Computers & Graphics 37 (2013) 21–32

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Computers & Graphics

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Technical Section

Material-aware cloth simulation via constrained geometric deformation$

Li Liu a,b,c, Zhuo Su a,b,n, Ruomei Wang a,b, Xiaonan Luo a,b

a National Engineering Research Center of Digital Life, State-Province Joint Laboratory of Digital Home Interactive Applications, School of Information Science & Technology, Sun

Yat-sen University, Guangzhou 510006, Chinab Shenzhen Digital Home Key Technology Engineering Laboratory, Research Institute of Sun Yat-sen University in Shenzhen, Shenzhen 518057, Chinac Faculty of Information Engineering and Automation, Kunming University of Science and Technology, Kunming 650500, China

a r t i c l e i n f o

Article history:

Received 25 June 2012

Received in revised form

27 October 2012

Accepted 29 October 2012Available online 29 November 2012

Keywords:

Cloth simulation

Material measurement

Cloth behavior

Mesh deformation

Deformation energy

93/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.cag.2012.10.006

article was recommended for publication by

esponding author at: National Engineering Re

ovince Joint Laboratory of Digital Home Inter

tion Science & Technology, Sun Yat-sen Un

el.: þ86 20 39943199; fax: þ86 20 3994 319

ail addresses: suzhuoi@gmail.com (Z. Su),

mail.sysu.edu.cn (R. Wang).

a b s t r a c t

Most real-world cloth consists of nonlinear material and exhibits anisotropic behavior. This paper

proposes an efficient and expressive mesh deformation method to obtain realistic cloth shapes with

various cloth materials. The key idea in this work is to model the cloth using a mesh-based deformation

energy that is composed of several energy terms and to fit the weighting coefficients of the terms from

real data. We first develop a direct geometrical material measurement method for testing the recovery,

stretching and bending behaviors of different real cloth samples. Then, we separate the geometric

deformation energy into three terms related to the vertex position, edge length and bending of the

dihedral angle, respectively, and the weights for the three energy terms are learned from the data

measured with real cloth. Reusing the weights for the geometric deformation by a numerical solution in

the least square sense can model similar cloth behavior. The experiments show that our method

effectively provides rich cloth simulation results that are able to capture distinctive material effects.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Cloth simulation is an important problem in computer gra-phics, with wide applications in geometric modeling, computeranimation, video games and virtual clothing. Different clothmaterials exhibit distinctive appearances in terms of draping,folding and wrinkling due to the properties resulting from themanner of weaving and the fibrous composition. As shown inFig. 1, shirts made of different cloth materials show distinctivewrinkles and folds when worn with the same posture.

Although the physically based methods can be used to model avariety of materials, it is often difficult to obtain the relevantphysical parameters, and the high computational costs of thesemethods prevent interactive applications. Geometrically baseddeformation methods are convenient for interactive manipula-tions because of their greater efficiency than physically basedmethods. While some of these geometric methods have beenapplied to cloth deformation, they usually ignore the specificmaterial properties of the cloth, making it difficult to generaterealistic results.

ll rights reserved.

Andrei Sharf.

search Center of Digital Life,

active Applications, School of

iversity, Guangzhou 510006,

9.

This paper proposes an efficient geometrically based clothmodel. To make the model aware of different cloth materials, wemeasure some key geometric features of the cloth material in realexperiments and set the parameters of the cloth model accordingto the experimental results. Fig. 2 presents an overview of ourmethod. The main contributions of this work are as follows:

�

A simple geometrical method to measure the recovery,stretching and bending behaviors of cloth samples. � A geometrically based energy model for cloth that is composed

of three weighted terms with respect to vertex position, edgelength and dihedral bending.

� A quadratic fitting method to fit the weight parameters in the

cloth model from the data measured with real cloth.

� Data-driven guidance for material-aware cloth simulation to

effectively provide visually realistic results.

2. Previous work

Cloth simulation is challenging because most real-worldclothes exhibit complex behavior. This section reviews the pre-vious work in this area.

Physically based simulation: A great deal of work has focused onimproving simulation results and collision handling. For example,Baraff et al. [1] proposed the implicit integral cloth simulationmethod based on the spring-particle system. Bridson et al. [2] first

Fig. 1. Shirts made of different cloth materials show wrinkles and folds when worn by the same human model in our simulation. (a) The red silk shirt has many small

wrinkles and folds. The simulation with blue nylon exhibits a distinctive appearance (b), while shirts made of blue white dot canvas (c) and blue washed star denim

(d) have fewer wrinkling details due to their weaving structures. (For interpretation of the references to color in this figure legend, the reader is referred to the web version

of this article.)

1 Geometrical Material Measurement

Real Cloth

Recovery

Stretching

Bending

Initial Mesh

Vertex Position

Edge Length

Dihedral Bending

2 Constrained Geometric Deformation

3 Material-aware Mapping

Deformed Mesh

5 Simulation With Material Effects 4 Numerical

Solution

Red Silk

Blue Nylon

Washed Star Denim

White Dot Canvas

Realistic Texture Mapping

Material Properties

Geometric Energy

Non Linear Optimization

Quadratic Fitting

Output

Parameter Settings

Material Learning

Cloth Modeling

…

Fig. 2. The overview of our five-step approach, including geometric material measurement, constrained geometric deformation, material-aware mapping, numerical

solution and simulation with material effects. Based on some real experiments with various types of cloth, we construct the geometric energy of the cloth according to the

parameters determined from the material properties. Finally, we achieve cloth simulations that accurately represent the material behavior by optimizing the geometric

energy and realistic texture mapping.

