Computer Graphics Techniques for Modeling Cloth - IEEE ...frey/papers/draping... · In computer...

Computer Graphics in Textiles and Apparel

odeling Cloth lexible objects are inherently more diffi- F cult to model than rigid objects. Until

about a decade ago, computer graphics was limited to modeling rigid objects, but subsequent advances in algorithms and graphics hardware now support the processing of flexible objects.

Nineteen cloth-modeling

techniques are summarized

and categorized by their

main theoretical method:

geometrical, physical, o r

hybrid. Recommendations

fo r future work consider the

different goals in textile

engineering and computer

graphics.

Cloth is one class of flexible objects that has received considerable attention in the computer graphics community. The appearance of clothing worn by modeled humans is of particular interest in animation. The textile and fashion industries also welcome tools that accurately generate the shapes of cloth objects and link them with CAD/CAM systems.

Cloth is normally woven, giving it special properties that generate interesting shapes when draped. These shapes have always interest- ed textile scientists (for excellent surveys of this work, see Breen’,’). Textile science has concentrated on studying a material’s behavior according to its mechanical properties. As a result, current methods for measuring the physical properties

of cloth are very sophisticated. The computation of shape based on these mechanical properties, however, is expensive.

In computer graphics or animation, appearance is generally more important than physical accuracy, so the emphasis has been more on visual realism than on physical accuracy. The textile and apparel industries, however, require both.

Nisse l~on~,~ discussed early applications of computer graphics in the fashion industry. In this survey, we present a contemporary overview of cloth-modeling techniques and conclude by speculating on future research directions that could optimize the agreement

Hing N. Ng and Richard L. Grimsdale University of Sussex

between the requirements of visual realism and physical accuracy.

Cloth-modeling techniques are usually classified into three categories: geometrical, physical, and hybrid. The techniques within each category do not follow well- defined patterns. We therefore generally report each work independently according to the chronology of pub- lication. Figure 1 presents an overview of the techniques to be discussed. At the end of the discussion of all techniques, we summarize their features in a table.

Geometrical techniques Geometrical models do not consider the physical

properties of cloth. Rather, they focus on appearance, particularly folds and creases, which they represent by geometrical equations. Geometrical techniques require a considerable degree of user intervention; they can be regarded as a form of advanced drawing tool.

The work of We;/ Weil’ was probably the first to apply geometrical tech-

niques to cloth visualization in computer graphics. He represented hanging cloth as a grid of points and simulated its shape by fitting catenary curves between the hanging or constraint points (see Figure 2).

A catenary curve is the natural shape of a hanging heavy cable and has the form

y = a cosh(x/a)

where a is the scaling factor. If two catenary curves intersect when projected on the x-z plane, as in Figure 2, a new resultant curve should be determined. Because this would require an extra computation, Weil simplified the process by removing the lower curve.

At this stage, his model consists only of a curved wire- frame that approximates the cloth’s structure. A proper simulation of the shape requires the addition of surfaces, which are generated by subdividing the triangles formed by the catenary curves connecting the constraint points.

Figure 3 shows how the surfaces, and thus details of

28 September 1996 0272-1 7-1 6196/$5.00 0 1996 IEEE

the shape, are generated by subdividing triangles. New catenary curves are fitted between these new triangles, and folds are subsequently produced by relaxing every point until an arbitrary offset position value is reached. Prior to rendering, spline curves are fitted to the surface of the cloth to make it smoother. Finally, the cloth is represented as a set of cylindrical objects and ren- dered by ray tracing.

Although Weil did not mention execution speed in his paper, we believe it would be fast, excluding rendering time, since it involves no complex numerical methods. The main limitation of the model is that it simulates onlyhanging cloth. And, of course, as a geometrical technique, it does not simulate the properties of real cloth.

The work of Agui et a/. Agui et aL6 presented a geometri-

cal method for modeling a sleeve on a bending arm. They represented the cloth as a hollow cylinder con- sisting of a series of circular rings, indicated as Ri in Figure 4 on the next page. They observed that folds form as a consequence of the differ- ences in curvature between the

Weil (1 986)

Agui et al. (1 990)

Hinds et al. (1 990, 1991, 1992)

Ng et al. (1 995) 1 Overview of cloth-modeling work.

Rudomin (1 990)

Kunii et al, (1 990)

Hybrid Taillefer (1 990)

Tsopelas (1 991)

Dhande et al. (1 993)

Feynman (1 986)

Terzopoulos et al. (1 987, 1988)

b Ng et al. (1 995)

Sakaguchi et al. (1 991)

Thalmann team Magnenat Tahalmann et al. (1 991) Lafleur et al. (1 991) Cariganan et al. (1 992) Yang et al. (1 992, 1993, 1994) Werner et al. (1 993) Magnenat Thalmann (1 994) Volino et al. (1 994, 1995)

Li et al. (1 993, 1994, 1995, 1996) Aono (1 990)

Breen et al. (1 992, 1993, 1994), Anderson (1 993)

Okabe et al. (1 992)

Provot (1 995)

inner and outer part of the bent sleeve. Figure 5 shows two cases, before and after bending.

