anisotropic deformation for local shape control...computational visual media doi...

Computational Visual MediaDOI 10.1007/s41095-017-0092-6 Vol. 3, No. 4, December 2017, 305–313

Research Article

Anisotropic deformation for local shape control

Matteo Colaianni1(�), Christian Siegl1, Jochen Sußmuth2, Frank Bauer1, and GuntherGreiner1

c© The Author(s) 2017. This article is published with open access at Springerlink.com

Abstract We present a novel approach to meshdeformation that enables simple context sensitivemanipulation of 3D geometry. The method is basedon locally anisotropic transformations and is extendedto global control directions. This allows intuitivedirectional modeling within an easy to implementframework. The proposed method complementscurrent sculpting paradigms by providing furtherpossibilities for intuitive surface-based editing withoutthe need for additional host geometries. We show theanisotropic deformation to be seamlessly transferable tofree boundary parameterization methods, which allowsus to solve the hard problem of flattening compressiongarments in the domain of apparel design.

Keywords anisotropy; modeling; as-rigid-as-possible (ARAP); deformation;parameterization

1 Introduction

When manipulating 3D geometry, artists oftenuse methods that rely on host geometries suchas cages, rigs, or control polygons. In contrast,approaches for articulated surface modeling existsuch as sculpting. All these methods—with andwithout a rig—exhibit a lack of directional controlwhile editing. To elongate parts of a mesh (e.g.,the legs of a dog, see Fig. 1), an artist can use,

1 Computer Graphics Group, University Erlangen-Nuremberg, 91058 Erlangen, Germany. E-mail:M. Colaianni, [email protected] (�); C. Siegl,[email protected]; F. Bauer, [email protected]; G.Greiner, [email protected].

2 Adidas AG,Adi-Dassler-Strasse1,91074 Herzogenaurach,Germany. E-mail: [email protected].

Manuscript received: 2017-03-07; accepted: 2017-05-30

amongst other methods, a cage attached to themesh. However, such linear methods often leadto undesired results, as features are not preserved(see Fig. 9, cage based). On the other hand,sculpting methods lack intuitive tools for directionaloperations such as elongation or thickening.In this work, we present a highly accessibleapproach to mesh modeling by combining a noveldirectional formulation with known surface-baseddeformation methods. We thus include direction-dependent transformations directly into deformationenergy formulations—namely anisotropic as-rigid-as-possible (AnARAP) and deformation gradientbased editing (AnDefGrad). Using our method, theartist is able to transform parts of an object alongon-surface directions while preserving essential shapefeatures. Affine transformations—such as scaling,shearing, or rotation—enable many aspects of locallyarticulated modeling. Deformation directions aredefined by generic vector fields or as user input.Using spatially varying directions (as used inexpression editing, see Fig. 10) even goes beyondthe possibilities of proxy-based or linear methods.Figure 1 shows a dog with different edits tolocally selected parts. By using the same basicintuition of on-surface directions, the presentedwork also transfers anisotropic deformation to thedomain of mesh parameterization. This tacklesan important problem in the apparel industry:meaningful shrinkage calculation for functionalcompression garments, along previously defineddirections. We show that the method worksfor different formulations of surface elements:triangles (for both deformation gradients and ARAPparameterization) and triangle-fans (for ARAP).This work contributes by enabling:• an enhancement to deformation methods that

305

306 M. Colaianni, C. Siegl, J. Sußmuth, et al.

Fig. 1 Anisotropic deformation paradigm enables directionally aligned modeling without the need for a rig. The original model (right) wasedited part-wise to change shape characteristics, such as the length and thickness of legs, body, ears, or tail (left).

includes intuitive direction dependent modeling;• a flexible way of defining deformation directions

via fields or local or global directions;• the possibility of flattening complex functional

garments.

