Obtaining a Canonical Polygonal Schema from a Greedy Homotopy Basis with Minimal Mesh Refinement

by   Marco Livesu, et al.

Any closed manifold of genus g can be cut open to form a topological disk and then mapped to a regular polygon with 4g sides. This construction is called the canonical polygonal schema of the manifold, and is a key ingredient for many applications in graphics and engineering, where a parameterization between two shapes with same topology is often needed. The sides of the 4g-gon define on the manifold a system of loops, which all intersect at a single point and are disjoint elsewhere. Computing a shortest system of loops of this kind is NP-hard. A computationally tractable alternative consists in computing a set of shortest loops that are not fully disjoint in polynomial time, using the greedy homotopy basis algorithm proposed by Erickson and Whittlesey, and then detach them in post processing via mesh refinement. Despite this operation is conceptually simple, known refinement strategies do not scale well for high genus shapes, triggering a mesh growth that may exceed the amount of memory available in modern computers, leading to failures. In this paper we study various local refinement operators to detach cycles in a system of loops, and show that there are important differences between them, both in terms of mesh complexity and preservation of the original surface. We ultimately propose two novel refinement approaches: the former minimizes the number of new elements in the mesh, possibly at the cost of a deviation from the input geometry. The latter allows to trade mesh complexity for geometric accuracy, bounding deviation from the input surface. Both strategies are trivial to implement, and experiments confirm that they allow to realize canonical polygonal schemas even for extremely high genus shapes where previous methods fail.


page 1

page 2

page 3

page 4

page 5

page 6

page 7

page 8


Topologically Trivial Closed Walks in Directed Surface Graphs

Let G be a directed graph with n vertices and m edges, embedded on a sur...

Recurrent Neural Networks as Optimal Mesh Refinement Strategies

We show that an optimal finite element mesh refinement algorithm for a p...

Robust Watertight Manifold Surface Generation Method for ShapeNet Models

In this paper, we describe a robust algorithm for 2-Manifold generation ...

Refinement strategies for polygonal meshes applied to adaptive VEM discretization

In the discretization of differential problems on complex geometrical do...

Error-Bounded and Feature Preserving Surface Remeshing with Minimal Angle Improvement

The typical goal of surface remeshing consists in finding a mesh that is...

Succinct Representation of Well-Spaced Point Clouds

A set of n points in low dimensions takes Theta(n w) bits to store on a ...

Quad layouts with high valence singularities for flexible quad meshing

A novel algorithm that produces a quad layout based on imposed set of si...

Please sign up or login with your details

Forgot password? Click here to reset