Geometry processing from an elastic perspective

Abstract

Triggered by the development of new hardware, such as laser range scanners for high resolution acquisition of complex geometric objects, new graphics processors for realtime rendering and animation of extremely detailed geometric structures, and novel rapid prototyping equip‐ment, such as 3D printers, the processing of highly resolved complex geometries has established itself as an important area of both fundamental research and impressive applications. Concepts from image processing have been picked up and carried over to curved surfaces, physically based modeling plays a central role, and aspects of computer aided geometry design have been incorporated. This paper aims at highlighting some of these developments, with a particular focus on methods related to the mechanics of thin elastic surfaces. We provide an overview of different geometric representations ranging from polyhedral surfaces over level sets to subdivision surfaces. Furthermore, with an eye on differential‐geometric concepts underlying continuum mechanics, we discuss fundamental computational tasks, such as surface flows and fairing, surface deformation and matching, physical simulations, as well as spectral and modal methods in geometry processing. Finally, beyond focusing on single shapes, we describe how spaces of shapes can be investigated using concepts from Riemannian geometry. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On $H^2$-gradient Flows for the Willmore Energy

“…are orthogonal with respect to the L 2 -inner product induced by g. For u ∈ W k+2,r (Σ; R m ) with −∆ g u = 0, we have by elliptic regularity (see Theorem A.1) that u ∈ W 2,2 (Σ; R m ) so that the same calculation as in (21) leads to du = 0. Hence we have u ∈ X 0 and ker(−∆ g ) = X 0 R m .…”

“…Indeed, some infinite dimensional problems related to curvature energies of higher-dimensional immersed submanifolds (such as surfaces in R 3 ), can hardly be put into an economic, strongly Riemannian context. 1 This is unfortunate as such energies occur frequently in practical applications, e.g., in mechanics as bending energy in the Kirchhoff-Love model for thin plates (see [14], [16]); in biology as Canham-Helfrich energy of cell membranes (see [5], [12]); and in computer graphics as regularizers for various geometry processing tasks (see [21] and references therein).…”

Elastic Correspondence between Triangle Meshes

“…Elastic shape modeling. Physically‐based elastic energy models have been widely used for computer graphics and geometry processing applications [RW14]. The classical model for elastically deformable surfaces is the shell model, originally introduced in a graphics context by Terzopoulos et al [TPBF87], for thin, flexible materials.…”

Functional Characterization of Deformation Fields

“…Our approach of considering the deformation via its induced metric distortion is also related to the work of [Eigensatz and Pauly 2009] and [Sela et al 2015] who manipulate shapes by explicitly editing their curvature properties. Moreover, our use of the strain tensor in characterizing metric distortion is closely related to the applications in various physically based deformation scenarios including [Thomaszewski et al 2009;Müller et al 2014] among many others (see also the surveys on physically based elastic deformable models [Nealen et al 2006;Rumpf and Wardetzky 2014]). Our approach is also related to the works that aim to design as-isometricas-possible shape deformations [Zhang et al 2015;Solomon et al 2011;Martinez Esturo et al 2013].…”