Frame Field Operators

TL;DR

This paper introduces a generalized class of anisotropic fourth-order differential operators, extending the Bilaplacian, parametrized by frame fields for enhanced control in geometry processing tasks.

Contribution

It proposes a novel discretization scheme for these operators, analyzes their properties, and explores potential applications in geometry processing.

Findings

Discretization converges reliably.

Operators exhibit predictable behavior under pullback.

Potential for improved geometry processing techniques.

Abstract

Differential operators are widely used in geometry processing for problem domains like spectral shape analysis, data interpolation, parametrization and mapping, and meshing. In addition to the ubiquitous cotangent Laplacian, anisotropic second-order operators, as well as higher-order operators such as the Bilaplacian, have been discretized for specialized applications. In this paper, we study a class of operators that generalizes the fourth-order Bilaplacian to support anisotropic behavior. The anisotropy is parametrized by a symmetric frame field, first studied in connection with quadrilateral and hexahedral meshing, which allows for fine-grained control of local directions of variation. We discretize these operators using a mixed finite element scheme, verify convergence of the discretization, study the behavior of the operator under pullback, and present potential applications.