AbstractIntroductionRelated workApproach overviewInitialization of coarse mapCoarse-to-fine map optimizationVertex optimizationDistortion metricApplications and resultsInter-surface mappingSimplicial parametrizationOctahedral parametrizationToroidal parametrizationDiscussionFuture workAcknowledgementsReferencesInter-Surface Mapping John Schreiner Arul Asirvatham Emil Praun Hugues Hoppe University of Utah University of Utah University of Utah Microsoft Research (a) Surface M1 with edges from M2 (notice density of edges from left wing) (b) Surface M2 with edges from M1 (see spike flattened on rear left knee)(c) M1 normals mapped onto M2 (lit using 2 antipodal light sources) (d) 50% morph Figure 1: Inter-surface map for two objects of genus 2, initialized with 8 user-specified feature points. (Symmetric stretch efficiency 0.311).Abstract We consider the problem of creating a map between two arbitrary triangle meshes. Whereas previous approaches compose pa-rametrizations over a simpler intermediate domain, we directly create and optimize a continuous map between the meshes. Map distortion is measured with a new symmetric metric, and is minimized during interleaved coarse-to-fine refinement of both meshes. By explicitly favoring low inter-surface distortion, we obtain maps that naturally align corresponding shape elements. Typically, the user need only specify a handful of feature corre-spondences for initial registration, and even these constraints can be removed during optimization. Our method robustly satisfies hard constraints if desired. Inter-surface mapping is shown using geometric and attribute morphs. Our general framework can also be applied to parametrize surfaces onto simplicial domains, such as coarse meshes (for semi-regular remeshing), and octahedron and toroidal domains (for geometry image remeshing). In these settings, we obtain better parametrizations than with previous specialized techniques, thanks to our fine-grain optimization. Keywords: surface parametrization, shape morphing, remeshing. 1. Introduction Surface parametrization refers to mapping a triangle mesh onto a simpler domain such as the plane, the sphere, or a coarse sim-plicial domain. The parametrization is represented by a map DMφ→ where M is the mesh and D is the simpler domain. In computer graphics, parametrization is central to texture mapping, whereby images placed in the domain are sampled on rendered surfaces to provide texture detail, place decals, encode shadows, record radiance transfer coefficients, etc. Surface parametriza-tions also appear in numerous applications, including digital geometry processing, morphing, surface editing, object recogni-tion, and geometry remeshing. We address the more general problem of directly constructing a continuous bijective map 12MMφ→ between two triangle meshes pologyM1 and M2 of the same to . (Continuity precludes maps between surfaces with different genus or number of boundaries.) Unlike previous approaches which compose parametrizations of M1 and M2 over some intermediate domain (as reviewed in Sec-tion 2), we directly optimize the quality of the overall map 12MMφ→. Our method works for arbitrary genus and does not the user to provide a simplicial complex (e.g. [Praun et al 2001]). The user may optionally specify corresponding feature points on Mrequire 1 and M2, and our construction guarantees that the map satisfies these constraints. Some parametrization schemes may require a large set of manu-ol to the Digital Geome- domain, to • terchange of • coarse-to-fine optimization algorithm to provide • A ile our motivating application is the n (for semi-regular remeshing): given • age remeshing): • ter results than the previous techniques specialized to these scenarios. ally specified features to guide the parametrization process to a good (or even valid) solution. As we shall show, our mapping method is robust even with few feature constraints. Moreover, directly minimizing the distortion of the inter-surface map tends to naturally align corresponding shape elements. Of course, a few user-specified constraints are helpful for overall registration and for linking semantically related regions. Our approach adds a new fundamental totry Processing toolbox. Its main contributions are: • Inter-surface mapping without any intermediatedirectly measure the distortion of the overall map. Symmetric distortion metric, i.e. invariant to the inM1 and M2. Symmetric robustness and convergence to a good solution. Initialization of map to robustly satisfy any user-specifiedfeature correspondences. dditional scenarios. Whcreation of maps between surfaces of comparable complexity, our framework can also be used in cases where M1 is a simpler mesh, possibly inferred from M2: • Simplicial parametrizatioa surface M2 and desired domain vertices on M2, we automati-cally create domain M1 and a parametrization. Octahedral parametrization (for geometry-im M1 is a regular octahedron, and feature points are unnecessary. Toroidal parametrization (for remeshing of genus-1 shapes). Our more general optimization framework actually obtains betBCADEBCAED2. Related work Planar parametrization. The traditional surface parametrization problem considers the ca2se where the domain D is a planar region PR⊂ (see survey in [Floater and Hormann 2003]). The map DMφ→ is represented by the parametric locations of vertices of M within the plane. Optimization can freely move the vertices the domain as long as bijectivity is maintained. Kraevoy et al [2003] present the Matchmaker scheme for satisfy-ing corresponding feature point constraints in D and M. We withinis that repre- e r o Mto surface M2, obtained by the construction of M1 from M2. extend their scheme to form a corresponding graph of paths on two surfaces M1, M2 of arbitrary genus g, possibly with bounda-ries. To guarantee the successful termination of the path insertion process, we impose ordering constraints on the neighbors of a feature vertex, and we trace a spanning tree and 2g non-separating cycles before completing the full graph. Consis-tent neighbor ordering is necessary to avoid partial graphs that are impossible to complete, as shown on the right (if D and E link to the same base vertex B or C, this will result in flipped triangles; if they link to different ones, edges will cross.) An important limitation of planar parametrization senting an entire surface requires
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