L. Liu et al. / Computers & Graphics 37 (2013) 21–3222

used the physically based modeling approach to generate realisticsimulations of clothes with folds and wrinkles. They also pro-duced a highly efficient algorithm [3] to accomplish robusttreatment of collisions, contact and friction in cloth animation.Kang and Lee [4] introduced a real-time cloth draping simulationalgorithm using conjugate harmonic functions and the FEMmethod to simulate wrinkling cloth. Popa et al. [5] defined thespace–time deformation to capture the real-time simulationresults of wrinkling garments, and they later presented analternative method [6] for realistic generation of dynamic wrin-kles with a post-processing stage. Kaldor et al. [7,8] demonstratedthat nonlinear anisotropic behaviors of knitted cloth can besimulated at the yarn level. However, as the behavior of thesimulated objects in the simulation should differ due to thedifferent nature of the fabric and the weave, creating accuratemodels requires finely tuned physical parameters, which iscomputationally expensive.

Geometrically based simulation: Typical geometrically basedmethods for wrinkling garments include the catenary model [9]for cloth modeling, a geometric model [10] based on interactiveclothing design, and the texture-based approach [11], whichenables the animation of wrinkles on textured clothes.

These methods of constructing the geometric constraints forwrinkling garments extract geometrically based conceptualrepresentations. A fully geometric approach to wrinkling gar-ments was proposed by Decaudin et al. [12], who used priorknowledge of cloth buckling behavior in the case of cylindricalcloth surfaces to generate pre-defined types of procedural folds.These methods are restricted to pre-defined deformations. Arobust and efficient algorithm [13] with the stretch constrainthas been defined directly in the continuum model. Chen et al. [14]also presented a fully geometric approach to a developable clothdeformation simulation to determine the final shape of the cloth.The method of Muller et al. [15] includes highly intuitive ways ofadding the appearance of wrinkles to dynamic meshes in a real-time setting, but the main drawback of this method is that thereis no simple way to control or specify the directions of thewrinkles. Recently, a deformation transformer method developedby Feng et al. [16] and other data-driven methods, such as thatdesigned by Wang et al. [17], have been proposed for real-timecloth animation. However, these transformer methods requiretraining data sets with similar examples, limiting the range ofapplication of the method. None of these methods are able toconsider the properties of the cloth material, making it difficult

L. Liu et al. / Computers & Graphics 37 (2013) 21–32 23

for them to distinguish between different cloth materials withinteresting wrinkling and folding effects.

Cloth measurement: Research have developed several approachesto directly capture the parameters from real cloth materials, as theisolation and measurement of each material parameter wouldrequire a complicated test environment and equipment. To simplifythis task, Bhat et al. [18] proposed a method for estimatingparameters based on data acquired from a video. Although theirexperiments do not require specific devices and can be easily set up,the unconstrained nature of the motions makes it difficult to definemeaningful features that can be tracked and used in the optimiza-tion process. Recently, Wang et al. [19] proposed a data-drivenpiecewise elastic cloth model through the physical measurement ofstretching and bending in real cloth samples, although this methodonly provides an approximation to the actual strain–stress relation-ship when the cloth is slightly deformed. If larger forces are appliedto the cloth, this approximation will become less accurate.

Mesh deformation: One of the advantages of subspace embed-ding approaches is the high efficiency and interactive usermanipulation, with examples including the skinning method[20], skeleton subspace deformation [21], free form deformation(FFD) [22], mean value coordinates [23] for closed triangularmeshes, and Harmonic coordinates [24] for character articulation.Julius et al. [25] and Huang et al. [26] used the iterative measuresto compute the global vertex coordinates encoded in subspaceand local rotation to obtain high quality local rigid and deformedresults. It is difficult to accurately represent the material effectsusing these approaches, which largely depend on user’s manip-ulations and experience. The local surface details may be dis-torted due to the failure to preserve the geometric features.

Compared to the methods based on the subspace embeddingapproach, the recently proposed gradient domain methods usedifferential coordinates for deformation to preserve surface detailsin the least squares sense. Yu et al. [27] perform 3D mesh deforma-tion via gradient manipulation. The positions of some anchor verticesare manually modified by the user, and the new vertex positions arecomputed using the Poisson equation. Other work uses the surfaceorientation of the unknown deformed mesh as the Laplaciandifferential coordinates for deformation computation [28–30]. Thesemethods are both efficient and intuitive to control. However, a meshsurface is continuously deformed while the Laplacian coordinates atall vertices are preserved as much as possible. These methods mainlycapture geometric features related to surface curvature, which leadto the elastic membrane effects.

Table 1Ten different cloth materials, each exhibiting distinctive properties of resilience, exten

document.

Color and name Red silk Blue white dot canvas B

Samples 0–4

Common usage Chirpaur, skirt Bag, curtain Je

Color and name Green woolen Blue poplin O

Samples 5 and 6

Common usage Overcoat Shirt C

Our work is inspired by the geometric deformation methodsdeveloped by Yu et al. [27], Popa et al. [31], Huang et al. [32], andXu et al. [33]. All of these methods consider material propertiesand use them to characterize the stiffness of the surface, andhence to provide continuous fine control of the surface behaviorduring deformation. The simple and intuitive method [31] is tocontrol the distribution of the bending and shearing throughoutthe model according to the local material stiffness. The distortionmeasures in these methods [32,33] are designed to combinelength stretch and bending penalties with different intensitiesto generate material effects on the mesh. However, these methodsare not designed to model cloth. Narain et al. [34] proposed anapproach for cloth simulation that efficiently models the aniso-tropic behaviors of cloth. The technique can dynamically refineand coarsen triangle meshes to automatically conform to thegeometric and dynamic detail of the simulated cloth. Differentlyfrom the above work, we propose the geometric method for clothsimulation with parameters determined from material measure-ments, making it directly suitable for geometric deformation.

3. Geometric material measurement

We propose a geometric material measurement for clothsimulation by directly measuring the geometric parameters ofvarious types of real cloth. The method relies on the threegeometrical variances, which are all easy to measure, as reflectedby the simple setup of the measurement device. We find optimalparameters that closely match the actual material properties ofeach cloth.

