Geometric optimization using nonlinear rotation-invariant coordinates

TL;DR

This paper develops a framework using nonlinear rotation-invariant coordinates (NRIC) for solving geometric optimization problems on discrete surfaces, enabling efficient and robust surface reconstruction and deformation analysis.

Contribution

It introduces a new approach to formulate and solve geometric optimization problems using NRIC, including integrability conditions, quaternion reformulation, and a fast reconstruction algorithm.

Findings

NRIC-based optimization is effective for near-isometric deformations.

The paper provides explicit derivatives for integrability conditions.

The approach simplifies solving geometric problems on triangular meshes.

Abstract

Geometric optimization problems are at the core of many applications in geometry processing. The choice of a representation fitting an optimization problem can considerably simplify solving the problem. We consider the Nonlinear Rotation-Invariant Coordinates (NRIC) that represent the nodal positions of a discrete triangular surface with fixed combinatorics as a vector that stacks all edge lengths and dihedral angles of the mesh. It is known that this representation associates a unique vector to an equivalence class of nodal positions that differ by a rigid body motion. Moreover, integrability conditions that ensure the existence of nodal positions that match a given vector of edge lengths and dihedral angles have been established. The goal of this paper is to develop the machinery needed to use the NRIC for solving geometric optimization problems. First, we use the integrability conditions to derive an implicit description of the space of discrete surfaces as a submanifold of an Euclidean space and a corresponding description of its tangent spaces. Secondly, we reformulate the integrability conditions using quaternions and provide explicit formulas for their first and second derivatives facilitating the use of Hessians in NRIC-based optimization problems. Lastly, we introduce a fast and robust algorithm that reconstructs nodal positions from almost integrable NRIC. We demonstrate the benefits of this approach on a collection of geometric optimization problems. Comparisons to alternative approaches indicate that NRIC-based optimization is particularly effective for problems involving near-isometric deformations.