L(i, j) is the distance betweenPij and&+ Deformation of the cloth-a fold-occurs when dL(i, j ) is kept to a minimum, where dL(i, j) represents the difference in distance before and after the deformation and is defined as

Using this idea, folds are simulated by transforming the circular rings into polygons. Figure 6a shows an undeformed ringR, on thex-y plane. The shaded area represents the cross section of the arm. The process is simplified by assuming that R, and the resulting polygons are symmetrical about they-axis.

In the first step, the process locates the point Na with

Step 1

Y

Step 4

IEEE Computer Graphics and Applications

2 Twocatenary curves.

3 Subdivision of triangles t o generate surfaces and details of shape.

29


4 Cloth sleeve represented as a hollow cylinder.

5 Asleeve section (a) before bending and (b) after bending.

6 The process of forming a polygon: (a) undeformed ring R;, (b) transformed to compute a sleeve fold.

7 Layered structure for associating cloth to skin and skeleton for animation.

30

&(i, j) set to greater than a predetermined threshold &,. The region with j < Na remains unchanged. The process continues for N1 with threshold dL1 and for N2 with threshold dLz where dL2 > dL,. Based on the

X

assumption that real folds are always flat on the top part of the sleeve, a straight line parallel to the x-axis is always assumed (indicated as the segmentbetweenj = J/2 and j = N2 in Figure 6b).

This process of creating polygons is applied to every ring on the sleeve. The resulting acute corners must then be smoothed by repositioning each point to be at the average of its four neighboring points.

The simulation described in Agui et a1.‘ targets just the specific application of a folding sleeve. They do not state how they determine thresholds.

The work of Hinds et a/. Hinds et al.7 targeted the automation of garment

manufacture. They obtained the shape of the human body by digitizing the upper torso of a mannequin. In their system, a garment is represented by a number of 3D surfaces (panels). A 3 D interactive editor allows the garment designer to position a panel on the body.

The panel is defined by its edges, and these are specified as offsets from the body. All remaining points of the panel are then generated by interpolation. e conti- nuity is maintained between the panels forming a garment by choosing the interpolation technique carefully. The designer can add textures and folds to achieve further realism. A fold is introduced as a harmonic function superimposed on the panels.

Hinds et al. addressed the early design stage of manufacturing in their early work. In later work,’ they reported how to translate the 3D panels to 2D patterns by using Calladine’s method.’ They returned later to refinements of interactive garment design.”

Geometrical work of Ng et a/. Intending to provide an animation tool that would

quickly produce images of clothed objects, Ng et a1.”J2 proposed a modeling technique that closely associated the cloth layer with the shape of the underlying skin layer. The two layers consist of a series of sections with identical numbers of vertices on each layer. Figure 7 shows the data structure.

To support the important role that folds play in giving cloth a realistic appearance, an algorithm was devised to generate them by using gaps between the cloth and skin layers. The algorithm is based on the idea that folds are likely to occur where the cloth is slack.

Once the position of a fold line has been nenerated. sinusoidal func-

September 1996

- tions are mapped along the line to mimic natural folds (see Figure 8).

The technique incorporates an interactive editor called Geoff (Geometrical Editor for Fold Formation), which gives the user considerable control over fold gen- eration. Folds also occur as a result of the underlying structural formations. For instance, when an arm is bent, folds are likely to occur on the sleeve near the bend. Therefore, a set of rules was developed to generate fold lines automatically. The user

can modify these fold lines via the editor. Attributes are allocated to each fold line to support individual patterns and simple animations. The technique is very fast since there are no complex equations to be solved. ....

Physical techniques We turn now to physical techniques. The techniques

in this category represent cloth models as triangular or rectangular grids, with points of finite mass at the inter- sections. The forces or energies of points on the cloth are calculated in relation to other points.

The formations or the number of neighboring points vary according to the technique. Energy-based techniques calculate the energy of the whole cloth from a set of equations and determine the shape of the cloth by moving the points to achieve a minimum energy state. Force-based techniques represent the forces among points as differential equations and perform a numerical integration to obtain the point positions at each time step. In general, energy-based techniques are used to produce static simulations while force-based techniques are used in dynamic simulations.

Physical techniques generally produce the behavior of a flexible object that resembles cloth. The degree of this resemblance varies from technique to technique.

The work of Feynman Feynman13 proposed a physical method for modeling

draped cloth. He represented the cloth in 3D space by a 2D grid, as shown in Figure 9. Observing that the equilibrium shape of the draped cloth is obtained when its energy is a minimum, he then derived an energy equation from the theory of elastic plates:

where k,, kb, and kg are elasticity, bending, and density constants, respectively. The energy terms included are the elasticity energyEeimt, the bending energyEbend, and the gravitational energy$,. The energy at point PIJ is calculated in relation to the eight surrounding points. The steepest descent method is used in searching for the energy minima. A multigrid technique accelerates the process.