3.1. Material properties

When people wearing clothes make various gestures, becausemost cloth materials are not elastic, the shape of the cloth isneither stretched nor compressed. As cloth is susceptible tobending, the drag and collision of cloth lead to different wrinkledetails on partial cloth surfaces. To realistically simulate clothdeformation, the influences of various materials on the clothshape must be considered. Different materials exhibit differentbehaviors when the cloth is suspended or wrinkled. Therefore, weselect 10 different cloth samples of size 460� 400 mm2 as shownin Table 1. The 10 materials in our experiments have different

sibility and flexibility. The full sets of parameters are shown in the supplemental

lue washed star denim Apricot linen Blue nylon

ans Dress, curtain Sports clothes

range ramie Gray fiber White satin

lothing Table-cloth Dress

L. Liu et al. / Computers & Graphics 37 (2013) 21–3224

fabric compositions including silk, denim, cotton, linen, nylon,polyester fiber and wool.

These materials are representative of common cloth materialsencountered in daily life, and they exhibit different woven orknitted patterns and different behaviors. The red silk and whitesatin materials are commonly used to make skirts and dresses, asthey are soft. In contrast, some materials exhibit stiffness inresponse to stretching, including the blue white dot canvas, grayfiber material and green wool, which are usually used to makebags, table-cloths or overcoats, respectively. The blue washed stardenim material is often used to make jeans, and the blue nylonmaterial has greater compressibility and is used to make sports-wear. The blue poplin and orange ramie bend easily, making themsuitable material for shirts.

3.2. Geometric measurement

All of these given cloth materials exhibit nonlinear and aniso-tropic behavior in the ease of recovery, stretching and bending dueto their fiber attributes, including resilience, extensibility andflexibility, respectively. These properties distinguish among sam-ples with distinctive appearances, e.g., the red silk material is notstretchy even though it is soft and not pliable or stiff. Ourmeasurement emphasizes the direct determination of the materialproperties in direct response to an efficient constrained geometricdeformation using a novel geometrically based experimental setup.

Recovery test: To capture the property of resilience or its abilityto maintain wrinkles, which reflects the ability of the cloth torecover from deformation, we designed a series of experiments totest the variance of the projected area between each initial clothsample and the deformed sample. We repeated these experiments10 times by compressing and unfolding each cloth sample,indicating where the material fell on the range from soft to hard.We configured a recovery tester as shown in Fig. 3(a). The firstmeasurement for the computation of projected area is carried outusing a Canon EOS 550D made in Japan, an 18.0 megapixel digitalsingle-lens reflex (DSLR) camera with continuous shooting at3.7 fps and a DIGIC 4 processor. The camera lens was positioned71.5 cm from cloth samples. The data obtained from the real clothare shown in Table 2, and the curve plotted in Fig. 4(a) exhibitsthe order of the resilience of the cloths studied here.

Fig. 3. The geometrically based measurement setup for recovery tests (a), stretching tes

10 real cloth samples considered here.

Table 2Mean material property values nðSmÞ, nðLmÞ and nðHmÞ are determined in real experime

Real fabrics Material properties

nðSmÞ nðLmÞ nðHmÞ

0-Red silk 0.9883 1.0209 0.9933

1-Blue white dot canvas 0.9755 1.0256 0.9993

2-Blue washed star denim 0.9896 1.1014 0.9968

3-Apricot linen 0.969 1.0337 0.9984

4-Blue nylon 0.9781 1.0218 0.9981

Stretching test: Our stretching test method is simpler than themethod used in studies such as Wang et al. [19], in which clothmaterials are tested by stretching in their warp and weft direc-tions. In our method, we pull the cloth up along the diagonal witha vertex around the corner by fixing the other three vertices of theflat cloth. Each sample is typically tested with three differentweights ranging from 400 g to 900 g with a 01 orientation using apointer tensiometer, and the diagonal length is gauged with ametric ruler as shown in Fig. 3(b). This experiment enableseffective comparison of the variation in stretching between eachinitial cloth sample and the stretched sample. Table 2 andFig. 4(b) show the results for the extensibility of each clothsample.

Bending test: Fig. 3(c) shows the experimental setup for thebending behavior of each cloth sample by recording the height ofthe bending angle. Unlike other methods for testing bendingproperties using technical devices and sensors, our experimentis convenient and it directly yields the geometrical results shownin Table 2 and Fig. 4(c).

The values representing the recovery, stretching and bendingbehavior of cloth samples can be obtained by the followingformula:

nðSmÞ ¼ Snm=Sini

m ,

nðLmÞ ¼ Lnm=Lini

m ,

nðHmÞ ¼Hnm=Hmax

m ,

8>><>>:

ð1Þ

where Sinim is the initial projected area of each cloth sample and Sn

m

is the average projected area of the deformed cloth sample. Linim is

the initial diagonal length of each cloth sample and Lnm is the

average diagonal length of the entire tested sample. For thebending measurement, Hmax

m represents the maximum bendingangle height of the wrinkles on each cloth sample and Hn

m is theaverage maximum bending angle height of the tested clothsamples. m and n indicate the numbers of cloth samples andmeasurements, respectively, with mA ½0,9�, nA ½0,9�. In total, 10recovery, 30 stretching and 10 bending tests were conducted foreach sample, for a total of 500 tests for all cloth samples.

This research is inspired by the goal of capturing real clothbehavior. Distinct from several other approaches, we develop theapproach of directly measuring the geometric parameters of

ts (b) and bending tests (c). These experiments yield the material properties of the

nts.

Real fabrics Material properties

nðSmÞ nðLmÞ nðHmÞ

5-Green woolen 0.9848 1.0488 0.999

6-Blue poplin 0.9883 1.0413 0.994

7-Orange ramie 0.9745 1.1242 0.9984

8-Gray fiber 0.9871 1.0613 0.9932

9-White satin 0.9833 1.0507 0.995

S−3 S−7 S−1 S−4 S−9 S−5 S−8 S−0 S−6 S−20.965

0.97

0.975

0.98

0.985

0.99

0.995

v(Sm

)

Sequence of cloth samples

Recovering measurement

interpolant−splineresilience variation

S−0 S−4 S−1 S−3 S−5 S−9 S−6 S−8 S−2 S−71.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

Sequence of cloth samples

Stretching measurement

v(Lm

)

interpolant−splineextensibility variation

S−8 S−0 S−6 S−9 S−2 S−4 S−3 S−7 S−5 S−10.993

0.994

0.995

0.996

0.997

0.998

0.999

1

1.001

Sequence of cloth samples

Bending measurement

v(H

m)

interpolant−splineflexibility variation

Fig. 4. Using more testing data, the fitting interpolation curves related to the recovery (a), stretching (b), and bending (c) can be obtained according to the appropriate

sequence for each cloth sample.

