Model Transduction for Triangle Meshes

Model Transduction for Triangle Meshes

Wu HY, Pan CH, Zha HB et al. Model transduction for triangle meshes. JOURNAL OF COMPUTER SCIENCE ANDTECHNOLOGY 25(3): 584–595 May 2010Model Transduction for Triangle MeshesHuai-Yu Wu 1,2 (), Chun-Hong Pan 2 (), Hong-Bin Zha 1 (), and Song-De Ma 2 ()1 Key Laboratory of Machine Perception (MOE), Peking University, Beijing 100871, China2 National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, ChinaE-mail:;;; December 1, 2009; revised March 5, 2010.Abstract This paper proposes a novel method, called model transduction, to directly transfer pose between differentmeshes, without the need of building the skeleton configurations for meshes. Different from previous retargetting methods,such as deformation transfer, model transduction does not require a reference source mesh to obtain the source deformation,thus effectively avoids unsatisfying results when the source and target have different reference poses. Moreover, we show othertwo applications of the model transduction method: pose correction after various mesh editing operations, and skeleton-freedeformation animation based on 3D Mocap (Motion capture) data. Model transduction is based on two ingredients: modeldeformation and model correspondence. Specifically, based on the mean-value manifold operator, our mesh deformationmethod produces visually pleasing deformation results under large angle rotations or big-scale translations of handles. Thenwe propose a novel scheme for shape-preserving correspondence between manifold meshes. Our method fits nicely in aunified framework, where the similar type of operator is applied in all phases. The resulting quadratic formulation canbe efficiently minimized by fast solving the sparse linear system. Experimental results show that model transduction cansuccessfully transfer both complex skeletal structures and subtle skin deformations.Keywordsretargetting, mesh deformation, mean-value manifold operator, cross-parameterization, model transduction1 IntroductionWith the significant increase in the number of 3Ddata produced by artists, scanning devices or visionbasedcapture and reconstruction, reusing existing 3Ddata has recently been popular in computer modelingand animation. Currently, 3D data retargetting techniquesmainly consist of two categories: motion retargettingand deformation retargetting.Gleicher et al. [1] present a technique for adapting themotion of one articulated figure to another figure withidentical structure but different segment lengths. Recently,[2-4] extend this work by using a hierarchical displacementtechnique, a physically-based motion transformationmethod, and a data-driven model approach,respectively. Except retargetting from one articulatedfigure to another articulated figure, the motion series ofan articulated figure can also be retargetted to a triangularmesh by using so-called skinning techniques, suchas skeleton subspace deformation (SSD) [5] , pose spacedeformation (PSD) [6] , EigenSkin [7] .Inspired by the motion retargetting techniques thatfocus on skeleton-based articulated body motions, [8-9] present the deformation retargetting techniques fortriangular meshes. Given a reference source mesh Sand a deformed source mesh S ′ , the source deformationis firstly computed by vertex displacements [9] orlocal affine transformations [8] . Then, the source deformationfrom S to S ′ is applied onto a different targetmesh T to generate the deformed target mesh T ′ .In this paper, different from motion retargetting anddeformation retargetting techniques, a novel skeletonfreepose retargetting technique, model transduction,is proposed. This method not only can directly retargetpose between different models without buildingthe skeleton configurations, but also can be usedfor pose correction of 3D models after various editingoperations, and skeleton-free deformation animationbased on 3D Mocap (Motion capture) data, etc.This method is a natural extension of the deformationtransfer work [8] . Another contribution of this paperis to present the mean-value manifold operator, whichis used for shape-preserving mesh editing and shapepreservingmesh correspondence.Regular PaperThis work is supported by the National Natural Science Foundation of China under Grant Nos. 60903060 and 60675012, theNational High-Tech Research and Development 863 Program of China under Grant No. 2009AA012104, and the China PostdoctoralScience Foundation under Grant No. 20080440258.©2010 Springer Science + Business Media, LLC & Science Press, China

Huai-Yu Wu et al.: Model Transduction for Triangle Meshes 585Below, we give a brief overview of works related toours.• Deformation Transfer and Model TransductionDeformation transfer [8-11] applies source mesh’s deformationonto a different target mesh. In order togenerate the deformed target model T ′ , one must provideall of the three models: the reference source modelS, the deformed source model S ′ and the reference targetmodel T , as showed in Fig.1(a). Note that in deformationtransfer, S is indispensable for obtaining thesource model’s deformation to be transferred to T . Onelimitation of deformation transfer is that the source andtarget reference models must have the same kinematicreference pose, since the reference target model T reproducesthe change in shape induced by the sourcedeformation between S and S ′ . Only when the tworeference poses are the same can this kind of transferringoperation be valid. Another drawback is thatif the source deformation itself has artifacts, evidentlythe deformed target model T ′ generated by the sourcedeformation can hardly be satisfying.Fig.1. (a) Deformation transfer [8] cannot produce satisfying resultswhen the source and target have different reference poses.