An Operator Approach to Tangent Vector Field Processing

Abstract

Figure 1: Using our framework various vector field design goals can be easily posed as linear constraints. Here, given three symmetry maps: rotational (S1), bilateral (S2) and front/back (S3), we can generate a symmetric vector field using only S1 (left), S1 + S2 (center) and S1 + S2 + S3 (right). The top row shows the front of the 3D model, and the bottom row its back. AbstractIn this paper, we introduce a novel coordinate-free method for manipulating and analyzing vector fields on discrete surfaces. Unlike the commonly used representations of a vector field as an assignment of vectors to the faces of the mesh, or as real values on edges, we argue that vector fields can also be naturally viewed as operators whose domain and range are functions defined on the mesh. Although this point of view is common in differential geometry it has so far not been adopted in geometry processing applications. We recall the theoretical properties of vector fields represented as operators, and show that composition of vector fields with other functional operators is natural in this setup. This leads to the characterization of vector field properties through commutativity with other operators such as the Laplace-Beltrami and symmetry operators, as well as to a straight-forward definition of differential properties such as the Lie derivative. Finally, we demonstrate a range of applications, such as Killing vector field design, symmetric vector field estimation and joint design on multiple surfaces.

Vector field processing on triangle meshes

“…Other ways to think of vector fields on manifolds in the continuous setting can also be leveraged to derive discrete representations. For instance, vector fields can be characterized as derivations of smooth functions on a manifold using directional derivative; this has led to an operator representation of vector fields, used first in fluid animation [Mullen et al 2009;Pavlov et al 2011;Gawlik et al 2011], and more recently in geometry processing [Azencot et al 2013;Azencot et al 2015].…”

“…In the discrete setting, Pavlov et al [2011] showed that if one defines a discrete notion of diffeormorphism that transfers scalar values between dual cells, then the resulting Lie derivative turns out to be (the Hodge star of) the discrete 1-form we used above, since it represents fluxes across dual cells. Since then, this functional point of view was shown to be very relevant to, e.g., the problem of finding correspondences between meshes [Azencot et al 2013;Azencot et al 2015]. We refer the reader to [Gawlik et al 2011] for a complete treatment of the dynamics of fluids using discrete diffeomorphisms.…”

Discrete Derivatives of Vector Fields on Surfaces -- An Operator Approach

“…Unlike the discretization of the directional derivatives of functions which can be reduced to computing gradients and is thus well established (e.g., Botsch et al [2010] and Azencot et al [2013]), there exists, to the best of our knowledge, no unified treatment of covariant derivatives of vector fields on meshes. Some derived quantities such as the divergence and the curl have received wide attention [Polthier and Preuss 2003;Wardetzky 2006;Hirani 2003;Meyer et al 2002], whereas the general case we are interested in, the Levi-Civita covariant derivative of a tangent vector field, has not been discretized directly.…”

“…We show that, for piecewise constant vector fields, under some mild conditions, it is not possible to define a discrete version of the covariant derivative operator which is both linear and fulfills the metric compatibility property. Finally, we propose a simple approach that is based on the recently introduced multiscale discretization of the directional derivative of functions [Azencot et al 2013], and we demonstrate experimental convergence of the previously mentioned properties under mesh refinement when both the vector fields and functions are smooth.…”

“…It has recently been shown in Azencot et al [2013] that it is possible to discretize the directional derivative of functionsD U f using a multiscale basis, such that the error in the product rule property experimentally decreases with the increase in the mesh resolution. We choose a similar discretization for the directional derivative of functions defined on the faces of the mesh, and thus get experimental convergence of the product rule for the component functions of the vector field.…”