Functional Characterization of Deformation Fields

TL;DR

This paper introduces a new coordinate-free operator-based representation for 3D shape deformation fields, enabling advanced analysis, processing, and applications in shape modeling and transfer.

Contribution

It proposes a novel metric-based, operator representation of deformation fields that allows for intrinsic analysis and manipulation without relying on pointwise correspondences.

Findings

Effective in shape deformation analysis and transfer

Applicable to both surface and volumetric shapes

Enables new applications like intrinsic shape symmetrization

Abstract

In this paper we present a novel representation for deformation fields of 3D shapes, by considering the induced changes in the underlying metric. In particular, our approach allows to represent a deformation field in a coordinate-free way as a linear operator acting on real-valued functions defined on the shape. Such a representation both provides a way to relate deformation fields to other classical functional operators and enables analysis and processing of deformation fields using standard linear-algebraic tools. This opens the door to a wide variety of applications such as explicitly adding extrinsic information into the computation of functional maps, intrinsic shape symmetrization, joint deformation design through precise control of metric distortion, and coordinate-free deformation transfer without requiring pointwise correspondences. Our method is applicable to both surface and volumetric shape representations and we guarantee the equivalence between the operator-based and standard deformation field representation under mild genericity conditions in the discrete setting. We demonstrate the utility of our approach by comparing it with existing techniques and show how our representation provides a powerful toolbox for a wide variety of challenging problems.