Feynman presented simulations that included hanging clothes and cloth draped over a sphere. No simulation times were reported.

The work of Terzopoulos et a/. Terzopoulos et al.14,15 introduced a deformable model

intended for generalized flexible objects, but the model has been used in several cases for cloth visualization. Referring to Figure 10, the position of a point on the body is represented by

where a is a point in the intrinsic coordinates of the body. In its undeformed state, the body can be represented by

Profile function J /Fold function

a2 Deforming body m

Pi,j

i -

P n d e f o r m e d body

ro

J rl

Lagrange's equations of motion are used to obtain

d2r h at2 at

p- + y - + Q(r) = f (r, t )

where p is the mass density, yis the damping density, &e is the internal elastic force that counteracts deformation, and f(r, t) is the net external force. The energy function e(r) for a deformable curve is

E@)= 1G-G' 12,+IB-Bo1$dalda2 I n

where I . I u and I . Ip are weighted matrix norms. The metric tensor G is defined as


8 Mapping process to determine fold lines and generate folds along them.

9 A 3D cloth represented as a 2D grid.

10 Definitions of the deformable model of Terzopoulos et al.

31


11 (a) Cloth represented as a grid and (b) point Pi,j

with its four neighboring points.

12 Three main parts of the Thalmann system: (a) 2D panel edit, (b) 3D seaming, and (c) 3D animation.

32

d r d r Gij(r(a)) = aai dai

The curvature tensor B is defined as

d2r Bij(r(a)) = n.- aaiaaj

where n is the unit surface normal at point a. To obtain the solutions, Equation 1 is first discretized

by a finite-difference or finite-element method, resulting in a large system of simultaneous ordinary differential equations. By integrating the equations through time, the model simulates the dynamics at each time step. The model is numerically expensive.

The work produced simulations of a waving flag, a carpet draped over various objects, and a sequence in which a piece of cloth is torn apart.

The work of Aono Aono16 used elasticity theory and DAlembert’s prin-

ciple to produce a wrinkle-propagation model. In his model, cloth is viewed as a medium that transmits wrinkles. A basic model is first produced by making the following assumptions:

In the initial state, the cloth is homogeneous, isotropic, and linearly elastic.

w By using D’Alembert’s principle, the cloth is in an equilibrium state at all times under the given applied forces.

w The cloth never stretches or shrinks along the surface normal direction.

Using the first assumption and the stress-strain equation, Aono obtains the following equation:

t ’ pi,j+l

I Lab

September 1996

b- Body

E _ 1 E

V _ _ E

0

E

E v _ _

1 E -

1 2G -

where eU is the strain, z$ is the stress, E is the Young’s modulus, vis the Poisson’s ratio, G is the modulus of rigidity, andG =E/(2(l+v)). Usingthesecondassump- tion and introducing damping, Aono obtains the next equation:

(3) %+++-+A az, az, =p-u,+c,-u, a2 a

at2

where5 is an applied force along thex, y , or z direction, cl is the damping constant, ul is the displacement and p is the density of the cloth. By using the strain-displacement equation, E~ = -((auJ/ax,) + (Wax,)), and elimi- nating z, from Equations 2 and 3, the following is obtained:

a d2 a &I at2

GV2u, + (A + G)-e + f, = p-U, + C, -u, (4)

where il= Ev/((l- 2v)(1 + v)) is the Lam& constant. This is a modified form of the equation for a wave prop- agating in an elastic continuum.

From the third assumption, no displacements occur along the z-axis. Therefore,

(5)

where u, v, and w are the displacements along the x, y , and z axes, By substituting Equation 5 into Equation 4, the equation for the intrinsic model is

-1 d d“ d GV;,p, + (A + G)-e,,y + f, = p-U, + C, -u, &I at2 at

where V:, = a2i&? + a2iayZ and e,,=au/ax + av/&. Aono then modified this intrinsic model to include

anisotropy and viscoelasticity. He considered two types of anisotropy, global and local. Global anisotropy affects the whole shape. Aono included it by adjusting the ratio of the Young’s modulus to the modulus of rigidity. Local anisotropy affects the shape of the fibers and is included by employing different displacements uxx along the warp and weft directions.

Aono modeled the viscoelasticity effect by a system of springs and dashpots connected in parallel. Finally, he compared the modifica- tions to the original model.

kBody (4

Aono’s model concentrated on the propagation properties of a wrinkle on a piece of cloth under an applied force. The cloth acts as the transmitting medium. The model does not consider draping or interaction with the underlying object.