2 Previous work

Modeling of surfaces based on skeletal rigging hasa long tradition in mesh deformation [1–3]. Thiseasily enables directed deformations with respectto bones while other directions may not be thatintuitive to model without deformation interpolationof affine matrices. In contrast, our method enablesdeformations along directions—even non-linear—onthe surface itself. An alternative approach is todeform meshes by using cages as low-resolutionproxies [4] as well as by replacing linear blendingby a biharmonic scheme presented by Jacobson etal. [5]. While offering more flexibility in the controlof direction they still are based on an additionaldeformation geometry. As-rigid-as-possible (ARAP)deformation was used for 2D shape modeling byIgarashi et al. [6], constraining a deformation tobehave in a rubber-like way. This was pickedup for 3D mesh manipulation by Sorkine andAlexa [7] and Wang et al. [8]. However, the issue ofanisotropic articulated modeling was not consideredby earlier works on ARAP modeling. Meshmanipulation based on the transfer of per-triangledeformation gradients was introduced by Sumnerand Popovic [9]. In the same fashion, arbitrary affinetransformations applied to nodes to provide directand intuitive deformation were presented by Sumneret al. [10]. More recently, modeling methods forarticulated, organic shapes based on polar and quad

representations have been presented in Refs. [11, 12].Parameterizing meshes using ARAP methods

was introduced by Liu et al. [13], extended forlength conservation by Zhang and Wang [14] andfurther extended by Smith and Schaefer [15].When developing functional garments based onpattern shrinkage during compression [16], directedgeometry manipulation is crucial. Their proposedscaling method is based on mechanical fabricproperties but does not inherently include directionaldeformation in order to achieve per surfacecompression control.

3 Anisotropic deformation

In order to allow intuitive control while deforminga model in a directional manner, e.g., scalingor shearing parts of an object, we introduceanisotropic transformation to two different surface-based deformation methods. An artist defines thepart he wants to modify by selecting a set ofvertices on the surface and provides a directionwith the desired amount of deformation. Thesystem then modifies the model in an articulated,feature preserving way while performing the desireddirected deformation. This deformation is not drivenby boundary conditions or handles, but insteadthe selected surface elements are directly deformedto locally alter the shape. This local shape shiftis then considered by the solver as an extensionto the energy formulation and changes the globalshape of the geometry. Since the basic intuitionof the presented work is to apply deformation todiscrete surface elements (i.e., triangles or triangle-fans), it is not restricted to a specific surfacerepresentation. Different transformations—suchas scaling, shear, and rotation—are introduced

Anisotropic deformation for local shape control 307

and embedded in two basic deformation methods:ARAP modeling [6, 7] and deformation gradients [9].

3.1 Anisotropic ARAP (AnARAP)

ARAP deformation minimizes the non-rigidremainder of a deformation. Therefore, the currentlydeformed instance of a mesh is compared tothe reference shape. In the original work (seeRef. [7]), the error is defined to be the differencebetween two rigidly aligned corresponding vertexneighborhoods. We present a method extending thisformula by including the anisotropy directly intothe deformation energy. Essentially, the referenceshape is transformed non-uniformly on a disjunct,per surface element level. For this, the anisotropicARAP energy is defined asE(v, R)=

∑i∈M

∑j∈N (vi)

wij‖(vj − vi)−TiRi(vj − vi)‖2

(1)

The matrix Ti defines the desired affine mapping fortransforming the local surface elements.3.1.1 Solving for anisotropic ARAPIn the spirit of the work by Sorkine and Alexa [7]the solution for the best vertex positions is found inan iterative flip-flop fashion with a local phase anda global phase. In the local phase, we achieve thisby additionally transforming the reference verticesby Ti when seeking the best rotations Ri—thisholds for transformations with no rotation part.To solve for the best global vertex positions,our method directionally transforms the referenceshape’s vertices as well. A sparse linear system isbuilt from the partial derivatives of the AnARAPenergy with respect to the reference shape’s positionsv. Regarding a vertex vi, the equation is written as∑j∈N (vi)

wij(vj−vi) =∑

j∈N (vi)

wij

2 (TiRi+TjRj)(vi−vj)

(2)

When using local transformations that includerotations, the local phase has to be modified. Solvingthis step seeks the rotation that aligns two fansrigidly. Performing this step for rotated targetgeometries TiRi will cause ARAP deformation tocounteract the rotation, which is not the desiredeffect. Instead, we compare the deformed fan inthe local phase to a transformed target without therotational portion.

