[Paper Review] Construction of Injective Mappings Of Meshes.

Yaron Lipman|arXiv (Cornell University)|Oct 3, 2013

Computational Geometry and Mesh Generation15 references3 citations

TL;DR

This paper presents three sets of sufficient conditions to construct injective simplicial mappings for manifold meshes, leveraging orientation consistency and boundary constraints. It generalizes global inversion theorems to piecewise-linear maps, enables injective mapping with flexible boundary conditions via linear constraints, and ensures full injectivity including the boundary when orientation is preserved—offering a practical, optimization-friendly framework for geometry processing.

ABSTRACT

This paper introduces three sets of sufficient conditions, for generating injective simplicial mappings of manifold meshes. A necessary condition for a simplicial mapping of a mesh to be injective is that it consistently preserves or inverts the orientations of all elements. However, these conditions are insufficient to guarantee injectivity. In this paper we provide additional simple conditions that, together with the above mentioned necessary conditions guarantee injectivity of the simplicial map. The first set of conditions generalizes classical global inversion theorems to the mesh (piecewise-linear) case. That is, proves that in case the boundary simplicial map is bijective and the necessary condition holds the map is a bijection. The second set of conditions is concerned with mapping of a mesh to a polytope and replaces the (often hard) requirement of a bijective boundary map with a collection of linear constraints that guarantees that the resulting map is injective over the interior of the mesh. These linear conditions provide a practical tool for optimizing an injective map of the mesh while allowing the boundary map to adjust freely. Allowing more freedom in the boundary conditions is useful for two reasons: a) it circumvents the hard task of providing a bijective boundary map, and b) it allows optimizing the boundary map together with the simplicial map to achieve lower energy levels. The third set of conditions adds to the second set the requirement that the boundary maps are orientation preserving as-well. This set of conditions guarantees that the map is injective on the boundary of the mesh as-well as its interior. Several experiments using the sufficient conditions are shown for mapping triangular meshes injectively. A secondary goal of this paper is to advocate and develop the tool of degree in the context of geometry processing and modeling of meshes.

Motivation & Objective

• To address the challenge of constructing injective simplicial mappings for manifold meshes, which are essential for geometry processing and modeling tasks.
• To overcome the limitation that orientation preservation alone is insufficient for injectivity, by identifying additional sufficient conditions.
• To provide practical tools for optimizing injective maps by relaxing the strict requirement of bijective boundary maps.
• To advocate for the use of the topological degree concept in mesh-based geometry processing and modeling.

Proposed method

• Generalizes classical global inversion theorems to the piecewise-linear (mesh) setting, proving that a bijective boundary map combined with consistent orientation preservation ensures global injectivity.
• Introduces a set of linear constraints on the boundary map that guarantee injectivity in the mesh interior, even when the boundary map is not bijective.
• Extends the second set by requiring orientation-preserving boundary maps, ensuring injectivity both in the interior and on the boundary of the mesh.
• Employs the concept of degree from algebraic topology to analyze and ensure the injectivity of simplicial maps over piecewise-linear meshes.
• Uses optimization frameworks that allow simultaneous adjustment of the boundary map and the interior map to minimize energy while maintaining injectivity.
• Applies the conditions to triangular meshes through multiple experiments, validating the theoretical guarantees with practical mappings.

Experimental results

Research questions

• RQ1 How can injective simplicial mappings of meshes be guaranteed using topological and geometric constraints?
• RQ2 What conditions ensure injectivity when the boundary map is not required to be bijective?
• RQ3 In what way can the concept of degree be effectively applied to verify injectivity in piecewise-linear mesh mappings?
• RQ4 How can boundary freedom be exploited in optimization without sacrificing injectivity?
• RQ5 What additional constraints ensure injectivity not only in the interior but also on the boundary of a mesh?

Key findings

• The paper proves that a bijective boundary map combined with consistent orientation preservation guarantees a globally injective simplicial map, extending classical inversion theorems to the mesh setting.
• Linear constraints on the boundary map can replace the need for bijectivity while still ensuring injectivity in the mesh interior, enabling flexible optimization.
• When the boundary map is both injective and orientation-preserving, the resulting simplicial map is injective across the entire mesh, including the boundary.
• The proposed conditions allow for energy-minimizing optimization of the map while preserving injectivity, even when the boundary is not fixed to be bijective.
• Experiments demonstrate the practical effectiveness of the conditions in generating injective mappings for triangular meshes.
• The use of topological degree provides a robust theoretical foundation for verifying injectivity in mesh mappings, advancing its application in geometry processing.