The work of Sakaguchi et a/. This team from the Kyoto University of Japan devel-

oped a system called Party17 to simulate clothes. The system used a grid to represent the cloth, as shown in Figure 11. The simulation was based on Newton’s law of dynamics:

where r is a point on the grid, m is the mass of that point, and F,, and Pint are respectively the external and internal forces acting on the point.

The internal forces included are

Fint = Fspring + Fvisco + Fplas

The components of the internal forces are based on Terzopoulos’ deformable model,15 where Fspring is the force due to the elongation of a spring, Fwsco is the force due to viscosity, and Fplas is the force due to plasticity.

Sakaguchi et al. determined the velocity and position of the cloth by using Euler’s method. They also considered the interactions of the cloth with the underlying human body. They based collision handling on the conservation of momentum. Since the cloth and the human body are represented by manyvertices, checking for collisions would be a very costly computational task. They sped up the process by using a finite bounding volume to limit the amount of space involved in a collision detection. The responses to collision detection included con- sideration of friction.

They presented simulations that included a cloth hanging from a pole and blowing in the wind and a dress on a model of a human body. No simulation times were given.

The work of the Thalmanns and their team The Thalmann research team has addressed cloth

visualization problems in an approach analogous to the production of a garment by a tailor. Their system consists of three main parts, shown in Figure 12. The garment is first represented as a polygonal panel (Figure 12a). Then its edges and vertices are seamed together in 3D (Figure 12b). Finally, the animation addresses the interaction between the body and the garment (Figure 12c).

The Thalmanns’ first paper” briefly surveyed cloth modeling, particularly comparing the models of Terzopoulos and Aono before describing their own work, which involves dressing synthetic actors. They focused on how to manage the interaction between the garment and the body.

Lafleur et a1.I9 investigated collisions between the garment and the body. This work recognized that the Moore-

NO

P surface

Wilhelms collision method” for dealingwith rigid objects did not apply to flexible objects such as cloth. Therefore, Lafleur et al. proposed a new method for creating a volume force field around the cloth surface.

Consider the triangle Po-P1-P2 in Figure 13-one of many triangles that make up the surface of a cloth. The normals (NO, N1, Nz) at each vertex of the triangle form a force field. When a point enters this volume force field, the Lafleur method applies a repulsive force so that, in the next step of the calculation, the point moves away from the cloth triangle.

Animation must handle many collisions between the cloth and the body and among the triangular facets in the cloth itself. This can be very time-consuming, since the cloth and body are normally represented by many small triangular facets. Yang et al.’l gave a more comprehensive treatment that increased its speed by using a bounding box around the facet and by employing an octree structure to search for facets that might intersect.

Carignan et a1.” presented the method of determin- ing the cloth deformation during animation and used the following dynamic equation:

where

Equation 6 is based on Terzopoulos’ deformable model with Equation 7 substituted for the original scalar term. Carignan et al. eliminated the bouncing effect by using the collision-handling method presented in Lafleur.19 An inelastic collision results from using conservation of momentum to determine self collisions and collisions between the cloth and the body. In addition, by introducing extra force constraints to the garment panel and the human body, this approach can achieve the effects of seaming and fixation of the garment to certain parts of the body.

The Thalmanns’ system required a special user interface.23 The immediate application of their work is in the garment ind~stry.‘~ A recent paperz5 summarized the work and outlined the possibility of using the system in a virtual environment.

Instead of Equation 1 proposed by Terzopoulos et Yangz6 used Newton’s law of motion


13 Force field around a cloth surface.

33


14 Compo- nents of F(r, t).

15 f i , j and its neighbors.

16 Cloth represented as triangles.

34

f . . .

fexternal

friction

fcurvature-elasticity

fcurvature-viscosity

F(r,t)

ti I

j+2

j+l

i

j-1

j-2 i

k 2 i-1 i i+l i+2

(8) d2r

F(r,t) = m(r)- At2

where F is the resultant force at the point r. The components of this force are quite comprehensive (see Figure 14).

Yang introduced thefcurva~re-viscosi~ force term to simulate the viscose nature of the cloth and computed the internal force on a point in relation to the 12 neighboring points (Figure 15). By applying Euler's method to Equation 8, he calculated the velocity and position of the cloth at a certain time. By employing a bounding box and an efficient hierarchical data structure, he reduced the time needed for collision detection. Collision response was calculated using inverse dynamics and conservation of momentum.

Yang also realized that the normal constraints used for rigid objects may not be suitable for flexible objects. Therefore, he developed special constraints for attach- ing and seaming the cloth and smoothing the seams.

September 1996

He produced several impressive computer-generated animations to demonstrate the capability of the technique. The computation time to calculate one frame of animation, in which a cloth is represented by a 25 x 40 grid, is about 20 minutes on an SGI Indy workstation. Yang indicated that 40 percent of the total computation time is spent handling collisions.26

Later, Volino et aLZ7 used elasticity theory and Newtonian dynamics to simulate deformable objects, including cloth. They represented cloth as a set of particles arranged as vertices of many small triangles (see Figure 16). Their model takes atwo-pass approach. The first pass considers interparticle constraints and external constraints such as gravity and wind. The second pass determines collisions by using the law of conservation of momentum.