3.2 Anisotropic deformation gradient

Modeling with anisotropy is not restricted to ARAPdeformation. Some shapes exhibit organic behaviourwhen deforming them using ARAP methods, andthis may not always be intended by the user.Therefore, we introduce the same intuition tothe method of deformation transfer [9] to enableanisotropic modeling. The original work representsa mesh’s deformation as a collection of per-triangleaffine transformations. This encoding subsequentlyis used to decode a topologically equivalent meshwhich leads to a deformation transfer. Our methodfollows this basic intuition and encodes a sourcewhich—in the decoding step—is applied to theoriginal mesh in order to deform it. In contrast, oursource transformations are not derived from a mesh,but we directly use the desired edits as an encoding.The initial transformation T j,init for a triangle j

is the identity I. A decoding of the mesh usingthis set of identities will not cause a deformation.As in the anisotropic ARAP extension, the userdefines a set of vertices and a direction dependenttransformation (e.g., scaling, rotation, or shear) todrive the deformation. Using AnDefGrad, we replacethe initially set identities for affected triangles by thedesired local transformation Qj,edit. As a result, theencoding for a deformation is S = {T1, . . . , T |M |}with M being the set of triangles and T j a 3× 3affine matrix for affected triangles; I is used forunaffected triangles. To decode the mesh with thedesired transformations, we solve the sparse linearsystem:

Av = c (3)

with the unknown deformed vertex positions v andthe weight-sensitive adjacency matrix:

A =

−w11 w12 . . . w1|V |

w21 −w22 . . . w2|V |...

... . . . ...w|V |1 w|V |2 . . . −w|V ||V |

(4)

Note that wij is zero whenever the vertices i and j

do not share an edge. Adjacency weights betweentwo vertices depend on the adjacent triangles tothis edge. wii is the summed edge weight forall edges adjacent to the vertex i. The right-handside c is the sum of each disjunct triangle cornerposition

∑j∈N (vi) T jvi. For efficiency reasons, we


pre-factorize the matrix A once and solve the systemfor spatially separated positions. Figure 2 depictsthe difference between the ARAP based and thedeformation gradient based anisotropic deformationmethods on an example, for a scaling of the orangevertices. In contrast to the modified ARAP method,this method leads to a more local solution andthe triangle deformation at the selection boundaryexhibits a higher discontinuity. Depending on theneed (a more organic versus a more articulated styleof feature preservation), one of the two methods canbe selected.

3.3 Local affine transformations

3.3.1 Anisotropic scalingOne possible transformation T i is a scaling along alocal vector field. With su and sv as the scale-factorsin vertex i’s tangent plane, the matrix is defined as

T i = C i

su 0 00 sv 00 0 1

CTi = C i Sscale CT

i (5)

C i is the local basis of the vertex vi aligned to thedesired direction of deformation ci:

C i =[ cu

i

‖cui ‖

,cv

i

‖cvi ‖

,cu

i × cvi

‖cui × cv

i ‖

](6)

Figure 3(top) shows the scaling in a local vertex–fanspace.

Anisotropic shearing. Using an affine shearingto deform the surface of a mesh follows the sameprinciple as directed scaling. The per-vertex appliedmatrix T i simply is modified to be a local shearing(exemplary for a shear along the x-axis):

T i = C i

1 shx 00 1 00 0 1

CTi = C i Sshear CT

i (7)

Fig. 2 Parts of a torus (left) are elongated using anisotropicdeformation gradients (middle) and ARAP deformation (right).While the deformation gradient based method exhibits a highdiscontinuity of triangle sizes, the deformation using anisotropicARAP deformation is distributed more globally over the selectionborder.

