On Smooth 3D Frame Field Design

TL;DR

This paper analyzes and formalizes methods for generating smooth 2D and 3D frame fields, showing that a unified function-based formulation simplifies the optimization process and improves initialization for 3D fields.

Contribution

It introduces a unified function-based formulation for 2D and 3D frame field optimization, simplifying the process and enhancing initialization strategies for 3D fields.

Findings

The 2D algorithm effectively finds good smooth fields.

The 3D problem shares similarities with 2D, enabling transfer of optimization techniques.

Using the new algorithm for initialization improves 3D field quality.

Abstract

We analyze actual methods that generate smooth frame fields both in 2D and in 3D. We formalize the 2D problem by representing frames as functions (as it was done in 3D), and show that the derived optimization problem is the one that previous work obtain via "representation vectors." We show (in 2D) why this non linear optimization problem is easier to solve than directly minimizing the rotation angle of the field, and observe that the 2D algorithm is able to find good fields. Now, the 2D and the 3D optimization problems are derived from the same formulation (based on representing frames by functions). Their energies share some similarities from an optimization point of view (smoothness, local minima, bounds of partial derivatives, etc.), so we applied the 2D resolution mechanism to the 3D problem. Our evaluation of all existing 3D methods suggests to initialize the field by this new algorithm, but possibly use another method for further smoothing.