Bijective Deformations in $\mathbb{R}^n$ via Integral Curve Coordinates

TL;DR

This paper introduces Integral Curve Coordinates, a bijective mapping method for shapes in any dimension based on gradient integral curves, useful for shape analysis and deformation.

Contribution

It proposes a novel coordinate system using integral curves of a function's gradient, applicable in arbitrary dimensions with a simple algorithm for shape deformation.

Findings

Applicable to shapes in any dimension with spherical boundary topology.

Provides a simple algorithm for generating suitable functions in any dimension.

Demonstrates practical tracing of integral curves on triangulated grids.

Abstract

We introduce Integral Curve Coordinates, which identify each point in a bounded domain with a parameter along an integral curve of the gradient of a function $f$ on that domain; suitable functions have exactly one critical point, a maximum, in the domain, and the gradient of the function on the boundary points inward. Because every integral curve intersects the boundary exactly once, Integral Curve Coordinates provide a natural bijective mapping from one domain to another given a bijection of the boundary. Our approach can be applied to shapes in any dimension, provided that the boundary of the shape (or cage) is topologically equivalent to an $n$-sphere. We present a simple algorithm for generating a suitable function space for $f$ in any dimension. We demonstrate our approach in 2D and describe a practical (simple and robust) algorithm for tracing integral curves on a (piecewise-linear) triangulated regular grid.