Vector Field Based Shape DeformationsWolfram von Funck∗MPI InformatikHolger Theisel†MPI InformatikHans-Peter Seidel‡MPI InformatikFigure 1: Volume-preserving deformation of the hand model (36619 vertices): no skeletal hand model is involved, no self-intersections occur.AbstractWe present an approach to define shape deformations by construct-ing and interactively modifying C1continuous time-dependentdivergence-free vector fields. The deformation is obtained by a pathline integration of the mesh vertices. This way, the deformation isvolume-preserving, free of (local and global) self-intersections, fea-ture preserving, smoothness preserving, and local. Different mod-eling metaphors support the approach which is able to modify thevector field on-the-fly according to the user input. The approachworks at interactive frame rates for moderate mesh sizes, and thenumerical integration preserves the volume with a high accuracy.CR Categories: I.3.5 [Computer Graphics]: Computational Ge-ometry and Object ModelingKeywords: shape-deformation, volume-preserving, vector fields1 IntroductionShape deformations is a well-researched area in computer graphicsand animation with many applications ranging from automotive de-sign to movie production. A variety of techniques have been devel-oped to transform an original shape into a new one under a certainnumber of constraints. These constraints can be for instance perfor-mance, detail preservation, feature preservation, volume preserva-tion, avoidance of (local or global) self-intersections, or local sup-port. In addition, different metaphors for an intuitive definition andhandling of the deformation exist, like the free movement of certainhandles [Singh and Fiume 1998; Bendels and Klein 2003; Pauly∗e-mail: [email protected]†e-mail: [email protected]‡e-mail: [email protected] al. 2003], a two-handed metaphor [Llamas et al. 2003], or themovement of a 9 dof object [Botsch and Kobbelt 2004].Most existing deformation approaches have in common that theyare defined as a map from the original to the new shape, i.e., itdoes not contain information about intermediate deformation steps.For many applications, the user wants to explore the deformation inan interactive manner, i.e., she wants to see a smooth change fromthe original to the desired shape moving along certain paths. Thismeans that the deformation has to be recomputed again and againat interactive frame rates. To do so, parts of the deformation canbe precomputed and reused for every intermediate deformation, seefor instance [Botsch and Kobbelt 2004; Botsch and Kobbelt 2005].In this paper we introduce an alternative approach to describe shapedeformations. We assume that the shape is given as a triangu-lar mesh. We construct a C1continuous divergence-free 3D time-dependent vector field v and obtain the new positions of every ver-tex p of the shape by applying a path line integration of v startingfrom p. This approach is motivated by two observations. First,it corresponds to the metaphor of smooth deformations by observ-ing the paths of the vertices over time. Second, due to the zero-divergence of v we get a number of desired properties of the defor-mation for free. In particular, the following properties hold:• No self-intersections (neither local nor global) can occur. Thisis due to the fact that path lines do not intersect in the 4Dspace-time domain [Theisel et al. 2005].• The deformation is volume-preserving. This is a well-knownproperty of divergence-free vector fields [Davis 1967].• The deformation preserves the smoothness of the shape to firstorder. This is due to the C1continuity of v: under a pathsurface integration, the normals of the evolving shape dependon ∇v. Hence, for a C1continuous v no discontinuities of thesurface normals appear during the integration.• The deformation preserves details and sharp features in asense that no smoothing due to an energy minimization oc-curs.Figure 2a gives an illustration of the main idea. In addition to theabove-mentioned properties of v, we construct it to be non-zeroonly in a certain area to obtain a local support of the deformation.Although divergence-free vector fields have been used to model thepath lines of voriginalshapedeformedshaperrirob01(a)(b)Figure 2: (a) Vector field based shape deformation: every vertex ofthe original shape undergoes a path line integration of v to find itsnew position. (b) Blending function b(r).flow of fluids [Foster and Fedkiw 2001], we are not aware of any ap-proach to use them for the interactive deformation of solid shapes.The rest of the paper is organized as follows: section 2 describes re-lated work to shape deformations. Section 3 describes how to con-struct the locally supported divergence-free vector field v. Section 4shows a number of modeling metaphors of our technique. Section 5gives implementation details. Section 6 gives an evaluation of ourtechnique and compares it with other approaches. Conclusions aredrawn in Section 7.2 Related workCurrent shape deformation approaches can be classified as surfacebased methods or space deformation methods. Surface based meth-ods define the deformation only on the shape’s surface. A com-mon approach is to specify a number of original and target ver-tices and compute the remaining vertex positions by a variationalapproach [Welch and Witkin 1992; Taubin 1995]. Multiresolutionmethods are well-established because of their ability to speed upcomputations and preserve features [Zorin et al. 1997; Guskov et al.1999; Kobbelt et al. 1998; Botsch and Kobbelt 2004]. More re-cently, approaches have been proposed which rely on the solutionof the Laplace/Poisson equations [Alexa 2003; Lipman et al. 2004;Sorkine et al. 2004; Yu et al. 2004; Lipman et al. 2005; Zayer et al.2005]. These approaches end up in the repeated solution of a largesparse linear system. Space deformation techniques modify objectsby deforming their embedded space. Prominent representatives ofthis are free-form deformation methods which can be classified aslattice-based [Sederberg and Parry 1986; Coquillart 1990; Mac-Cracken and Joy 1996], curve-based [Barr 1984; Singh and Fiume1998], or point-based [Hirota et al. 1992; Hsu et al. 1992]. Differentbasis functions to define the space deformation have been applied,like radial basis functions [Botsch and Kobbelt 2005] or swirls [An-gelidis et al. 2004a]. [Zhou et al. 2005] extends the Laplacian ap-proach from surface based techniques
View Full Document
Unlocking...