Discrete exterior calculus for variational problems in computer vision and graphics

Abstract

Abstract-This paper demonstrates how discrete exterior calculus tools may be useful in computer vision and graphics. A variational approach provides a link with mechanics.

Multigrid convergent principal curvature estimators in digital geometry

“…Digital Geometry is a perfect domain for such 7 Implicit surface is 81x 4 + 81y 4 + 81z 4 − 45x 2 − 45y 2 − 45z 2 − 6 = 0. bound and has demonstrated efficient algorithm in practice with low computational costs. Convergence speed are obtained with a weak constraint on the distance betweenx and x (which just needs to be lower that h for the l ∞ metric).…”

“…They generally fall into three categories: fitting, discrete methods, curvature tensor estimation. We may refer to [5] and [6] for comprehensive evaluations, and Desbrun et al [7] or Bobenko and Suris [8] for a more general theory. Most of them have not theoretical convergence guarantees even without noise on the mesh.…”

Modern Trends in Hypercomplex Analysis

“…The same can and will be studied in the case of Script Geometry although some of necessary tools need to be developed since Script Geometry is a more loose concept than working with simplicial complexes. In [14] the authors describe their approach to the theory of discrete exterior calculus (DEC). They introduce notions of discrete vector fields and operators acting on them, e.g.…”

“…Hereby, one chooses first the mesh and introduces a boundary operator given by the mesh which induces the corresponding discrete variational formulation. From a practical point of view this is even more interesting since finite element meshes are also widely being applied in other fields, such as computer graphics [13], [14]. In this framework notions of discrete vector fields and operators acting on them, e.g.…”

“…One of the major problems with this approach is that there is no a priori connection between the choice of the mesh and the variational formulation. The modern approach lifts the problem and, therefore, the finite element modelation directly on to the mesh, resulting in the so-called Finite Element Exterior Calculus [1,14]. Hereby, one chooses first the mesh and introduces a boundary operator given by the mesh which induces the corresponding discrete variational formulation.…”