Technical Report MSC-2016-05

Title: Tangent Vector Fields on Triangulated Surfaces - An Edge-Based Approach
Authors: Alexandre Djerbetian
Supervisors: Mirela Ben Chen
Abstract: Since Tron in 1982, Computer-Generated Imagery (CGI) gradually infiltrated most of today’s productions. From purely animated movies like Ratatouille to integrated special effects in Star Wars, CGI is present in almost every movie. Partly, the immense visual progress from Tron to Ratatouille can be attributed to advances made in Geometry Processing.

Geometry Processing covers several subfields, such as texture handling, mesh animation or remeshing. In many of those applications, the algorithms rely on the ability to design and handle vector fields, and more precisely, tangent vector fields to surfaces. Even though tangent vector fields are intuitive to visualize, their handling has been, and still is, extensively studied. Indeed, the discretization of curved surfaces into triangulations leads for instance to ill-defined tangent planes on edges and at vertices. The main question at the origin of this work is therefore the question of representation: how do we discretize / sample a vector field on a discrete surface?

The first and most popular choice of representation is face-based sampling. The representation requires one vector per face, and considers the vector field constant on each face. Since the tangent plane of a face is the face itself, this representation alleviates the problem of tangency and therefore has simple formulations of most of the common operators. However, it also leads to lower accuracy and to difficulties in defining derivatives and smoothness energies.

Another choice is to sample the vector field on vertices, and then “linearly” interpolate the values inside faces. This representation yields better results in terms of accuracy and allows the direct computation of derivatives and smoothness energies. However, one needs to redefine the tangent plane at the vertex, which is a nonlinear operation and therefore leads to a complicated formulation.

Finally, another representation used in Geometry Processing comes from Discrete Exterior Calculus (DEC), and stores on every edge the integrated projection of the vector field along the edge. This representation is linear, and therefore simple and accurate but has one major drawback: contrary the previous representations, it is not clear how DEC can represent N-RoSy vector fields. N-RoSy fields are N Rotationally Symmetric vectors associated to one point. For instance, a 4-RoSy vector field assigns a "cross" to every point of the mesh. N-RoSy vector fields are very useful in remeshing and non-photorealistic rendering, for instance.

In our work, we propose a simple yet powerful new representation which stores vectors on edge mid-points and linearly interpolates inside faces. Interestingly, the tangency problem can be trivially resolved at the edges, and the resulting vector fields are linear per face. Our representation is therefore simple, accurate and can handle N-RoSy vector fields and therefore provides a simple unified framework for tangent vector field processing.

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