A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing

Abstract

Spectral mesh analysis and processing methods, namely ones that utilize eigenvalues and eigenfunctions of linear operators on meshes, have been applied to numerous geometric processing applications. The operator used predominantly in these methods is the Laplace‐Beltrami operator, which has the often‐cited property that it is intrinsic, namely invariant to isometric deformation of the underlying geometry, including rigid transformations. Depending on the application, this can be either an advantage or a drawback. Recent work has proposed the alternative of using the Dirac operator on surfaces for spectral processing. The available versions of the Dirac operator either only focus on the extrinsic version, or introduce a range of mixed operators on a spectrum between fully extrinsic Dirac operator and intrinsic Laplace operator. In this work, we introduce a unified discretization scheme that describes both an extrinsic and intrinsic Dirac operator on meshes, based on their continuous counterparts on smooth manifolds. In this discretization, both operators are very closely related, and preserve their key properties from the smooth case. We showcase various applications of our operators, with improved numerics over prior work.

A Curvature and Density‐based Generative Representation of Shapes

“…Ye et al. [YDT*18] create a framework, which consistently discretized the extrinsic Dirac operator and an intrinsic Dirac operator. In this paper, we improve the reconstruction based on [CPS11, YDT*18] by solving an equation with a larger solution space and introducing an area calibration (see Section 3.5).…”

“…A curvature‐to‐shape reconstruction algorithm with high accuracy is critical for generating plausible shapes. We follow the basic idea in [CPS13] and [YDT*18]. The deformation between the domain and target shapes is given by the solution of the Dirac equation.…”

Analysis of Schedule and Layout Tuning for Sparse Matrices With Compound Entries on GPUs

“…Later publications based on the quaternionic Dirac operator defined by Crane et al . result in quaternionic matrices as well [CPS13, LJC17, YDT*18]. Chern et al .…”

Controlling Meshes via Curvature: Spin Transformations for Pose-Invariant Shape Processing

“…It is more generally related in spirit to large diffeomorphic frameworks [30,21] that can flow a shape from a template and (pose-equivariant) vector field. Our work expands on the framework of discrete spin transformations as introduced by Ye et al [32]. One of the appeals of a discrete framework is to bypass discretization errors by design and to offer a consistent definition of discrete geometric concepts such as curvature.…”

“…The task is now to reconstruct ("extrude") a shape of interest back from a reference sphere, given its mean curvature h and area A mapped onto the sphere surface. There is to our knowledge very little done in that direction, even in related work [6,32]. To emphasize, we only wish to recover the original mesh up to pose Fig.…”