrization distortions.
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