Device Bhat et al. [18] Wang et al. [19] Our Method

Data source

Theory foundation

Complexity

Cost

Accuracy

Data reusing

Biaxial tensile testers

Videos and images

Physically-based measurement

Geometric measurement

Motion capture system

Data-driven elastic model

Materials science Textile engineering

Geometric deformation

High Low

Weak Strong

Fig. 5. Comparison between our geometric measurement and the other main

measurement approaches to capture real cloth behavior.Fig. 6. (a) The initial cloth mesh with 441 vertices, 2480 half edges and 800 facets.

(b) The target mesh is the deformed result obtained by fixing the yellow points

and dragging the pink colored control vertex. (For interpretation of the references

to color in this figure legend, the reader is referred to the web version of this

article.)

L. Liu et al. / Computers & Graphics 37 (2013) 21–32 25

different types of cloth. Fig. 5 compares the mechanics of thesemethods.

4. Material-aware cloth simulation

Based on the material properties, our work defines a globalnonlinear energy minimization procedure that includes the vertexposition, the edge length and the dihedral bending terms. We usematerial learning in the simulation process by setting the weightsof the three energy terms according to the material-awaremapping. Then, we solve for the energy by assembling theseterms into a nonlinear optimization framework. The details of ourmethod are described in the following text.

4.1. Constrained geometric deformation

We formulate our constrained geometric deformation in termsof material resistance to recovery, stretching and bending. Thisresistance is described by the vertex position, edge length anddihedral bending scalar fields defined on the mesh.

At the beginning, we construct an initial cloth triangle meshM¼ ðV ,E,FÞ, where V denotes the set of mesh vertices, E describesthe set of mesh edges, and F is the set of mesh faces.V ¼ fv1, . . . ,vng, viAR3, i¼ 1,2, . . . ,n, and V 0 ¼ fv00,v01, . . . , v0ng,v0iAR3, i¼ 1,2, . . . ,n, represent the vertex set of the initial meshM and the target set M0, respectively. An example is shown inFig. 6(a) and (b).

L. Liu et al. / Computers & Graphics 37 (2013) 21–3226

Vertex position: The recovery stiffness is associated with themesh vertex position, affecting whether each individual vertexmaintains its wrinkles. Thus, after optimization, the final targetmesh M0 should be correlated with its initial mesh M due to thetesting results for maintenance of wrinkles in real cloth samples.The first approximation of energy can be obtained from

Ep ¼Xn

i ¼ 0

Jv0i�viJ2: ð2Þ

This energy term stipulates that the current vertices vi areapproximated to the target mesh vertices v0i. The differencebetween the current and target vertices can be obtained by theEuclidean distance. When the edge length constraint strategy isnot identified, only the elastic membrane effect illustrated inFig. 6(b) occurs, which does not conform accurately mimic thebehavior of natural cloth. Therefore, the explicit edge lengthconstraint is introduced to achieve a realistic cloth effect.

Edge length: A reasonable cloth simulation method should becapable of conducting the inextensible mesh deformation. Thestretching stiffness is associated with the mesh edges, reflectingthe extensibility of the cloth. Thus, the stretching scalar field isdefined by a set of energy defined on the mesh edges, which canbe obtained by the following minimization that resembles thatof the quasi-developable mesh interpolation presented byTang [35]:

El ¼Xði,jAEÞ

9‘ðe0ijÞ�‘ðeijÞ92, ð3Þ

where E is the edge set, ‘ðeijÞ is the edge length eij of the initialmesh M, and ‘ðe0ijÞ is the length of edge e0ij on the target mesh M0.The length of an edge eij is determined by its two verticesvi and vj. The length of edge e0ij depends on the other two verticesv0i and v0j. The numeric value of ‘ðeijÞ and ‘ðe0ijÞ can be determinedby the Euclidean distances ‘ðeijÞ ¼ Jvi�vjJ and ‘ðe0ijÞ ¼ Jv0i�v0jJ,respectively.

Dihedral bending: Many disperse bending energy models suchas Bergou’s isometric bending models [36] introduced the dihe-dral angle with regard to the quadric energy of vertex coordi-nates, but these methods are clearly scale-dependent. In ourmodel, we determine the dihedral bending energy from the localbending energy of Huang [32] and Popa [31]. According to thebending properties of different cloth materials, a geometricenergy term that can reflect material flexibility needs to be added.The dihedral bending term can be formulated as

Ea ¼Xði,jAEÞ

Jyðe0ijÞ�yðeijÞJ2, ð4Þ

where yðe0ijÞ and yðeijÞ are the magnitudes of the dihedral anglesbetween two adjacent triangles on the initial mesh M0 and thetarget mesh M, respectively. Given an edge eij, let vertices vk andvl be the third vertices in the two triangles that share eij (Fig. 2).We can calculate the magnitude yðe0ijÞ and yðeijÞ with ðrv0iþ

ð1�rÞv0jÞ�ðsv0kþð1�sÞv0lÞ and ðrviþð1�rÞvjÞ�ðsvkþ ð1�sÞ vlÞ.

The coefficients r and s are chosen to make yðe0ijÞ2yðeijÞ parallelto the normal of edge eij. This energy term can measure the localbending in a similar way as the other energy terms.