(b) Using model transduction, a lion model successfully imitatesthe pose of a cat model even if the reference source mesh is absent.Now, let us consider a more challenging problem:with the absence of the reference source mesh S (Is Sessentially redundant?), how to get the deformed targetmesh T ′ which acts like the deformed source mesh S ′while looking like the reference target mesh T ? Weshow that a feasible solution to this problem is themodel transduction method introduced in this paper(see Fig.1(b), Fig.2, Fig.12). This method is a naturalextension of the deformation transfer work [8] . However,[8] does not explicitly consider surface details’ preservation,and requires a reference pose for the source mesh’sdeformation. Our method address both issues.Similar to the human perception, the transferringprocess is like first inducting the knowledge from theobservation, and then deducing it to a new example.Can we make a shortcut, and go from an existing exampledirectly to a new example? This shortcut is called a“transduction” [12] . Similarly, by using the model transductionmethod, we can directly produce the deformedtarget model T ′ while not needing the reference sourcemodel S.• Cross-ParameterizationEstablishing cross-parameterization (or consistentcorrespondence, inter-surface mapping) [8,13-17] betweendifferent shapes is a fundamental task in a vast numberof applications. Most existing cross-parameterizationmethods belong to indirect schemes, i.e., an intermediateparameterization domain is required. The differencesamong algorithms lie in the choice of the intermediatedomains, such as the plane [14] , the sphere [13,18] ,the cylinder, the triangle patch [15] , the quadranglepatches. However, for indirect schemes, the difficulty tocompatibly construct well-shaped patch layouts makesthese methods hard to keep balance between efficiencyand robustness. Tricky topological operations are requiredto deal with genus, intersections, blocking, cyclicorders, branch mismatch etc. Furthermore, besidesthe drawback of discontinuity when transiting interpatchboundaries, the error of sub-mappings also canbe amplified through the mapping composition, so thefinal inter-surface mapping may have large errors somewhere.Except building intermediate domains, crossparameterizationcan also be constructed directly, i.e.,using the target mesh as the common domain, thusavoids explicit cross-parameterization. In [17], bysmoothing local affine transformations, Allen et al.use the template fitting technique to directly constructcross-parameterization for a set of human models.Sumner et al. [8] propose a similar algorithm tobuild correspondence map between meshes. However,direct schemes so far do not explicitly take the shapepreservationproperty into account, thus will introducelarge approximation errors when the input models havesignificantly different geometries.In this paper, we apply a novel shape-preserving operator,the mean-value manifold operator, to directlyconstruct the shape-preserving cross-parameterizationwithout needing to build the intermediate domains andpartitions for input meshes, which effectively avoids erroramplification, discontinuity along partition boundaries,and large approximation distortion.• Differential Mesh DeformationShape deformation [19-30] has various applications incomputer modeling, simulation and animation. Recently,differential information as local intrinsic featuredescriptors (e.g., Laplacian coordinates or gradientfield) has been used for mesh processing, especially

586 J. Comput. Sci. & Technol., May 2010, Vol.25, No.3for detail-preserving mesh editing. However, Laplaciandifferential coordinates are the average differencevectors of adjacent vertices, which are not rotationinvariantand must somehow be transformed by heuristicmethods (or user’s adjustment) [10,20-21,23] to matchthe desired new orientations, otherwise the detailsin the deformed mesh are distorted. Though somerotation-invariant representations [22,31-35] have beenproposed, they are usually involved in complicated andtime-costuming nonlinear optimizations.In this paper we adopt the novel mean-value manifoldoperator to construct the minimal-magnitude(close to zero) vector field. With the nice properties(fair encoding weights and small ratios) of this meanvalueshape-preserving representation, our editing systemis efficient and stable even when the handles areunder large angle transformations or moved rapidly bythe user.The rest of the paper is structured as follows. Wefirst describe the overall flowchart of the model transductionmethod in Section 2. Then, in Section 3 andSection 4, we concretely describe the two basic ingredientsof model transduction: model correspondence andmodel deformation. Section 5 demonstrates more applicationsof model transduction. Section 6 presents theexperimental results. We conclude the paper in Section7.2 Model TransductionModel transduction is designed to directly transferpose between different meshes without the need ofbuilding the skeleton configurations for meshes. Differentfrom deformation transfer [8] , our method does notrequire an extra reference source mesh to obtain thesource deformation, as shown in Fig.1.Fig.2. The flowchart of the model transduction method. (a)Matching mesh. (b) Deformed source mesh. (c) Reference targetmesh. (d) Pose “mesh”. (e) Deformed target mesh.Fig.2 describes the overall flowchart of model transduction,which is composed of three sequential steps.First, we establish the consistent correspondence betweenthe deformed source mesh S ′ and the referencetarget mesh T by using the cross-parameterizationmethod to be proposed in Section 3. Thus, we get acompatible matching mesh ˜T having the identical connectivitywith T and the same geometry with S ′ . Then,T ’s triangles are rotated and translated to appropriatepositions on ˜T to generate the pose “mesh” T p ,which extracts rigid transformation components fromthe mesh T and retains the overall pose