Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes

We consider an algorithm called FEMWARP for warping triangular and tetrahedral finite element meshes that computes the warping using the finite element method itself. The algorithm takes as input a two- or three-dimensional domain defined by a boundary mesh (segments in one dimension or triangles in two dimensions) that has a volume mesh (triangles in two dimensions or tetrahedra in three dimensions) in its interior. It also takes as input a prescribed movement of the boundary mesh. It computes as output updated positions of the vertices of the volume mesh. The first step of the algorithm is to determine from the initial mesh a set of local weights for each interior vertex that describes each interior vertex in terms of the positions of its neighbors. These weights are computed using a finite element stiffness matrix. After a boundary transformation is applied, a linear system of equations based upon the weights is solved to determine the final positions of the interior vertices. The FEMWARP algorithm has been considered in the previous literature (e.g., in a 2001 paper by Baker). FEMWARP has been succesful in computing deformed meshes for certain applications. However, sometimes FEMWARP reverses elements; this is our main concern in this paper. We analyze the causes for this undesirable behavior and propose several techniques to make the method more robust against reversals. The most successful of the proposed methods includes combining FEMWARP with an optimization-based untangler.Comment: Revision of earlier version of paper. Submitted for publication in BIT Numerical Mathematics on 27 April 2010. Accepted for publication on 7 September 2010. Published online on 9 October 2010. The final publication is available at http://www.springerlink.co

Patch-based image vectorization with automatic curvilinear feature alignment

Raster image vectorization is increasingly important since vector-based graphical contents have been adopted in personal computers and on the Internet. In this paper, we introduce an effective vector-based representation and its associated vectorization algorithm for full-color raster images. There are two important characteristics of our representation. First, the image plane is decomposed into nonoverlapping parametric triangular patches with curved boundaries. Such a simplicial layout supports a flexible topology and facilitates adaptive patch distribution. Second, a subset of the curved patch boundaries are dedicated to faithfully representing curvilinear features. They are automatically aligned with the features. Because of this, patches are expected to have moderate internal variations that can be well approximated using smooth functions. We have developed effective techniques for patch boundary optimization and patch color fitting to accurately and compactly approximate raster images with both smooth variations and curvilinear features. A real-time GPU-accelerated parallel algorithm based on recursive patch subdivision has also been developed for rasterizing a vectorized image. Experiments and comparisons indicate our image vectorization algorithm achieves a more accurate and compact vector-based representation than existing ones do. © 2009 ACM.link_to_subscribed_fulltex