Large Mesh Deformation Using the Volumetric Graph LaplacianKun Zhou1Jin Huang2∗John Snyder3Xinguo Liu1Hujun Bao2Baining Guo1Heung-Yeung Shum11Microsoft Research Asia2Zhejiang University3Microsoft ResearchAbstractWe present a novel technique for large deformations on 3D meshesusing the volumetric graph Laplacian. We first construct a graphrepresenting the volume inside the input mesh. The graph need notform a solid meshing of the input mesh’s interior; its edges sim-ply connect nearby points in the volume. This graph’s Laplacianencodes volumetric details as the difference between each pointin the graph and the average of its neighbors. Preserving thesevolumetric details during deformation imposes a volumetric con-straint that prevents unnatural changes in volume. We also includein the graph points a short distance outside the mesh to avoid lo-cal self-intersections. Volumetric detail preservation is representedby a quadric energy function. Minimizing it preserves details ina least-squares sense, distributing error uniformly over the wholedeformed mesh. It can also be combined with conventional con-straints involving surface positions, details or smoothness, and effi-ciently minimized by solving a sparse linear system.We apply this technique in a 2D curve-based deformation systemallowing novice users to create pleasing deformations with littleeffort. A novel application of this system is to apply nonrigid andexaggerated deformations of 2D cartoon characters to 3D meshes.We demonstrate our system’s potential with several examples.Keywords: differential domain methods, deformation retargeting,local transform propagation, volumetric details.1 IntroductionMesh deformation is useful in a variety of applications in computermodeling and animation. Many successful techniques have beendeveloped to help artists sculpt stylized body shapes and deforma-tions for 3D characters. In particular, multi-resolution techniquesand recently introduced differential domain methods are very effec-tive in preserving surface details, which is important for generatinghigh-quality results. However, large deformations, such as thosefound with characters performing nonrigid and highly exaggeratedmovements, remain challenging today, and existing techniques of-ten produce implausible results with unnatural volume changes.We present a novel deformation technique that achieves convincingresults for large deformations. It is based on the volumetric graphLaplacian (VGL), which represents volumetric details as the dif-ference between each point in a 3D volume and the average of itsneighboring points in a graph. VGL inherits the strengths of recentdifferential domain techniques [Yu et al. 2004; Sorkine et al. 2004].In particular, it preserves surface details and produces visually-pleasing deformation results by distributing errors globally through∗This work was done while Jin Huang was an intern at Microsoft Re-search Asia.Figure 1: Large deformation of the Stanford Armadillo. Left: original mesh;middle: deformed result using Poisson mesh editing; right: deformed resultusing our technique. Poisson mesh editing causes unnatural shrinkage es-pecially in the model’s right thigh.least-squares minimization. But by working in the volumetric do-main instead of on the mesh surface, VGL can effectively imposevolumetric constraints to avoid unnatural volume changes and localself-intersections (Figure 1). Volumetric constraints are representedby a quadric energy function which can be efficiently minimizedby solving a sparse linear system, and easily combined with otherwidely-used surface constraints (e.g., on surface positions, surfacedetails [Sorkine et al. 2004], and surface smoothness [Botsch andKobbelt 2004]).To apply the volumetric graph Laplacian to a triangular mesh,we construct a volumetric graph which includes the original meshpoints as well as points derived from a simple lattice lying insidethe mesh. These points are connected by graph edges which area superset of the edges of the original mesh. The graph need notform a meshing (volumetric tessellation into tetrahedra or other fi-nite elements) of the mesh interior. This flexibility makes it easy toconstruct. The deformation is specified by identifying a limited setof points on the original mesh, typically a curve, and where thesepoints go as a result of the deformation. A quadric energy functionis then generated whose minimum maps the points to their specifieddestination while maintaining surface detail and roughly preservingvolume.Our main contribution is to demonstrate that the problem of largedeformation can be effectively solved by using a volumetric dif-ferential operator. Previous differential approaches [Yu et al. 2004;Sorkine et al. 2004] considered only surface operators. A naive wayto extend these operators from surfaces to solids is to define themover a tetrahedral mesh of the object interior. However, solidlymeshing a complex object is notoriously difficult. To our knowl-edge, available packages remesh geometry and disturb its connec-tivity, violating a common requirement in mesh deformation. Solidmeshing also implies many constraints (e.g., that no tetrahedron beflipped and that each interior vertex remain in the visual hull of itsneighbors) that make it harder to economically distribute interiorpoints and add an “exterior shell” as we do to prevent local self-intersection. Our key insight is that the volumetric Laplacian op-erator can be applied to an easy-to-build volumetric graph withoutmeshing surface interiors.Using the method, we have developed an interactive deformationsystem based on 2D curves. Manipulating vertices in 3D spaceis tedious and requires artistic skill; our system allows novices tocreate pleasing results with a few, simple operations. A novel ap-plication of this system is to transfer the exaggerated deformationsof 2D cartoon characters to 3D models by specifying a set of cor-responding curves between the images and models. Our techniquedoes not require the skeletons and key poses of the 3D models asinput and can handle a wide range of nonrigid deformations.2 Related WorkMesh Deformation Energy minimization has long been used todesign smooth surfaces [Welch and Witkin 1994; Taubin 1995].Recently, a freeform modeling system allows users to define basisfunctions customized to a given design task [Botsch and Kobbelt2004]. The resulting linear system handles arbitrary regions andpiecewise boundary conditions with
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