informationof the mesh S ′ . Finally, T p ’s triangles are pieced togetherto generate the final conforming mesh T ′ , bysolving an optimizing process subject to a series of energyconstraints (including the detail-preserving deformationconstraint to be proposed in Section 4).In the following subsections, we describe each stepof model transduction in detail.2.1 Step One: Generating the Matching MeshIn the first step, we need to establish a correspondencebetween the source mesh S ′ and the targetmesh T . We adopt the shape preserving crossparameterizationmethod (to be proposed in Section3) to generate the compatible matching mesh ˜T . Notethat our correspondence method is a kind of iteratedclosest point algorithm. To accelerate the closest-pointmatching, we adopt the ANN library [36] for both exactand approximate nearest neighbor searching, whichperforms quite efficiently in linear time.2.2 Step Two: Obtaining the Pose “Mesh”In the second step, our aim is to obtain a nonconforming“mesh” T p to be used for pose representation(see Fig.2(d)). This nonconforming “mesh” is generatedby firstly detaching all the triangles of the referencetarget mesh T without distorting them, and thenreattaching each triangle onto its corresponding triangleon the matching mesh ˜T . In order to achieve therigid transformation (i.e., without distorting), we firstcompute an affine transformation for each triangle, thenextract the translation and rotation components fromthe affine transformation.This step can be described as follows. First, let D kbe a triangle in the reference target mesh T , ˜Dk beD k ’s corresponding triangle in the matching mesh ˜Tgenerated in Step One. Q k is a 3×3 affine matrix [17,37]which describes the local affine transformation from D kto ˜D k .Then, we adopt the singular value decomposition(SVD) method to extract the rotation part R k and theshearing-scaling part S k from the affine matrix Q k :Q k = A k D k B k = (A k B k )(B T k D k B k ) = R k S k (1)

Huai-Yu Wu et al.: Model Transduction for Triangle Meshes 587where all of A k , B k , R k are rotation matrices; D k =diag(λ k1 , λ k2 , λ k3 ) is the diagonal matrix resulted fromSVD, λ k1 , λ k2 and λ k3 are the eigenvalues of Q k .For each pair of the corresponding triangles D k and˜D k , we rotate the triangle D k with the matrix R k , andthen translate it onto the triangle ˜D k to produce thetriangle D p k, called pose triangle:v p k i= R k (v ki − c k ) + ˜c k i = 1, 2, 3 (2)where c k (˜c k ) is the centroid of the triangle D k ( ˜D k );v p k iis v ki ’s corresponding vertex on D p k .Thus we get a set of disconnected pose triangles D p k s,which constitute the pose “mesh” T p .2.3 Step Three: Translating Pose Triangleswith the Detail-Preserving Constraintand the Smoothness ConstraintIn Step Two, we have generated T p by rotatingand translating the triangles of T to appropriate positions.The integrity of the triangles of T p representsS ′ ’s pose to be retargetted onto the final resultT ′ . However, these triangles are disconnected and scattered,and therefore do not form a real “mesh”. So,in Step Three, in order to piece together these trianglesto form the final mesh, we translate the adjacenttriangles close to one another, and meanwhile imposeboth the detail-preserving constraint and the smoothnessconstraint on adjacent triangles to form the finalmesh T ′ .Conceptually, what we want to solve is the followingerror minimization problem:E(V ′ ) =|T |∑k=1 i=1|T |∑w sk=1|T |∑w tk=13∑‖Q k δ ki − δ ′ k i‖ 2 +( ∑j∈adj (k)( ∑(i,j)∈{(1,2),(2,3)})‖Q k − Q j ‖ 2 F +‖v ′ k i− v p k i−v ′ k j+ v p k j‖ 2) (3)where k is the index of a triangle of the mesh, i is theindex of a vertex of this triangle; δ ki (δ ′ k i) is the detailencodeddifferential coordinates (see Section 4) of thei-th vertex of mesh T (T ′ )’s k-th triangle D k (D ′ k); thematrix norm ‖ · ‖ F is the Frobenius norm; adj (k) is theset of facet neighbors of the triangle k; w s = 0.001 andw t = 5 are the item weights.The first term indicates the shape details after transformationsare preserved. The second term indicatesthat the change in transformations for adjacent trianglesshould be smooth. The last term enforces translationalmotions to the nonconforming triangles. Specifically,we want to minimize the displacement differencebetween the three vertices of each triangle in T ′ andthe three vertices of the corresponding triangle in T p .That is to say, under ideal conditions, we should have:v ′ k 1− v p k 1= v ′ k 2− v p k 2= v ′ k 3− v p k 3(4)where {vk ′ 1, vk ′ 2, vk ′ 3} is the triangle D ′ k on the mesh T ′ ,{v p k 1, v p k 2, v p k 3} is the triangle D p kon the pose “mesh”T p .As a whole, the minimization problem (3) amountsto the solution of a sparse linear system and it returnsoptimal vertex positions. The final results of modeltransduction are shown in Figs. 1, 2, 12.3 Shape Preserving Cross-ParameterizationNow we describe the first ingredient of themodel transduction method, shape preserving crossparameterization,which builds the consistent correspondence(inter-surface mapping) between the sourcemesh S and the target mesh T . Note that the symbolS in this section is equivalent to the symbol S ′ inFig.2. The resulting compatible mesh ˜T will have theidentical topology (i.e., identical number of vertices andtriangles) with the target mesh T and have the similargeometry with the source mesh S.Our cross-parameterization technique is based onthe energy minimization framework, which containsthree terms: mean-value energy term E m , global constraintterm E g , and data fitting term E d . In the following,we formulate the three terms, respectively.The first term is the mean-value energy item E m .Fig.3. The example of mapping a lion model (a) to a cat. Compared with the Laplacian representation (b), the mean-value manifoldoperator can obtain a shape-preserving correspondence (c).