The approach ofVolino et al. is similar to that of Breen et al. (next section), but in the reverse order. The interparticle constraints are derived from the classical elasticity theory of an isotropic surface. The force due to the stresses at the edgej of the ith triangle is

where Tis the thickness, L, is the length of the edge j, m, is a unit vector perpendicular to the edge j, o, and Tare the stresses in the corresponding direction. The force on the edge is then translated equally to the vertices (or particles) joining this edge.

Volino et a1.28 also improved collision detection in the Thalmanns' system. By exploiting the geometrical regularity of the cloth surface and representing it hierar- chically, the time to detect collisions became roughly proportional to the number of colliding polygons, rather than the total number of polygons. Several cloth simulations of one minute were developed, and the time to produce them was on the order of hours on an SGI workstation.

The work of Breen et a/. Breen et al.1329 used particles3' to model the draping

behavior of woven cloth. This method treats the cross- ing points of the warp and weft threads as particles (see Figure 17). The Breen et al. simulation was in two stages. In the first, particles are allowed to fall freely (see Figure 18a). Any collisionwith the object or the ground is determined during this step. The particle positions can be obtained using the equation

m a + cv = m g

where a is the acceleration, v is the velocity, m is the mass, c is the air resistance, and g is the gravitational acceleration constant.

The result is a rough shape of the draped cloth. In the second stage (Figure 18b), an energy mini-

mization process is applied to the interparticle energy functions to generate fine detail in the shape of the cloth. The total energyEtorallJ of the particle P,, is defined as

where Erepellj iS the repulsion energy among particles, EscretchIJ is the stretching energy, EbendIJ is the energy due to interthread bending out of the plane, EtrellIsIJ is the energy due to interthread bending in the plane, and EgrayltylJ is the gravitational energy. The final equilibrium shape of the cloth occurs when there is minimum energy over the whole of the cloth.

To search for the energy minima, Breen et al. developed an algorithm called stochastic gradient descent.’ SGD is an improved simulated annealing method to determine the global minima of a system, but it takes considerable time. The simulation time used in their examples ranged from three days to a week on an IBM RS6000 workstation. They believe that the message- passing environment, on which the model is built, imposes a large overhead.

The parameters used in calculat- ing the energy terms are determined on an ad-hoc basis. In a 1994 Siggraph paper,3’ Breen et al. used piecewise polynomial and approximation techniques to derive the bending energy termEbendlJ and trel- lis energy term Ecre~,,, from empiri-

Warp threads

$ 2 ‘ $

(a)

4 x o I Yo A - J\,

-_ cal data. They presented several simulations, including cloth draped over an armchair, a square table, and a circular table.

Anderson3’ implemented Breen’s model on the trans- puter-based Edinburgh Concurrent Supercomputer. He reported obtaining results within hours, instead of days.

Y

The work of Okabe et a/. The Okabe et al. cloth visualization system33 is direct-

ed toward automating the traditional garment manufacturing process. The user supplies a rough 2D paper pattern, which is scanned and then mapped onto a 3D object (a model of a mannequin was used). The system first forms a 3D structure that roughly resembles the underlying body. To add the drape effects of real cloth, the system formulates the cloth’s mechanical properties as energy equations and uses a minimization technique to determine the shape of the cloth at eq~i l ibr ium.~~ The 3D structure, with the drape effects, then undergoes another mapping process to generate a more accurate 2D paper pattern.

In the mechanical analysis of the drape effects, their system represents cloth by a triangular mesh, as shown in Figure 19. Parametric values of u and v (representing the cloth’s warp and weft) are used so that a pointp(u, v) has a corresponding pointp(x,y, z) in the model.

The equilibrium shape of a cloth can be determined by finding its energy minima. The set of energy equation in this case is

where E, is the elongation energy, Esh the shearing energy, Eb the bending energy, E, the twisting energy, Eg the gravitational energy, and Ep the penalty energy function to prevent the cloth from penetrating the body.

The energy terms Ee, Esh, Eb, and Em are derived from

I

i +

V

the elasticity theory of shell structures. Every point on the cloth has a certain energy value that can be calculated according to Equation 9. Therefore, by moving every point toward the energy minima, the cloth achieves an equilibrium shape. The energy minima was defined by a steepest descent method called Adaptive Coefficient Gradient Method (ACGM) .

