Geodesy

Quantum mechanics

Injective function

Algorithm

Benchmark (surveying)

Philosophy

Pure mathematics

Discrete mathematics

Computer graphics (images)

Dimension (graph theory)

Computer science

Mathematical optimization

Simple (philosophy)

Embedding

Physics

Mathematics

Geography

Maxima and minima

Epistemology

Domain (mathematical analysis)

Mathematical analysis

Artificial intelligence

Combinatorics

Polygon mesh

Topology (electrical circuits)

Volume (thermodynamics)

ACM Transactions on Graphics

Mapping a source mesh into a target domain while preserving local injectivity is an important but highly non-trivial task. Existing methods either require an already-injective starting configuration, which is often not available, or rely on sophisticated solving schemes. We propose a novel energy form, called Total Lifted Content (TLC), that is equipped with theoretical properties desirable for injectivity optimization. By lifting the simplices of the mesh into a higher dimension and measuring their contents (2D area or 3D volume) there, TLC is smooth over the entire embedding space and its global minima are always injective. The energy is simple to minimize using standard gradient-based solvers. Our method achieved 100% success rate on an extensive benchmark of embedding problems for triangular and tetrahedral meshes, on which existing methods only have varied success.

Lifting simplices to find injectivity