45 research outputs found
Discrete exterior calculus (DEC) for the surface Navier-Stokes equation
We consider a numerical approach for the incompressible surface Navier-Stokes
equation. The approach is based on the covariant form and uses discrete
exterior calculus (DEC) in space and a semi-implicit discretization in time.
The discretization is described in detail and related to finite difference
schemes on staggered grids in flat space for which we demonstrate second order
convergence. We compare computational results with a vorticity-stream function
approach for surfaces with genus 0 and demonstrate the interplay between
topology, geometry and flow properties. Our discretization also allows to
handle harmonic vector fields, which we demonstrate on a torus.Comment: 21 pages, 9 figure
Small grid embeddings of 3-polytopes
We introduce an algorithm that embeds a given 3-connected planar graph as a
convex 3-polytope with integer coordinates. The size of the coordinates is
bounded by . If the graph contains a triangle we can
bound the integer coordinates by . If the graph contains a
quadrilateral we can bound the integer coordinates by . The
crucial part of the algorithm is to find a convex plane embedding whose edges
can be weighted such that the sum of the weighted edges, seen as vectors,
cancel at every point. It is well known that this can be guaranteed for the
interior vertices by applying a technique of Tutte. We show how to extend
Tutte's ideas to construct a plane embedding where the weighted vector sums
cancel also on the vertices of the boundary face
The Complexity of Drawing a Graph in a Polygonal Region
We prove that the following problem is complete for the existential theory of
the reals: Given a planar graph and a polygonal region, with some vertices of
the graph assigned to points on the boundary of the region, place the remaining
vertices to create a planar straight-line drawing of the graph inside the
region. This strengthens an NP-hardness result by Patrignani on extending
partial planar graph drawings. Our result is one of the first showing that a
problem of drawing planar graphs with straight-line edges is hard for the
existential theory of the reals. The complexity of the problem is open in the
case of a simply connected region.
We also show that, even for integer input coordinates, it is possible that
drawing a graph in a polygonal region requires some vertices to be placed at
irrational coordinates. By contrast, the coordinates are known to be bounded in
the special case of a convex region, or for drawing a path in any polygonal
region.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
Grid computing in image analysis
Diagnostic surgical pathology or tissue–based diagnosis still remains the most reliable and specific diagnostic medical procedure. The development of whole slide scanners permits the creation of virtual slides and to work on so-called virtual microscopes. In addition to interactive work on virtual slides approaches have been reported that introduce automated virtual microscopy, which is composed of several tools focusing on quite different tasks. These include evaluation of image quality and image standardization, analysis of potential useful thresholds for object detection and identification (segmentation), dynamic segmentation procedures, adjustable magnification to optimize feature extraction, and texture analysis including image transformation and evaluation of elementary primitives
Hierarchical Techniques for Global Illumination Computations -- Recent Trends and Developments
Since the beginning of computer graphics, one of the primary goals has been to create convincingly realistic images of three-dimensional environments that would be impossible to distinguish from photographs of the real scene. The goal to create photo-realistic images has lead to the development of completely new software techniques for dealing with the inherent geometric and optical complexity of real world scenes. This paper gives an overview of advanced algorithms for photorealistic rendering and in particular discusses hierarchical techniques for global illumination computations