This approach took about 20 minutes of computation time on an IBM RS6000/320 workstation to model a cloth represented by 1,300 nodes.35

The work of Li et a/. Li et a1?6 considered the particular situation of a piece

of cloth immersed in an airflow. The cloth is represented as an impermeable stream surface in the form of m x n small patches. They used the simplified Reynold’s aver- aged Navier-Stokes equations to find the forces on each patch and thus to determine distributed forces on the whole cloth. Cloth deformations are determined by using these forces and Terzopoulos’s model.

The reported simulations used 200 patches to represent the cloth and required about 3 seconds on an SGI Power workstation to complete a 1,000-step calculation of the distributed forces. The model’s memory requirements will increase enormously if larger numbers of patches are used.

Their early work36 treated the airflow as quasi-steady, that is, the instantaneous airflow velocity is unchanged. In later work,37 Li et al. changed their model to treat the airflow as unsteady, that is, with a constant change of flow velocity. This modified model is suitable for simulating large cloth movements with rapid changes of air velocity. The simulation time is longer than with the quasi-steady version. For a cloth represented by 20 x 20


17 A woven cloth as particles.

18 Thetwo stages of the Breen et al. simulation.

19 Represen- tation of cloth as a triangular mesh.

35


20 Themass- spring model.

36

Flexion springs . Pi,j+2

Pi,j+l

Structural L springs

Pi,j P.+I,,

“i,j+l pi+l ,j+l

Sheer X springs

Pi,, Pi+1 ,j

patches, it took about 32 seconds on the same platform to perform a 1,000-step simulation. The memory requirement to represent a cloth with a large number of patches remains a limitation.

This work formed Li’s doctoral d i~ser ta t ion ,~~ a con- densed version of which appeared in The Visual Computer.39 The model targeted the particular application of cloth in an airflow and was not concerned with draping effects. The simulation of a flag blowing in the wind was impressively realistic.

The work of Provot Provot4’ used the classic mass-spring model shown in

Figure 20a to simulate cloth under constraints. He classified the springs as flexion, structural, or sheer springs, as shown in Figure 20b, and used Newton’s law of dynamics in the simulation:

where m is the mass of the pointP(i,j) and a(i,j) is the acceleration of that point. Fint(i, j) represents the internal forces at P(i, j) :

f

where xis the stiffness of the springs, R is a set of neighboring points surrounding Pi,, and

is the undeformed length between the point P,, andPk,l. This model considered FexiiJ to be composed of three

external forces. First, the gravitational force is

where g is the gravitational acceleration constant Second is the damping force

Fdamp(Pi,) =-Wij

where c is the damping constant and VIJ is the velocity of l?,.

Third is the viscosity force

September 1996

where k is the viscosity constant, n,, is the unit normal at%, and ufluld is the fluid velocity. Provot used the Euler method to integrate Equation 10 through time so that the position ofP,, at each time step can be obtained.

Provot observed that when a hanging cloth is simulated using the classical model, unrealistic deformations will occur at the constraint points. To reduce this effect, the stiffness could be increased. However, this would require more iterations to yield the solution. Provot solved this problem by detecting the deformation rate in those springs around the constraint points to determine if any exceeded the predetermined threshold. If so, their elongation was limited to 10 percent. Provot illustrated the effect of this modification by giving several simulations, including a cloth held by two corners and a flag in the wind. The computation time was about 1 minute to compute 1 second of animation for a cloth represented by a 17 x 17 mesh on a Sun Sparc 10 workstation.4l Collisions were not considered.

The physical work of Ng et a/. Ng et al.42 presented a cloth-visualization system that

determined the cloths shape by an energy minima gov- erned by a set of energy functions. These are modifica- tions of Feynman’s functions.13 An improved multigrid method and searching method for the energyminimum allow the system to run on a PC. The results are presented in DXF format, commonly used by many CAD software packages.

Simulation examples included hanging clothes and a cloth draped over a square surface. The simulation time in each case was under two minutes on a 486 33MHz PC.

Hybrid techniques

metrical methods. The hybrid techniques combine physical and geo-

The work of Rudomin R ~ d o m i n ~ ~ suggested reducing the computation time

required by traditional physical techniques by using a geometrical approximation as a starting condition. His method simulates drape effects by first effectivelywrap- ping the object in the cloth to determine the object’s convex hull (Figure 21a). Next, the lower part of the convex hull is removed by introducing a lower point along the axis of the object’s center of mass (Figure 21b) and recal- culating the hull. This computed upper hull may be larger than the actual cloth.

Rudomin then developed an algorithm to map this upper hull onto the cloth space (a simple 2D rectangle) and to trim any points lying outside the cloth. All the remaining points inside the cloth are then recomputed to form a 3D cloth hull. This gives the approximate shape of the draped cloth. Terzopoulos’ deformable is then used to refine the shape.