4.2. Geometric energy in cloth modeling

We obtain the final deformation energy function by combiningthe above three energy terms to model the deformation ofthe cloth mesh. The deformation process eventually reaches thecondition in which Ep, El and Ea are at minima. Therefore, thecloth simulation task becomes a constrained geometric deforma-tion. We aim to solve the following global nonlinear energy

minimization problem:

EðM0Þ ¼ arg minVðaEpþbElþgEaÞ: ð5Þ

It is difficult to obtain an explicit expression for the functionEðM0Þ in Eq. (5). The vertex coordinates v0i and v0j on mesh M0 withregard to Ep, El and Ea are unknown variables, e.g., the ‘ðe0ijÞ in Ep isan unknown function of the vertex coordinates of v0i, v0j on meshM0. Thus, we can optimize this objective function through numer-ical solution, and the variables in the objective function EðM0Þ areonly v0i, v0j, v0k and v0l. The optimization of the vertex coordinates ofthe mesh M0 uses an iterative optimization scheme instead of anonlinear optimization. To solve the stability and efficiency, weensure that the nonlinear conditions in terms of coordinatedifferences are linearized and the step length is small enough.

4.3. Material learning

Unlike other dynamical adjustment strategies [32,35] in whichthe weight parameters are optimized to determine the progress ofthe geometric energy reduction, the three weights a, b and g informula (5) are determined from the real cloth using a quadraticfitting method. We model the deformation effects of cloths withdifferent material properties by setting the scalar weights in frontof the vertex position, edge length and dihedral bending terms inthe geometric deformation energy. Our goal is to obtain theweight parameters with respect to the resilience, extensibilityand flexibility properties of cloth materials, that is, the scalarweights for the three energy terms are learned from the databased on some measurements with various types of cloth. Ourmethod of automatically acquiring the material properties fromthe measured data is described below.

Given the mean values nðSmÞ, nðLmÞ and nðHmÞ of the geometricproperties based on the real material measurements, we hypothe-size that the maximum (minimum) in the test data set of eachcloth sample is not greater than (less than) the maximum(minimum) value in the training data set of each cloth sample.We first use the normalization method to map all the data from0 to 1, mainly due to the inconsistency among units and theinconvenience of data processing. The normalized results of therecovery tests for each cloth sample are as follows:

jðSmÞ ¼vðSmÞ�min vðSmÞ

max vðSmÞ�min vðSmÞ: ð6Þ

Similarly, the normalized results jðLmÞ and jðHmÞ of thestretching and bending tests for each cloth sample are presentedin Table 3. Using the first normalized results, we can capture thecorrelation between the three weights a, b and g for the threeenergy terms as well as the geometric variations in terms ofprojected area, length and height. The model generates fittedconfigurations that are sufficiently well controlled to enable thedeformation framework to model different cloth materials. Withthe further normalization, a correlation can be achieved byquadratically mapping the weights in ½0,1� based on the learnedresults jðSmÞ, jðLmÞ and jðHmÞ as follows:

a¼ jðSmÞ

jðSmÞþjðLmÞþjðHmÞ,

b¼jðLmÞ

jðSmÞþjðLmÞþjðHmÞ,

g¼ jðHmÞ

jðSmÞþjðLmÞþjðHmÞ:

8>>>>>>>><>>>>>>>>:

ð7Þ

The weights are calculated as percentages that always sum to 1,so we obtain the relative ratio between the three weights shown inTable 3. The three weights a, b, g were determined from real dataon the recovery, stretching and bending resistance of each cloth

85

87

89

91

93

95

Sample_0 Sample_1 Sample_2 Sample_3 Sample_4 Sample_5 Sample_6 Sample_7 Sample_8 Sample_9

shape-1 shape-2 shape-3 shape-4 shape-5 shape-6 shape-7 shape-8 shape-9 shape-10

Cloth samples

Fig. 7. The accuracy of matching between the simulated results for each cloth sample and the real photographs captured during the material evaluation. The simulated

results match the real cloth behavior very well in the range of [85%, 95%].

Table 3

The relative ratio between the three weights a, b and g for each cloth sample is learned from the normalized data jðSmÞ, jðLmÞ and jðHmÞ with respect to the mean values

nðSmÞ, nðLmÞ and nðHmÞ.

Sample 0 Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Sample 7 Sample 8 Sample 9

jðSmÞ 0.9370 0.3156 1.0001 0.0001 0.4418 0.7671 0.9370 0.2671 0.8787 0.6943

jðLmÞ 0.0001 0.0456 0.7794 0.1240 0.0088 0.2702 0.1976 1.0001 0.3912 0.2886

jðymÞ 0.0165 1.0001 0.5903 0.8526 0.8084 0.9509 0.1312 0.8526 0.0001 0.2952

Real fabrics The scalar weights Real fabrics The scalar weights

a b g a b g

0-Red silk 98.26 0.01 1.73 5-Green woolen 38.58 13.59 47.83

1-Blue white dot canvas 23.19 3.35 73.46 6-Blue poplin 74.02 15.61 10.37

2-Blue washed star denim 42.2 32.89 24.91 7-Orange ramie 12.6 47.18 40.22

3-Apricot linen 0.01 12.7 87.29 8-Gray fiber 69.19 30.80 0.01

4-Blue nylon 35.23 0.7 64.06 9-White satin 54.32 22.58 23.1

L. Liu et al. / Computers & Graphics 37 (2013) 21–32 27

sample shown in Fig. 4. For example, the relative ratio b of silk andnylon indicate that these materials are more extensible than othermaterials. Apricot linen has less resilience than other materials,and gray fiber has less bending resistance. In addition, based on theanalysis of the data in Table 3, we would like to discuss a specifictype of material with a specific configuration of these weights.Combining these weights with other positional constraints andmeshes enables the realistic modeling of cloth behavior. To verifythe computational weights based on the real measured data, wetest each specific configuration of the weights in our geometricdeformation. Algorithm 1 gives the parameters set by material-aware mapping.

Algorithm 1. The material-aware mapping scheme.