Fig. 3 Top: instead of aligning the currently deformed vertex fan(green) to the reference mesh’s fan (blue), we scale the neighborhoodvi by the directed scaling Ti (red). Bottom: the triangle case for theparameterization is shown accordingly.

With this, a geometry can be sheared, as depicted inFig. 4.3.3.2 Anisotropic rotationIn the same fashion, local rotations are performedby a simple modification to T i. Replacing the localtransformation S by a rotation matrix leads to

T i = C i

cos − sin 0sin cos 00 0 1

CTi = C i Srot CT

i

(8)The organic character of ARAP leads to lesspredictable results due to the organic fashion of

Fig. 4 The original geometry (left) is sheared using AnARAP(middle) as well as AnDefGrad (right) deformations. The organicappearance of ARAP is visible. The modified deformation gradientmethod gives more control. The orange regions are involved in thedeformation while the blue ones are unaffected.


the deformation (see Fig. 5(middle)). For certainscenarios and low strengths, this appearance ispreferable to the results of the gradient-basedmethod (see Fig. 5(right)).

3.4 Deformation directions

The local frame to which the transformationsare aligned is spanned by the normal ni andthe tangent vectors cu

i and cvi . In the present

work we give several possibilities for creatingdeformation directions. A vector field followingprincipal curvature directions [17] is used for thesnake example in Fig. 8. The projection of screenspace directions onto the surface enables an intuitiveway of modeling without the need for a previouslydetermined vector field. However, this methodis obviously limited to directions which are notparallel to the surface’s normal direction. Finally,a global direction can immediately be used as the

Fig. 5 The post (left) is rotated locally at the selected verticesto become a hook. The AnARAP approach (middle) exhibits moreglobal deformation while AnDefGrad (right) has more local control.

Fig. 6 The selected (orange) vertices of the post are twisted usinganisotropic deformation gradients along a global direction. The twistangle increases with height. The torsion frequency is increased fromleft to right.

Fig. 7 Anisotropic deformation is performed by selecting vertices(orange) and applying different classes of deformations. Top: astretch transformation performs anisotropic scaling along the desireddirection. Bottom: a local rotation of the surface elements isperformed for two different rotation axes. The blue regions remainunaffected and, therefore, act as deformation constraints.

deformation direction (see Figs. 4, 5, and 6).Elongation of parts can be achieved using a

skeleton which involves tedious and unintuitiverigging. By interpolating the bone transformations,it is even possible to achieve a spatial dependency ofdeformation directions. Our method not only easesthis process by allowing on-surface direction control,but it expands this possibility by allowing arbitrary,even procedural or user-painted deformationdirections. This is demonstrated in facial expressionmodeling in Fig. 10(top), where the direction fieldvaries its orientation across the lips.

4 Applications and results

4.1 Modeling with the directional method

Using the presented directional modeling approach,a user can select parts of an object and performdeformation along arbitrary directions. In doingso, a set of vertices is selected and differentlocal transformations can be applied. The shapeintuitively deforms accordingly. In Fig. 7, localscaling and rotation transformations are depicted.The deformation is shown to follow the desireddirection while the features are preserved. Incontrast, the snake in Fig. 8 was deformed alongan automatically-defined vector field along thesurface. A deformation parallel to the field elongatesthe snake maintaining its diameter, while scalingorthogonal to the field preserves its length. Wecompare the two different incarnations of our


Fig. 8 A curvature-aligned direction field is used to deform the snake. Top: the snake is scaled parallel to the field to elongate it—itsthickness is well preserved. Bottom: deformation is performed perpendicular to the field to achieve a thickening—its length is well preserved.