588 J. Comput. Sci. & Technol., May 2010, Vol.25, No.3Fig.4. With a high-quality cross-parameterization between meshes, a women’s face surface is being gradually transformed into a man’sface surface just by the linear interpolation.Initially, we consider the Laplacian energy. As will bedescribed in Section 4, the Laplacian coordinate of avertex v i is the average difference vector of its adjacentvertices to this vertex. If setting the vector tozero at each vertex, we build a smoothness energy, i.e.,smoothly distributing each vertex as close as possibleto the barycenter of its immediate neighbors. In matrixform, the formulations for all the vertices can berewritten as:E l = ‖LV ‖ 2 (5)where V = [v 1 , v 2 , . . . , v N(VT )] T , L is the Laplaciancoefficient matrix of the target mesh T , i.e., L mn =−δ {m=n} + 1/d i · δ {(m,n)∈ET }, δ is the Dirac function.However, as shown in Fig.3(b), this representationcannot reflect geometric properties of the target mesh.In the following, we try to incorporate informationabout the original shape, for instance encoding the informationabout the size, angle and orientation of localsurface shape. Here, we adopt the mean-value manifoldoperator to construct the matrix:E m = ‖MV ‖ 2 (6)where M is the mean-value manifold (see Section 4)matrix of the target mesh T .As shown in Fig.3(c), it can be seen that comparedwith Laplacian matrix L, the mean-value shape preservationrepresentation successfully captures the shapeinformation of the target mesh, such as the thread-likestripes on the legs and belly of the lion model.The second term E g focuses on the global constraintvertices. A few pairs of markers are specified by the useron both input meshes, which serve as the global constraints,i.e., a priori, to initialize the correspondence.The global constraint term E g is defined as:E g = ∑ i∈G|v i − g i | 2 (7)where G is the index set of global constraint vertices,g i is the position of the correspondent marker on thesource mesh.Note that except user-specified markers, after havingobtained an initially aligned compatible mesh usingthose initial markers, the important feature verticesFig.5. The initial compatible mesh (a) may not well align the targetmesh (c) on the boundary region, if only a few marker pairsare provided by the user. We detect important feature vertices onthe object mesh and then project them onto (a) to generate moreglobal constraints. Thus, the profiles of the compatible mesh willnot collapse, as shown in (b).(e.g., high curvature vertices) of the target mesh canfirst be automatically detected by the sorting algorithmand then projected to the initially aligned compatiblemesh to act as extra global constraints for the finerfitting. Take Fig.5 for instance, which is the correspondenceresult of the source and target models in Fig.4.The user specified only 29 marker pairs on both models,and the boundary of the models has few markers.Therefore, the initial compatible mesh (Fig.5(a)) maynot well align the object mesh (Fig.5(c)) on the boundaryregion.To overcome this, we detect salient feature verticeson the object mesh based on curvature. Thus, importantfeatures of the object mesh are automatically obtained,e.g., nose tip, eye corner, and the boundary ofthe face model. Then, we map the salient vertices ofthe object mesh onto the initially aligned compatiblemesh to ensure sufficient matches. First, for each salientvertex, we choose the closest vertex on the compatiblemesh as its counterpart, thus extra global constraintsare generated (701 salient points are detected for modelsin Fig.3, 946 salient points are detected for models inFig.4). Similar to foregoing global constraint item, wealso formulate the salient constraint as the quadraticenergy function. Once the compatible mesh approximatesthe object mesh at the salient vertices (e.g., theboundary vertices), the profiles of the compatible meshwill not collapse, as shown in Fig.5(b).