Input: Snm, Ln

m, Hnm: the measured data for each cloth sample, m:

cloth samples, n: tests

Output: a, b, g: the scalar weightsBegin

1: for m¼0 to 9 do normalize the data 2: for n¼0 to 9 do the values based on testing data 3: nðSmÞ’Sn

m=Sinim

4:

nðLmÞ’Lnm=Lini

m

5:

nðHmÞ’Hnm=Hmax

m

6:

end for 7:

jðSmÞ’vðSmÞ�min vðSmÞ

max vðSmÞ�min vðSmÞ

8:

jðLmÞ’

vðSmÞ�min vðLmÞ

max vðLmÞ�min vðSmÞ

9:

jðLmÞ’

vðSmÞ�min vðHmÞ

max vðHmÞ�min vðSmÞ

10:

end for 11: for m¼0 to 9 do the relative ratio between three

weights

12:

a’ jðSmÞ

jðSmÞþjðLmÞþjðHmÞ

13:

b’ jðLmÞ

jðSmÞþjðLmÞþjðHmÞ

14:

g’ jðHmÞ

jðSmÞþjðLmÞþjðHmÞ

15:

end for 16: return End

To further demonstrate the effectiveness of our approach, wedesign a user investigation with subjective experiments. Usingthese fitted parameters with other positional constraints andmeshes enables the modeling of 100 final shapes of each clothsample. In total, there are 1000 deformed shapes of all clothsamples and 10 shapes that are most similar to the real photo-graphs of each cloth material. Therefore, we administered somequestionnaires to 100 people (50% men and 50% women rangingin age from 18 to 78 years old) and asked them to evaluate howrealistic the 10 selected shapes for each cloth sample appeared toobtain an accuracy rating capturing how well these shapes meetexpectations. As shown in Fig. 7, the specific configuration of theweights was closely correlated with perceptions reported in thequestionnaires, with an agreement above 89%.

4.4. Numerical solution

Nonlinear optimization is generally performed using iterativeschemes. We iteratively linearize the objective function withrespect to variables and changes in a nonlinear least squaresproblem to convert it to a linear problem. We conduct secondaryunfolding of the objective function near the current approximatesolution at each step of the iteration. To obtain the optimizeddeformation result in a short time, and based on the method

L. Liu et al. / Computers & Graphics 37 (2013) 21–3228

raised in the literature [35], we use the nonlinear D-value factorin linear coordinates to ensure a suitable convergence rate andstep length. There are 3�n variables in Eq. (5). The correspondingcoordinates of the n vertices in vertex set V are ðxi,yi,ziÞ. Thesevariables are expressed by the column vector of X ¼ ½x1,y1,z1,x2,y2,z2, . . . ,xn,yn,zn�

T . Hence, the three geometric energy terms areexpressed as geometric constraints on X:

mðviÞ ¼ v0i�vi ¼ 0, iA ½1,n�,

dðeÞ ¼ ‘ðe0ijÞ�‘ðeijÞ ¼ 0, i,jA ½1,n�,

fðvivjÞ ¼ yðe0ijÞ�yðeijÞ ¼ 0, i,jA ½1,n�:

8>><>>:

ð8Þ

All terms on the left side of Eq. (8) are functions of X. These arehighly nonlinear equations, and we solve the resulting linearsystem based on the least squares method. Supposing a givenfunction f(x) in the current iteration, we need to update its valuef ðx0þeÞ to minimize the minimum the change in e. We cancalculate f ðx0þeÞ via a Taylor expansion at e : f ðx0þeÞ ¼ f ðx0Þþ

ef 0ðx0ÞþOðe2Þ.For an edge eij on the initial mesh M, its length is determined

by its two vertices vi and vj, with coordinates denoted as ðxi,yi,ziÞ

and ðxj,yj,zjÞ. We can calculate ‘ðeijÞ ¼ Jvi�vjJ, which is linearizedby a Taylor expansion as

‘ðDðxi�xjÞÞ ¼ ‘ðeÞþðxi�xjÞ

‘ðeÞðDxi�DxjÞþOð9Dðxi�xjÞ9

2Þ: ð9Þ

All the equations in Eq. (9) are linear functions of Dxi and Dxj

omitting the second order. Setting these three functions to zeroand moving the constant terms to the right-hand side, weconstruct the following sparse linear system of equations:

AX ¼

a½P�b½L�g½A�

0B@

1CA X ¼ b, ð10Þ

where X ¼ ½Dx1,Dx2,Dx3, . . . ,Dxn�, P is an n1 � 3n matrix (n1 is thenumber of vertices in V), L is an n2 � 3n matrix (n2 is the numberof edges in E), A is an 3m� 3n matrix, and b is an ðn1þn2þ3mÞ � 1column vector. The linear system is over-determined and we solve

Fig. 8. Draping cloth simulation: the initial piece of rectangular cloth (a) with red contro

and dihedral bending energy. (c) and (d) exhibit the deformation effects from the mem

marks, appearing prominently when the geometrically based energy model is optimize

referred to the web version of this article.)

it in the least squares sense

X ¼ ðAT AÞ�1AT b, ð11Þ

the matrix AT A represents a sparse matrix, and the sparse linearequation system can be computed through the Cholesky decomposi-tion of AT A, that is, AT A¼ RT R, where R is an upper-triangular sparsematrix. After the decomposition, X is obtained through back substitu-tion. The pseudocode of the iterative simulation strategy to clothmesh deformation is given in the table of Algorithm 2.

Algorithm 2. Cloth deformation scheme.

l points

brane t

d. (For

Input: M: initial mesh, l: iterations

Output: M0: target meshBegin

1: if (Termination) then Exit 2: else do Algorithm 1 determine the three weights a, b, g 3: for n¼0 to l do 4: M’ variable of vi coordinates on initial mesh 5: M0’ variable of v0i coordinates on target mesh

6:

‘ðeijÞ’ Euclidean distance between vi and vj

7:

‘ðe0ijÞ’ Euclidean distance between v0i and v0j 8: yðeijÞ,yðe0ijÞ’ construct the dihedral angle

9:

end for 10: for n¼0 to l do 11: v0i�vi linearize vertex position energy

12:

‘ðe0ijÞ�‘ðeijÞ linearize edge length energy

13:

yðe0ijÞ�yðeijÞ linearize dihedral energy

14:

X ¼ ðATAÞ�1ATb least squares numerical solution 15: Mn

¼Mnþ1

16:

end for 17: return End

The total energy function for cloth simulation can obtainphysically plausible effects of real fabrics, though this simulation

and green fixed vertices. The membrane effect (b) ignoring the edge length

o the cloth, and the folds and wrinkles are shown with yellow rectangular

interpretation of the references to color in this figure legend, the reader is

L. Liu et al. / Computers & Graphics 37 (2013) 21–32 29

is based on the geometric deformation method. The majorreasons for the accurate results are described here. The entireiterative process can be simulated by dragging and wrinkling eachshape in ways that correspond to realistic cloth deformation. Thedeformation iteration gradually decreases as certain vertices onthe cloth mesh are dragged from one direction to another untilthe energy function is minimized, representing the finaldeformation shape.