The simulation examples reported for this work used small grid sizes and did not include enough detail to assess the method. Finding the convex hull of a polyhe- dral object can be quite time-consuming, and therefore

the benefit of using this geometrical initial input was not clearly demonstrated.

w

The work of Kunii et a/. Although Kunii et al.44 recognized the realistic

images possible from using physical techniques to model flexible objects like garments, they also recognized the limitations. First, flexible objects are usually represented by many small elements, so the computation time is very long. Second, it is difficult to obtain the right parameters for the differential equations representing the objects.

Since geometrical techniques have their own limitations-namely, the difficulty of incorporating dynamic constraints and representing local structure accurately-Kunii et al.44,45 therefore proposed a hybrid method. The first step is a physical simulation of the cloth in which the two boundary edges are gradually brought together. The simulation represents the cloth as a grid with springs connecting the points, as shown in Figure 22a. Two kinds of energy are defined, metric and curvature, shown in Figure 22b and 22c, respectively. The shape of the cloth is obtained by using a gradient descent method to find the energy minima.

After this step, singularity theory is used to charac- terize the resulting wrinkles. Subsequent simulations use the captured features as modeling primitives. These features include the position and types of characteristic points, and the contours associated with the characteristic points.

In the second step, these features are specified on a garment and surfaces are then constructed between the characteristic points to form the wrinkles. A sinusoidal curve is used to construct the surface:

nrc f(s) = xsin(-s) a

where x is a small constant, n is a natural number, A is the length of the curve, s is the parameter for arc length (0 5 s 2 A), andf(s) is the vertical displacement of the points. Energy minimization is then applied again to the garment with the wrinkles added.

Kunii et al. produced several animation sequences of a sleeve on a bending arm using the following assumptions:

The number of characteristic points remain constant. The contours get sharper as the animation progresses.

H The wrinkles get deeper as the animation progresses.

There was no strategy for specifymg the characteristic points, and interactions with the underlying object were not considered.

The work of Taillefer Taillefer% categorized the folds in a hanging cloth into

two types, horizontal and vertical, as shown in Figure 23. He modeled the horizontal folds using catenaries, as suggested inWeL5 The offset of each horizontal fold (z value) from the plane containing the hanging points PI and PZ is calculated by

Convex hull ,d

Upyer hull

i

/ Horizontal folds

arctan(lO(l- PIPz / L ) ) rc/2 \ z = z m a x

\ \

The vertical folds are modeled using a relaxatioh process similar to Weil's but with more external constqaints. These constraints include stretching, bending, g rT ta - tional, and self-repulsion energies. To reduce the COY- putation time, the relaxation is only carried out on the contours of the vertical folds.

Taillefer's work is an extension of Weil's and addresses only hanging cloth.

The work of Tsopelas T ~ o p e l a s ~ ~ adopted a hybrid technique to model folds.

He treats garments as thin cylindrical tubes under axial loads and simulates garment folds using thin-walled deformation theory. This process focuses on regions where folds are most likely to appear, that is, regions with large curvatures. These occur at the back of the knees or at seams.

In processing thin-walled objects under axial load,


21 Definitions of the hulls: (a) convex hull from wrapping and (b) upper hull recomputed from dropped lower point.

22 Physical model of cloth using (a) spring connecting points, (b) metric energy, and (c) curvature energy.

23 Two types of folds: horizontal and vertical.

37


24 Parameters of a buckled cylindrical object under axial force.

25 Generating a swept surface.

38

Force

I Generatrix curve’

diamond patterns occur (see Figure 24) and the buck- ling wavelength il can be calculated using

1=1.74t /D

where tis the fabric thickness and D is the diameter of the tube.

Once these diamond patterns have been located, the process traces several principal curve lines (which Tsopelas refers to as threads) in each diamond and then fits them with elasticas. The deformation of an individual elastica is calculated using the equation

where B is the bending rigidity, ko is the initial curvature of the elastica, Ris the reactive force, w is the weight per unit length, @is the angle between the tangent of the elastica and the line joining the endpoints after the force F is applied, and s is the length of the elastica.

The deformed surfaces can now be constructed by interpolating between the points on the elasticas using nonuniform B-splines. When the garment moves, the points that form the folds also move. For each step in the motion, new principal curve lines (threads) are traced and elasticas are fitted. The force needed to produce such a movement is calculated using an inverse dynamics process and is subsequently applied to other threads to form crumples.

September 1996

Tsopelas presented several diagrams of symmetric and unsymmetric collapses of sleeves. Neither the collisions between the garment with itself nor with the underlying body were considered. The work was implemented on a Sun Sparc2 workstation. Although he did not mention computation time in his paper, we estimate that Tsopelas’s approach would obtain results in the order of minutes.

The work of Dhande et al. Dhande et al.48 used a swept surface to model the

draping effects of a fabric. They use a generatrix and a directrix curve to generate a biparametric surface (see Figure 25). The surface is defined by

where r (u , v) is any point on the surface, r2(v) is the directrix curve defining the surface of the underlying object, M is the transformation matrix for the generatrix curve rl from the local n-b-t coordinate system to the global x-y-z coordinate system, and r l (v , (u)) defines the shape of the fabric when it is outside the underlying object.