Huai-Yu Wu et al.: Model Transduction for Triangle Meshes 589The third term is the data fitting term E d , whichfits T to the source mesh S as close as possible. E d isformulated as:E d = ∑ i∈D|v i − d i | 2 (8)where D is the index set of the vertices in T except theglobal constraint vertices, d i is the closest valid pointon the source mesh to vertex v i .Our complete objective function E is the weightedsum of the three error functions:As mentioned above, the Laplacian coordinate ζ i ofa vertex v i is the average difference vector of its adjacentvertices. Though translation-invariant, the vector representationsare not rotation-invariant and they mustsomehow be transformed locally to match the desirednew orientations. Furthermore, the uniform Laplacianoperator only reflects the topology information of itsneighboring vertices, while not capturing the local parameterizationinformation (such as local size, angle,and orientation) of the mesh.min E(V ) = E m + w g E g + w d E d (9)where w g , w d are weights. We solve the objective functionin two phases. First, we ignore the data fittingterm E d by using weights w g = 0.3, w d = 0, and obtainan initial mapping result. Then, we increase w deach time and update ˜T ’s vertices. In our experiments,increasing w d step by step from 0.001 to 0.01 generatesgood results. Each time the objective function is minimized,˜T is updated from its original position and moreclosely approximates S. Note that our correspondencesystem is actually designed to deform the target meshT into the source mesh S to produce the compatiblemesh ˜T , thus implicitly guaranteeing that ˜T has theidentical connectivity with T .4 Mesh Deformation with the Mean-ValueManifold OperatorThen we describe the other ingredient of modeltransduction, detail-preserving model deformation, indetail. Our transduction method is based on the differentialframework for mesh deformation. We first reviewthe common differential representation: Laplaciancoordinate. The Laplacian linear operator [20] providesa differential representation of the mesh, and allows efficientconverting between absolute and intrinsic representationsby solving a sparse linear system. As shownin Fig.6(a), the Laplacian differential coordinates ζ i ofvertex v i are represented by the difference between v iand the average of its neighbors:v i − 1 d i∑j∈N(i)v j = ζ i = (ζ (x)i, ζ (y)i , ζ (z)i ) (10)where N(i) = {j|(i, j) ∈ E} are the edge neighbors,d i = |N(i)| is the valence of a vertex, i.e., the numberof edges that emanate from this vertex. Note that insteadof uniform weights, geometric discretizations ofthe Laplacian will have better approximation qualities,such as the cotangent weights proposed by Pinkall andPolthier [38] .Fig.6. The illustration of Laplacian coordinates, mean-value coordinates,and mean-value manifold operator.The mean-value coordinate [14] , a generalization ofthe barycentric coordinate, is derived from an applicationof the mean value theorem for harmonic functions.Let v i and its neighboring vertices {v j |j ∈ N(i)} bepoints in the plane. The vertices {v j |j ∈ N(i)} form astar-shaped polygon with v i in its kernel. The meanvalueweights of v i are defined as:∑j∈N(i)w ij (v j − v i ) = 0 (11)w ij = tan(α ij/2) + tan(β ij /2)‖v j − v i ‖(12)where w ij is the mean-value coefficient, α ij and β ij arethe angles shown in Fig.6(a). The weights can be guaranteedto be positive, and have the nice properties of dependingcontinuously and smoothly on the vertices [39] .The right-hand side in (11) is just the constant zerovector, i.e., a “weakened” vector. In other words, whileflattening v i into the 1-ring plane, the mean-value coordinatevector vanish, i.e., become zero. However,(11) is strictly satisfied in the star-shaped planar case.For the common 2-manifold case (e.g., the 3D trianglemesh), the right-hand side cannot guarantee to be thezero, that is, it may become a non-zero vector too. Todeal with this, we present a novel mean-value shapepreservingrepresentation, called the mean-value manifoldoperator, to obtain a “weakened” (close to zero)vector field which is more suitable for the manifoldmesh operation. The mean-value manifold operator insome sense can be seen as the LLE [40] (Locally Linear

590 J. Comput. Sci. & Technol., May 2010, Vol.25, No.3Embedding) of the original differential representation.As shown in (13), similar to the mean-value weightsin 2D star-shaped case, it is desirable for the meanvaluemanifold representation {M ij |j ∈ N(i)} in 3Dto satisfy the following requirements: (a) they arepositive, because the negative weights may lead to“foldover” in the mapping [14] and therefore undesiredconvergence performance [41] , especially for detailed andhighly irregular meshes; (b) the distribution of relativemagnitudes is fair, e.g., in the bad case ofmin{M ij } ≪ max{M ij }, min{M ij } will become negligible;(c) σ i / ∑ j∈N(i) |M ij| is small enough to makethe vector field as “weak” as possible (at best zero), asone of our main goals is to reduce the magnitude of thedifferential coordinate vector. All these conditions willeffectively avoid possible degeneration or disappearanceof the triangles during varied complex mesh operations,especially for irregular meshes.