In this article, 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 one 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 provides both 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. shape distortion via shape difference operators [44] . One advantage of these representations is that linear operators can be naturally composed, which makes it easy to define, for example, the push-forward of a vector field with respect to a mapping, if both are represented as linear operators, or to solve for Killing vector fields, by composition between a derivation and the Laplacian operator. Moreover, by using a consistent functional representation, these techniques often alleviate the need for pointwise correspondences, which can be difficult to obtain, as shown very recently, for example, in a work on joint cross-field design [3] . While tangent vector fields are classically understood as operators (derivations) in differential geometry, extrinsic vector fields
For human shape correspondence, the authors in REF 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.
Functional Characterization of Deformation Fields
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Computer Science Mathematics