method to commonly used approaches in directionalmodeling: cage-based deformation (see Fig. 9)and handle-based modeling via isotropic ARAPdeformation. As the chess piece is elongated,significant features, e.g., the rims below the headand at the bottom, should remain as similar aspossible to their original shapes. The cage basedapproach stretches the shape uniformly, leadingto less pronounced features. Likewise, AnARAPdeformation produces a result which does notstrongly preserves features—the shape becomesorganic. This is an inherent property of ARAPmethods, where edges are smoothed by three-dimensional rotations of the fans. Naturally, theoriginal ARAP method’s performance is poor aswell. Using AnDefGrad preserves sharp featuresmuch better with respect to the original object,due to the highly local deformation approach ofdeformation gradients. In addition, the benefit ofusing an anisotropic formulation of the deformation

Fig. 9 The original geometry (left) is linearly elongated usingsingle cage-based deformation. Regular ARAP deformation andanisotropic ARAP deformation (AnARAP) results are shown incomparison. Note that feature sharpness is lost using an isotropicenergy formulation. Overall, ARAP—with or without anisotropy—does not perform as well at sharp edges as the anisotropic deformationgradient modeling (AnDefGrad).

in contrast to the regular (isotropic) energy is alsoshown in Fig. 9. The original ARAP method is usedto calculate the deformation for an elongated chesspiece without considering anisotropy while usingthe inherently anisotropic deformation elongates thegeometry in a more feature-preserving way, becausethe penalty is restricted orthogonally to the desireddeformation energy.4.1.1 Facial sculptingWe present another strength of the methodby altering the facial expression of an ancientRoman bust using anisotropic deformation (seeFig. 10 (top)). A vector field is defined across the lipswhich changes direction depending on the distance tothe corners of the mouth. AnARAP deformation isused to stretch the surface along the altering field

Fig. 10 Top: the vector field (left) is bent along the lips to causethe Roman bust (middle) to smile (right). Bottom: the beard iselongated along a vector field on the surface.


leading to a different but still articulated expression.This result is achieved with little modeling input—avector field changing over surface and a deformationstrength is defined—and thus, it is highly accessibleto the user. To show that even high-frequencyfeatures are well preserved, we elongate the beard ofthe face as shown in Fig. 10(bottom). Here, the beardbecomes longer following a constrained directiondefined on the surface.

4.2 AnARAP parameterization forcompression garments

Our method is not restricted to mesh deformationfor the purpose of sculpting-inspired modeling. Weshow that the directional scaling can be appliedto the parameterization method introduced byRef. [13]. Since our directional scaling is invariantto the surface elements (triangles or fans) used,the method can easily be used to also enrichthe above formulation by adding anisotropy(see Fig. 3(bottom)). Using anisotropy for freeboundary parameterization solves an importantissue in apparel design: flattening cut pieces forcloth production with proper compression scaling.Functional garments are important in the fieldof sports, and garment development pipelines aremore and more influenced by virtual technologies.Including compression into the cloth is usuallyachieved by shrinkage of the flat cut piece. Toincrease compression while not shortening thegarment, this scaling currently is performed non-uniformly along one single direction in 2D. Thismakes it hard to properly shrink patterns withmore complex shapes deviating from this singledirection. Our current methods allow us to definethe shrinkage for local positions independently, andin 3D. Forcing the directions ci for some targetvertices vi on the surface gives the user control overthe compression behavior at desired positions (seethe arrows below the armpit in Fig. 11 (left)). In thisexample, compression is applied to support the areabelow the arms. The constrained compressions arepropagated over the whole surface patch and the cutpiece is computed by flattening the original surfacesubset. Solving for the best parameterizationwhile taking the local, directed shrinkage intoaccount leads to the desired compression scalingeven for complex shapes. In contrast to state-of-the-art shrinkage (Fig. 11 (scaled)), our approach

Fig. 11 The desired compression directions are defined for apatch (left). The original method of scaling fails to distributethe compression well (middle). Our method maintains the desiredcompression over the whole surface.

distributes compression along defined fielddirections (Fig. 11(AnARAP)). The anisotropicparameterization for garment pattern developmentwas used to extend the pipeline presented inRef. [18]. Figure 12 shows a short pipeline overviewof how user-defined shrinkage-directions resultin a sewn functional garment with the desiredcompression behavior.