∑j∈N(i)M ij (v j − v i ) =σ i (13)where M is the mean-value manifold operator.As illustrated in Fig.6(b), we describe this formulationin detail. Specifically, starting from a vertex v i ,we look for a new point v ′ i related to v i in the normaldirection n i :v ′ i − v i = λ i n i (14)where λ i is a factor used to decide v i v ′ i ’s length.Since the mean curvature flow direction ξ i lies exactlyin the linear space spanned by the normals of theincident triangles, we set the normal direction n i =ξ i /|ξ i |:ξ i =∑j∈N(i)w ξ ij (v j − v i ) (15)w ξ ij = cot θ ij + cot γ ij (16)where w ξ ij is the mean curvature flow coefficient[38,42] ;θ ij and γ ij are the angles shown in Fig.6(b).Finally, we encode the new point v i ′ with mean-valueweights:∑w ij (v j − v i) ′ = σ i (17)j∈N(i)where w ij is the mean-value coefficient in (12).Thus, the vertex v i ’s mean-value manifold representationis rewritten as:∑ ( ∑(w ij − λ i w ik)w ξ ij )(v j − v i )j∈N(i)= ∑j∈N(i)k∈N(i)M ij (v j − v i ) =σ i (18)where M ij is the mean-value manifold operator; thevariable λ i has been weighted by the normalization |ξ i |of the normal component. Note that in 2D plane case,the mean-value manifold operator M ij is actually themean-value coordinate w ij .Fig.7. The mesh editing results for various models, such as thedinosaur model, the feline model, the armadillo model, and thelion model. (a) illustrates the metaphor. The red region indicatesfixed global constraints. The blue region indicates the manipulationhandle.Fig.8. The input model (a) has irregular sampling quality (leftmostcolumn). The cot Laplacian iterative editing framework willproduce a poor deformation result, as shown in (c) [41] . In contrast,our mean-value manifold operator can achieve a satisfyingresult.From (18), it can be seen that we need to find asuitable single-variate parameter λ i that satisfies theabove constraint conditions. In fact, it can be easilyachieved just by dynamic search along the normal direction.Note that λ i can be a positive, zero, or negativevalue.Due to just the single-variate search, this optimizationprocess is very fast. And in our tests, comparedwith the Laplacian coordinates, the magnitude of themean-value manifold vector σ i is usually “weakened”(reduced) to about 1/10.

Huai-Yu Wu et al.: Model Transduction for Triangle Meshes 591Now, we describe the mesh editing algorithm basedon the mean-value manifold representation. Mathematically,differential reconstruction of the resulting meshcan be formulated as the following energy minimization:E(V ) =‖ MV − σ(V ) ‖ 2 + ∑i∈C mw 2 ‖ v i − c i ‖ 2 (19)where M is the differential operator (M ij ) matrix (n ×n, n is the number of vertices on the mesh) constructedfrom the original mesh before editing; σ(V ) are the differentialcoordinate vector; C m is the global constraint.Solving the quadratic minimization problem in (19)results in a sparse linear system:[AV =MwI m×m |Θ] [ σd (V )]V = = b (20)wC dwhere d ∈ {x, y, z}. Note that this system is fullrankand thus has a unique solution in the least-squaressense, which can be solved with V = (A T A) −1 A T b.Then, similar to [41], we further perform a simpleand fast iteration process for better results by automaticallyadjusting the weakened vector field (our editingmethod is therefore an iteratively linear system). Theiteration process is to first pre-compute the factorizationof the system matrix A for once and iteratively performefficient updating of the differential coordinatesduring editing. Note that during the iterative process,M is fixed, therefore A is also fixed. That is, we onlyneed to solve λ i for the original mesh just for once andmake it fixed.Each iteration includes two steps: updating the vertexpositions and updating the differential coordinates.In the first step, we use the current differential coordinatesto compute the vertex positions. That is,we enforce the handle constraints and update the vertexpositions so that the rotation-invariant information(e.g., the local parameterization) encoded in M is similarto the original mesh. In the second step, we updatethe differential coordinates to match the current deformedmesh. In order to preserve the original featuresizes, we keep the magnitudes of the differential coordinatesunchanged by performing normalization. To conclude,the two-step iteration minimizes the change ofrotation-invariant information (e.g., parameterizationdistortion) while keeping the rotation-variant features(that have been weakened by the mean-value manifoldoperator) similar to those of the original mesh.The iterations stop when the maximum ratio of thechanges in vertex positions between two successive iterationfalls below a certain threshold. Usually very fewiterations (t < 5 in our experiments, in comparisonwith t ≈ 20 in [41]) suffice to converge and achievevisually satisfying deformed results. From our analysis,the level of nonlinearity of a vertex depends on theratio of the magnitude of its differential coordinate tothe support size of its one-ring neighbors. The iterativeprocess will be faster and more stable (see Fig.8) ifthe vertices have small ratios and optimized (e.g., positive)weights. Though existing weighting schemes (e.g.,cotan Laplacian) address the detail preserving so as totake irregularity in sampling into account, the level ofnonlinearity still cannot be well controlled. As a result,they may be not robust in some complex conditions.On the other hand, our mean-value manifold editingmethod is designed to put the details information oflocal surface into the rotation-invariant component asmuch as possible, while the remaining rotation-variantfeatures are weakened (to zero or near-zero) as muchas possible by the mean-value manifold operator. Andfor those near-zero features, we further keep the magnitudesof the differential coordinates unchanged in orderto preserve the original feature sizes.5 More Applications of Model TransductionExcept retargetting pose between different meshes,the model transduction method also has more applicationsin shape modeling and processing, such as, posecorrection