Functional Characterization of Deformation Fields

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.

Robust Shape Collection Matching and Correspondence from Shape Differences

“…We believe that our approach can serve as an important new tool in the shape analysis and correspondence toolbox. Future work and generalizations include using other inner product metrics to tailor the shape differences for specific applications [CO19], and using better regularization constraints [RPWO19]. Moreover, it might be beneficial to use multiple base shapes, and to learn the shape differences operators from data.…”

A Literature Review: Geometric Methods and Their Applications in Human-Related Analysis

“…For articulated objects, like human bodies, extrinsic properties are not capable of describing their intrinsic properties, like shapes and symmetric properties. Although suffering from topological noise, isometry-preserving properties are widely used for human-related analysis, e.g., the methods from [80,81,82,83,84,85] are utilized for human shape analysis, and the method from [86] is utilized for human shape recognition. Geometric methods are also isometry-preserving methods.…”

“…For HSA, the paper reviews the related works on human shape correspondence, human model symmetry analysis, and human shape recognition. For human shape correspondence, the authors in [83] represented a deformation field as a linear operator on real-valued functions on the shape and gave the state-of-the-art performance on human shape correspondence. An exemplary result is illustrated in Figure 25.…”

“…The figure shows exemplary point-to-point maps from one human body model to another. The overall performance of the proposed geometric method (right) is working better than the compared SHOT(left) method on the entire shape [83]. Republished with permission of ACM, from ACM Trans.…”

A Smoothness Energy without Boundary Distortion for Curved Surfaces

“…Moreover, our simple Crouzeix-Raviart discretization of the oneform Dirichlet energy containing covariant derivatives from Section 6.2 offers an interesting approach to discretize the vector Dirichlet energy in a wide variety of applications. Potential applications include vector field design [Knöppel et al 2013], parallel transport of vectors [Sharp et al 2018], and many more [Azencot et al 2015;Corman and Ovsjanikov 2019;Liu et al 2016]. Stein et al [2018] (α = 2.5⋅10 -9 ) our Hessian energy (α = 1.8⋅10 -9 )…”