5. Experimental results and discussion

This section provides several examples used to assess theperformance of the simulation. We have implemented our pro-posed cloth simulation method and applied it to simulate varioustypes of cloth. All the tests were performed on a PC with an IntelCore Duo 3.1 GHz CPU, an Intel G41 Express Chipset, 4 GB DDR3Ram, Visual Studio 2010 software and a Graphite [37] platform.

The first example illustrates the effectiveness of the threeenergy terms for cloth simulation: vertex position, edge lengthand dihedral bending (Fig. 8). We input the initial square meshshown in Fig. 8(a) with the fixed vertices colored green aroundthe four corners. Lifting the red colored control vertices for directmesh deformation without any detail preservation produced themembrane deformation effect shown in Fig. 8(b). As a compar-ison, Fig. 8(c) shows the final shape of the cloth with the samefixed and control vertices. The vertex position and edge length

Fig. 9. Simulation behaviors with different resolution and triangulation surface meshes

48 vertices shown in (a) and the effect with more wrinkling details (b) when input the m

same initial mesh (329 vertices) exhibit similar deformation results shown in the two

Ram

ieP

oplin

Fibr

e

Fig. 10. Simulation results with the fitted parameters to model cloth behaviors. Differe

final deformed shapes of the cloths shown in each figure (right) from (a) to (f). Compare

ability of the simulation results to model similar realistic cloth behaviors using the res

energy terms enable penalties for shrink and edge stretch, andmake the cloth effects possible. Fig. 8(d) shows the optimizationresult with different wrinkling details when dihedral bendingenergy is added to the deformation.

To test the influence of the mesh triangulation and samplingon the cloth simulations, we make a comparison using the twogroups of examples including the two notable different resolutionmeshes which are both regularly tessellated, the two meshes withregular tessellation and irregular delaunay triangulation, respec-tively. Based on the same configuration of the weight parameters,we set up the same control vertices in the blue rectangular regionon the input different meshes. The deformation effect (Fig. 9(a))lacks the apparent wrinkling details compared to the mesh shownin Fig. 9(b), which illustrate that the notable discrepancy ofresolution on input meshes can affect the deformation details.However, we do not use the low resolution meshes with fewvertices in cloth simulation to model the different wrinklingdetails. As shown in Fig. 9(c) and (d), the same control verticesshown in the blue rectangular region were set up on thetessellated and delaunay triangulation meshes, with observedbehavior varying widely depending on the parameter settings.When the configuration of weights is held constant, changes inthe vertices of the input mesh apparently do not affect thedeformation results of a simple input model.

Using different weight parameters can model various types ofmaterial behavior. To demonstrate the effectiveness of thelearned weight ratio on each cloth sample (Table 3), Fig. 10 shows

. With the same parameters setting, the deformed results of the initial mesh with

esh (441 vertices). The delaunay triangulation (c) and the tessellation (d) with the

insets, respectively.

Line

nW

oole

nS

atin

nt patterns of control vertices and their target positions are applied to obtain the

d with the real photographs shown in each figure (left), respondents evaluated the

ults for ramie (a), linen (b), poplin (c), woolen (d), fiber (e), and satin (f) materials.

Table 4Performance statistics of the input mesh and the average computing time for each

example. (Unit of time: s.)

Fig. Vertices Triangles Edges Initial Linear Solution

1 1714 3296 10,024 0.683 0.025 0.013

8 961 1800 5520 0.156 0.0032 0.016

10 2000 3840 11,680 0.863 0.0415 0.191

11 2601 2500 10,200 1.154 0.0485 0.245

12, 13 1361 2500 7650 0.373 0.0125 0.045

L. Liu et al. / Computers & Graphics 37 (2013) 21–3230

the comparisons between the real photographs and the finaldeformed shapes including ramie, linen, poplin, woolen, fiber,and satin materials. The model was also objectively evaluated(Fig. 7) in terms of the ability of the parameters determined fromreal measurements to model realistic cloth behavior.

In our algorithm, the weights of the three energy terms aredetermined from the measured data. A specific configuration ofthe three weights a, b and g can model a specific type of clothmaterial. Fig. 11 shows the simulation results for different clothmaterials, such as silk, denim, nylon and canvas. These photo-graphs demonstrate the real effects of four types of 460�400 mm2 cloths covering an 11 mm high semisphere 12 mm indiameter. Table 4 lists the data of the initial cloth mesh in thisexperiment. The different patterns of the control vertices and thedifferent parameter settings can be used to obtain, the final andstable deformed cloth shapes of cloths. The model is able tosimulate not only the soft material effects of silk and nylon butalso the hard materials, including canvas and denim.

In the next test, a clothing design example is used to illustratehow our constrained deformation method can be utilized tomodel cloth surfaces with material effects. There are someconstrained points (yellow colored) and control vertices (redcolored) on the original skirt. As shown in Fig. 12(a) and (b), theLaplacian and space-based methods cannot preserve the geo-metric features well because they are not designed to modelcloth. Fig. 12(c) and (d) shows the results of our algorithm usingthe same input mesh with the same manipulation. Instead of asmooth surface design, our method relies on nonlinear energyminimization to model similar anisotropic cloth behaviors. There-fore, this method should be compared with the other minimiza-tion approaches such as the method of [14] shown inFig. 12(e) with the same fixed and control points (Fig. 12(f)).Differently from this previously published approach, our methodconsiders the material properties of the real cloth. Using the threeweights determined by material measurement, the results withwool and denim are shown in Fig. 12(g) and (h), respectively.