An elastica that includes the fabric’s bending and shear properties is used to represent rl. We therefore classify the technique as hybrid. It produced a good approximation of the way fabric drapes over a rigid flat surface. However, as the authors admitted, it will be difficult to simulate the effects of cloth draped over complex objects.

Conclusions Table 1 summarizes the essential features of the work

on cloth visualization surveyed here. Execution time, an important issue, was not always reported. Where tim- ing information was not available, we estimated it. We classify a technique as fast if it requires seconds to complete a simulation, medium if it requires minutes, and long if it requires days. This roughly indicates the computational requirements of the individual technique.

Our survey shows that physical techniques normally involve solving a large set of differential equations or iterating the process many times to search for an ener- g y minima. Hence they require longer computation times than other techniques. The time required ranges from minutes to days.

Some physical parameters are better expressed in the form of energy (for example, strain energy of the cloth); others are better expressed as forces (for example, wind). The nature of the application therefore drives the choice between these two forms of physical repre- sentation. Otherwise there is little to choose between them. Physical techniques allow intuitive control in selecting the parameter values, hence their popularity. For example, an increase in the mass value will result in a heavier cloth.

The alternative geometrical techniques do not attempt to model the cloth properties faithfully. Instead, they produce satisfactory shapes that give the appearance of the cloth sample. Geometrical techniques do not provide as intuitive control over parameters selection as

\

do physical techniques. They have the advantage, however, of short execution times.

Hybrid techniques exploit the advantages of the other two techniques. They usually employ a geometrical technique to determine a rough shape for the simulated cloth and then use a physical technique to refine the structure.

Cloth is a nonlinear material. Most models employed in the physical techniques are based on elasticity theory, which considers macroscopic behavior and thus does not accurately describe a cloth. However, representing every thread would require too large a model. Garment

manufacturing and textile engineering require an accurate description of cloth. Moreover, garment design, animation, and visualization require a model that produces results rapidly yet simulates the behavior of cloth satis- factorily.

Can we obtain a scalable model that satisfies both worlds? This is still an open question. The degree of accuracy can depend on the number of iterations. We foresee this scalable model using some form of hybrid technique.

We also foresee the need for a unified model that can simulate a cloth in all situations, from a hanging cloth to

IEEE Computer Graphics and Applications 39


40

one draped over a complexobject. Currently, most models can do well in one situation but cannot deal with other situations without modification.

Textile engineers investigating cloth modeling problems have generally focused on deriving accurate models or methods to measure the properties of cloth. The overall shape of a cloth was not of major interest. Recent work has focused on the use of finite-element meth- o d ~ . ~ ~ - ~ ~ Cloth is a complex material and to describe it accurately and quickly requires long-term research. Research workers in the textile and graphics community need to tackle these problems collaboratively.

Current cloth visualization techniques lack the means for fashion designers to input their requirements. Models should help designers visualize their ideas during the garment design process. Before a garment is manufactured, fashion designers already have a con- cept of what it will look like when it is worn. They have hitherto expressed their designs with pencil sketches. If they can move their design stage to computers, the time and cost of the whole garment manufacturing cycle will go down. People at different physical locations can par- ticipate in the design process via communication links. We believe that VR technology can be applied success- fully in this area.

Cloth visualization is an exciting branch of computer graphics. We hope our review will stimulate further interest.

Acknowledgments We would like to thank Mamiko Murao, Akria Ijuin,

and Kazuhira Kondo for their help in translating the Japanese references. Many authors sent their reports, papers, and theses, and answered our queries in com- piling this overview. We thank them. Thanks also go to the reviewers for their helpful comments. The first author is supported by scholarships awarded from the Croucher Foundation and the University of Sussex; their help is gratefully acknowledged.

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Hing N. N g is a technical specialist a t CentroDigital Pictures, HongKong. He received his PhDfrom the Univer- sity of Sussex, Center f o r VLSI and Computer Graphics, in May 1996. In his dissertation, he developed two fast techniques f o r visualizing cloth. His

research interests include computer animation, physically based modeling, multimedia, and VR. He is a member ofACM, IEEE, and a n associate member of IEE.

Richard L. Grimsdale is aprofes- sor of electronic engineering a t the University of Sussex, and founder of the Center f o r VLSI and Computer Graphics. His research interests are in rendering, expert systems in computer graphics, animation, and VLSI

architectures f o r graphics accelerators. He obtained his Ph.D a t Manchester University where he designed the first transistor computer and contributed to theAtlas Machine. He is a chartered engineer and fellow of the Institution of Electrical Engineers, British Computer Society, and Euro- graphics Association.

Readers may contact Ng at HKlTC #601, 72 Tat Chee Avenue, Kowloon Tong, Hong Kong, e-mail krates@ centro.com.hk.

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