after various mesh editing operations, andskeleton-free deformation animation based on 3D Mocapdata.5.1 Additional Application I: Pose Correctionof 3D Model after Various EditingOperationsWe first describe how to implement pose correctionfor 3D model after various editing operations. Thismotivation originates from the fact that although variousmesh editing methods have been presented, to ourknowledge, there are few techniques proposed to correctexisting dissatisfactory results produced by various deformationmethods. In this subsection, we introduce ageneral refinement approach that requires no knowledgeof the actual method used to edit the original shape.Our pose correction method can be taken as an effectivepost-step toward the better deformation result.The model transduction approach can be applied topose correction straightforwardly. Fig.9(b) shows aninitially deformed result generated by IK (inverse kinematics)skeletal technique in MAYA. It can be noticedthat some geometric details of the triceratops modelare lost and visual artifacts can be found, especially inthe leg parts. Nevertheless, the integrity of the meshin Fig.9(b) has contained the overall pose information(referred to the original mesh in Fig.9(a)). Note thatvarious mesh editing methods usually do not change the

592 J. Comput. Sci. & Technol., May 2010, Vol.25, No.3Fig.9. (a) is the original triceratops model. (b) is an initially deformedresult generated by IK (inverse kinematics) skeletal techniquein MAYA. It can be noticed that some geometric details ofthe triceratops model are lost and some artifacts can be found,especially in the leg parts. With our approach, these artifacts aresuccessfully corrected and the satisfactory result (d) is obtained.(c) is the intermediate result.topology of the mesh, so there naturally exists a consistentcorrespondence between the original and deformedmeshes. Thus, similar to the transduction process, first,an intermediate nonconforming “mesh” is generated byextracting the rigid components from the original meshand mapping them onto the initially deformed mesh.Then this nonconforming mesh is pieced together toform a 2-manifold mesh while satisfying both the translatingconstraint and the differential constraint. As aresult, the corrupt mesh is successfully corrected anda better deformation result is achieved (as shown inFig.9(d)).5.2 Additional Application II: Skeleton-FreeDeformation Animation with 3D MocapDataAnother application of our transduction method isthe skeleton-free deformation animation based on 3DMocap data. By using our skeleton-free skinning technique,the overall pose of marker points from motioncapture (Mocap) is used to directly drive a 3D avatarmodel, without needing to build skeleton structures forthese points. Moreover, the computation complexity ofour technique is independent of the number of markers,Fig.10. Our stereo-vision motion capture the markers just serve as the space constraints of thesolution system in a least squares sense.Although the skeleton structure contains abundantpose information and provides intuitive controls [5-7] ,defining and manipulating a skeleton structure for a3D model, which is usually represented by triangularmesh, is not a trivial task [43] . In the motion retargettingscenario, it is usually cumbersome to constructthe skeleton structure for lots of marker points. Moreover,the number of markers used in Mocap often variesin different scenes, so does the skeleton structure builtfrom those markers. In addition, many 3D objects donot have obvious skeleton structures or their metamorphosescannot be described in terms of a skeleton.The skeleton-free retargetting method can effectivelyovercome the above drawbacks. In the following foursteps, we discuss how to use our transduction techniqueto drive a 3D mesh model while not needing to buildthe skeleton structures for the Mocap data.1) Given a mesh model and the Mocap data, the userfirstly builds a correspondence between the markers ofthe Mocap data and the counterpart vertices on themesh. Note that these feature markers are typicallyplaced on the subject at anthropometric landmarks,such as the shoulders, elbows and wrists. The correspondenceprovides the landmark positions for a subsetof the model’s vertices in each frame of the motion sequence.Therefore, we adopt markers’ positions as thespatial constraints of our mesh deformation system.2) Then we generate an initial result by using somemesh deformation technique. Here, we adopt the meanvaluemanifold deformation method proposed in Section4. Note that in the case of the overall deformation ofhuman motion, the user need to neither specify the desiredregion of interest (ROI), nor adjust the transformationof the handle frame to the transformation of thehandle position such as [21].Note that artifacts may appear in the initially deformedmesh ˜M because of incorrect vertex/marker correspondences,large deformations, or exaggerated movements.So in the following steps, we need to adjust theinitial result that has already captured the overall poseinformation.3) In the third phase, we will generate an intermediate“mesh” M temp , which acts the same role as thepose mesh (Fig.2(d)) in model transduction.4) Having obtained the intermediate mesh M temp ,we now piece together the triangles in M temp accordingto the differential constraint and the translating constraint.The final 2-manifold mesh M ′ is produced.Thus, after the above four steps, the final drivenresult is generated. Fig.11 shows an example whenthe identical Mocap data is retargetted to different