The results of our cloth simulation are predicted by theiteration procedure. For a single frame, the major computationaldemand of our system involves the initialization of the three

Fig. 11. Simulation results for four types of cloth material using different configuratio

covering the hemisphere. With the same initial mesh, the deformed cloths with the co

denim and canvas are shown in (c) and (d) with different patterns of control verti

interpretation of the references to color in this figure legend, the reader is referred to

weights, the user interface response, the numerical solution andthe display of the cloth model. The duration of a deformationsession depends on two factors: the frame-rate, which indicatesthe computational cost of each iteration, and the number of totaliterations l required to reach a stable state. l is an importantfactor but is difficult to quantify because it depends on the shapeof the input model and the locations of the constraints. To ensurethe stability of the iteration procedure, the actual output of thel-th iteration is 15. As shown in Fig. 13, our solver converges fastand stably during the iteration procedure. For all models tested inthis paper, the procedure converges to a visually stable resultwithin approximately 15 iterations. Therefore, the satisfactoryresults can always be obtained at an interactive rate.

The experiments presented above contain different numbers ofvertices, triangles and edges on the input meshes. Table 4 lists thegeometric data of the specific meshes and the computing statis-tics of the test examples. The efficiency of the whole algorithmmainly depends on three factors. The first term is the initial timeto compute the three coefficients in front of the vertex position,edge length and dihedral bending energy terms, as specified inAlgorithm 1. This task occupies a large proportion of the totaltime, and we list the average time required to compute the threeweights for each iteration. The average linear times are also listedin Table 4, as we propose to improve our cloth deformationscheme by linearizing the nonlinear energy terms. The averagetimes for the least squares numerical solution specified inAlgorithm 2 are shown in Table 4.

ns of the learned weights. The real photographs show four types of cloth samples

ntrol vertices show the simulated results for silk (a) and nylon (b). The results for

ces and different parameter settings. The control vertices are colored red. (For

the web version of this article.)

Fig. 12. Comparison between our method and other approaches on an initial skirt mesh with yellow colored vertices as constrained points and red colored vertices as

control points. (a) The output of the Laplacian method [29] based on membrane-like effects with compression. (b) The result of a space-based deformation method [23] on

the local surface of an overskirt with weak preservation of the geometric details. The local geometric features are preserved in our method with different specific weights,

as shown in (c) and (d). (e) The effects related to the minimization method [14] on a skirt model. With the same model with 1361 vertices, fixed points and control vertices

in our method (f), different cloth behaviors between green wool (g) and blue washed star denim (h) can be obtained by altering the three weights a, b and g. (For

interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 13. The convergence curve of cloth deformation. Iterative draping of a simple

skirt model with the three weights capturing the material properties a¼ 12:6,

b¼ 47:18, g¼ 40:22. (a) is the original skirt model. This curve shows the variation

in the geometric energy over the iterative solution procedure. The horizontal axis

indicates the number of iterations. (b) shows results of the 3rd iteration. (c) and

(d) illustrate the results of 15th (final) and 45th iterations, respectively.

L. Liu et al. / Computers & Graphics 37 (2013) 21–32 31

6. Conclusion and future work

We have presented a new data-driven method for clothsimulation capable of modeling different materials. This novelmethod has several important advantages. First, we measure thematerial properties from real-world cloth samples without usingexpensive or complex devices, and the measured results aredirectly suitable for application to geometric deformation. Sec-ond, we use a global nonlinear energy minimization includingvertex position, edge length and dihedral bending terms corre-sponding to the properties of recovery, stretching and bending ingeometric measurements. Third, we propose a material learningapproach to determine the optimal weights of these energy termsbased on real measured data. The three weights can be easilyincorporated into our geometrically based cloth deformationmethod to generate distinctive wrinkles and folds, which are

often difficult to model in current simulation models that rely onmanual parameter setting. Finally, various simulations havedemonstrated that this method of material-aware cloth modelingis able to realistically simulate the different behaviors of a varietyof cloth materials.

Although our experiments show that the proposed simulationframework successfully models the behavior of various types ofcloth samples, it also has a few limitations that need to beaddressed in the future. First, we have only measured staticparameters. Our method is only able to model the cloth behaviorby finding the final shape and is not able to model its dynamicbehavior. Thus, this method is not suitable for cloth animation orfor varying vertices constrained through time to address colli-sions. Additionally, testing errors and the numbers of clothsamples inevitably affect the performance and behavior of clothmaterials with distinctive behavior. Our method only provides anapproximation of the material properties of resilience, extensi-bility and flexibility of the actual cloth samples.

We hope that the proposed method for cloth deformationbased on geometric measurements encourages further research incloth simulation along this new direction. In future work, wewould like to extend our cloth modeling to more cloth materialsand address the limitations in the current method as describedabove. More investigations and measurements are required, inparticular for extreme situations such as large-scale deformations.We are also interested in constructing data-driven collisionmodels for dynamic cloth behavior and thus extending ourmaterial-aware cloth modeling to more applications.

Acknowledgements

This research is supported by NSFC-Guangdong Joint Fund(U0935004, U1135003), the National Key Basic Research andDevelopment Program of China (973)(No. 2013CB329505), theNational Natural Science Foundation of China (61232011,61073131), the National Key Technology R&D Program (No.2011BAH27B01), the Industry-academy-research Project of

L. Liu et al. / Computers & Graphics 37 (2013) 21–3232

Guangdong (No. 2011A091000032), and the Innovative TalentsTraining Program for Doctoral Students of Sun Yat-sen University.

Appendix A. Supplementary data

Supplementary data associated with this article can be found inthe online version at http://dx.doi.org/10.1016/j.cag.2012.10.006.

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