5 Discussion

Limitations. The method presented exhibitsa major limitation in the inconvenient needfor a deformation direction as a user input.Although the method is based on a vectorfield distributed on surface, one can exchangethe direction generation for more convenient andintuitive methods. As with many modeling methods,a high amount of deformation results in aninhomogeneous distribution of differently sizedtriangles. This is solvable by re-triangulation ofthe mesh but not considered further here becausean expensive recalculation of the system matrixis required. AnDefGrad and AnARAP exhibit

Fig. 12 The user defines the desired compression-directions on thesurface (a). The pattern is flattened (b, red contour) by retaining thedesired compression value along the given direction. The traditionalgarment shrinkage process (scaling in the 2D pattern domain) isalso shown (b, gray pattern). The sewn garment with the desiredcompression is shown in (c).


similar performance with respect to conservation oflow-frequency shape features (see Figs. 7 and 8).For high-frequency features (e.g., creases and sharpborders) AnARAP performs worse than AnDefGrad(see Fig. 9); it should only be used for organic shapes.However, AnARAP results in more continuoustriangle sizes, which is well suited for low-frequencymodeling (see Fig. 2). Finally, the projection ofglobal or screen directions onto the surface includesan obvious drawback: directions orthogonal to thesurface result in undetermined behavior and needspecial treatment.

Conclusions. In this paper we have presenteda method to enrich sculpting- and modeling-basedmesh deformation, providing directional control ofsurface deformations. We have shown that themethod allows intuitive deformation of surfaces,while maintaining semantic features of the originalshape well. In contrast to other methods, thepresented deformation paradigm does not rely onhandles or constraints. Surface regions are selectedand the deformation is directly applied to theselection. Different affine transformations: scaling,shearing, and rotation, are implemented in theframework for two exisiting deformation paradigms:ARAP deformation and deformation gradients. Themethod of generating deformation directions caneasily be replaced. We have presented shapemanipulation along different kinds of on-surfacedirections as well as surface-aligned anisotropy forfacial modeling. As anisotropic scaling is applicableto mesh parameterization, it is well suited tosolving an important restriction in functional appareldevelopment.

Acknowledgements

We want to thank Blendswap artists Calore for thecobra, Metalix for the dog, and Nerotbf for theRoman bust.

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[18] Colaianni, M.; Siegl, C.; Sußmuth, J.; Rott, F.;Greiner, G. Shape adaptive cut lines. In: Proceedingsof the Eurographics Workshop on Graphics for DigitalFabrication, 49–55, 2016.

Matteo Colaianni is a Ph.D.candidate in the Computer GraphicsGroup of the University of Erlangen-Nuremberg. His research focus isgeometry processing in the field ofapparel design and statistical shapeanalysis.

Christian Siegl is a Ph.D. candidatein the Computer Graphics Group of theUniversity of Erlangen-Nuremberg. Hisresearch focuses on mixed-reality usingprojection mapping, medical imageprocessing, and the virtual creation ofapparel.

Jochen Sußmuth completed hisPh.D. degree on geometry processing inErlangen University in 2011. After that,he joined the Adidas group where he iscurrently working as a researcher in thefield of computer graphics.

Frank Bauer is a research fellow inthe Computer Graphics Group of theUniversity of Erlangen-Nuremberg.His research focuses on 3D scenereconstruction, augmented, mixedand virtual-reality applications, andaccessible human–machine interactions.

Gunther Greiner is the head ofthe Computer Graphics Group of theUniversity of Erlangen-Nuremberg. Hisresearch focuses on geometry processingand geometric modelling.

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