Huai-Yu Wu et al.: Model Transduction for Triangle Meshes 593high sampling rate is advisable when fast motions arecaptured. Then the motion capture data are inputinto our 3D skeleton-free deformation animation program.We do not construct the skeleton structure forthe Mocap data. In Fig.11, the identical Mocap datais retargetted to different characters, a woman and aman, respectively.Fig.11. The identical Mocap data (a) is retargetted to two different3D characters, a woman (b) and a man (c).characters, a woman and a man, respectively.6 Results and DiscussionThe model transduction results are demonstrated inFigs. 1, 2, 12. It can be seen that even though thereference source mesh is absent, our method still successfullyretargets both gross skeletal structure and subtleskin deformation, and is effective for large deformations.Fig.1(a) and Fig.1(b) give a comparison betweendeformation transfer and our method, by transferring acat’s pose to a lion. As discussed in Section 1, the limitationof deformation transfer is that the source andtarget reference meshes must have the same kinematicreference pose, since the reference target mesh T reproducesthe change in shape induced by the source deformationbetween S and S ′ . Only when the two referenceposes are similar can this kind of transferring operationbe valid. While our method successfully overcomes thisdrawback. Another advantage of model transductionis that because of explicitly handling surface details’preservation (see (3)), model transduction can achievebetter results than deformation transfer when visualartifacts already exist on the source deformed model.As for the limitation of our method, similar to deformationtransfer [8,10] , model transduction currently isdesigned for the case where there is a similar semanticcorrespondence between the source and target meshes.Our method may not yield convincing results for modelswith very different semantics (e.g., one animal witha tail versus another animal without tail, or very shortlegs versus very long legs).Except for pose correction after various mesh editingoperations (Fig.9), our transduction method is alsoused for skeleton-free deformation animation based on3D Mocap data. The motion data are firstly obtainedby our stereo-vision motion capture system (Fig.10),and then formatted into TRC point format files. Thesample rate of our Mocap system is up to 100 Hz. ThisFig.12. With the model transduction method. (a) Old manmodel directly imitates the expression of a young man model. (b)Muscular man model directly imitates the fetus model’s pose.We also test our model correspondence method andmodel deformation method, respectively. First, fromFig.3, it can be seen that our mean-value manifoldcross-parameterization scheme successfully establishesa shape-preserving correspondence between input surfaces,such as the thread-like stripes on lion’s legs andbelly. In this example, eighteen markers are provided bythe user to serve as the global constraints. Fig.4 showsa few snapshots from the morphing series of two models(a women face and a man face). Thanks to the well establishedcross-parameterization, the gradual changesof the shape are natural and visually appealing resultis obtained.Then, we showcase the detail-preserving mesh editingresults in Fig.7. Meanwhile, Fig.7(a) illustrates themetaphor. In our system, point handles (Fig.7(a)) aresupported to provide a simple interface such that thereis no need for the user to specify the orientations (localframes) of the handles. That is, the local orientationat the point handle is automatically decided bythe system. Owing to the nice properties of the meanvaluemanifold operator, our editing system is effectiveand stable, and visually pleasing deformation resultsare achieved.Our transduction method is numerically efficient,because the solution to the optimization problem canbe obtained by fast solving a sparse linear system. And

594 J. Comput. Sci. & Technol., May 2010, Vol.25, No.3the linear system is separable in the three coordinatesof the vertices, which reduces system’s scale to 1/3.We first compute the factorization of the normal equationsand then find the solution by back-substitution.With a sparse LU decomposition solver [44] , for example,5K/14K/20K vertices only require 0.281/0.829/1.407seconds for factorization and 0.008/0.016/0.032 secondsfor back-substitution on an Intel P4/3.0 GHz. Moreover,for very large models, running the algorithm firston a simplified model and then using the hierarchicalcorrespondence between the simplified mesh and theoriginal mesh can significantly reduce the times.7 ConclusionsThis paper presents an effective method for posetransfer between triangle meshes. Our approach ispurely mesh-based. Using model transduction, the poseinformation is obtained from the source model and directlyretargetted to the target model, without the needof building skeleton configurations or extracting sourcedeformation with an extra reference source mesh. Experimentalresults show that our method can successfullytransfer both complex skeletal structures and subtleskin deformations. We also demonstrate another twoapplications of the model transduction method: posecorrection after various mesh editing operations, andskeleton-free deformation animation based on 3D Mocapdata. Furthermore, we propose the novel meanvaluemanifold operator to perform shape-preservingcross-parameterization and mesh deformation as thecomponents of model transduction. The results showthat our methods are effective and efficient for commonapplications.Acknowledgment We would like to thank Dr.Yong Wang for his valuable advice.References[1] Gleicher M. Retargeting motion to new characters. 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