TRIGONOMETRIC CURVE-BASED HUMAN MODELING

L. H. You

, X. S. Yang

, X. Y. You

and Jian J. Zhang

National Centre for Computer Animation, Bournemouth University, Dorset, U.K.

Faculty of Engineering and Computing, Coventry University, Coventry, U.K.

Keywords: Human modeling, Cross-section curves, Trigonometric series, Surface creation.

Abstract: In this paper, we present a modeling method to build human models. A human model is divided into

different parts. For each part, cross-section curves are created and approximated with trigonometric series.

All parts are constructed from these trigonometric curves and assembled together to create the whole human

model. Since surface creation of human models is transformed into generation of cross-sectional curves and

few design parameters are required to describe these cross-sectional curves, our approach can decrease the

data size of geometric modeling greatly and is especially suitable for reconstruction of human models from

scanned point clouds.

1 INTRODUCTION

Virtual humans are widely applied in various

situations such as computer games, virtual reality

and digital films. As pointed out by (Thalmann N.M.

and Thalmann D., 2005), in order to create realistic

and believable virtual humans, three techniques

should be developed. They are realistic appearance

modeling, realistic, smooth and flexible motion

modeling and realistic high-level behaviour

modeling.

Among them, realistic appearance achieved by

creation and deformation of human models has

attracted a lot of research attention.

In the work of (Beylot et al., 1996), the issues of

image data for extraction of 3D shapes, surface

reconstruction, topological modeling of different

anatomical elements and potential applications of

topological data base were addressed. (Scheepers et

al., 1997) considered the influence of the

musculature on exterior form, developed anatomy-

based models of muscles which responds to the

changes of the posture of an underlying articulated

skeleton, and applied them to the torso and arm of a

human figure. Modeling muscles, bones, and

generalized tissue as triangle meshes or ellipsoids,

treating muscles as deformable discretized cylinders

whose shapes change as the joints move, creating

skin by voxelizing the underlying components,

filtering, and extracting a polygonal isosurface,

(Jane and Allen, 1997) proposed an improved,

anatomically based approach to modeling and

animating animals. (Allen et al., 2002) introduced an

example-based method to capture human body

scans, estimate poses and kinematics, reconstruct a

complete displaced subdivision surface in each pose

and combine the surfaces using k-nearest-neighbors

scattered data. Based on anatomy concepts, (Nedel

and Thalmann, 1998) presented a method which

divides a human representation into three different

layers: the rigid body from a real skeleton, the

muscle design and deformation, and the skin

generation. Still using the anatomic model of

deformable human bodies consisting of skeleton,

muscles and skin, and presenting muscles by the

action lines and the muscle shape, they introduced a

mass-spring system with angular springs to

physically simulate muscle deformations (Nedel and

Thalmann, 2000). Defining the space deformed by

the control surface by a distance function around the

surface, (Singh and Kokkevis, 2000) proposed a

surface-oriented FFD which is more suitable for the

automated skinning of characters. Deriving muscle

motion and deformation from one or several action

lines and deforming each action line by a 1D mass-

spring system, (Aubel and Thalmann, 2001)

proposed a muscle model based on physiological

and anatomical considerations. Using a layered

canonical model to represent the animal's skeleton,

muscles, and skin, generating feature points to

deform the attached mesh skin representation,

(Maryann et al., 2002) studied a semi-automatic

technique for creating 3D models of creatures.

(Mohr and Gleicher, 2003) presented a framework

for extending linear blending skinning which can

capture detailed skin deformations and developed an

H. You L., S. Yang X., Y. You X. and Zhang J..

TRIGONOMETRIC CURVE-BASED HUMAN MODELING.

DOI: 10.5220/0003316500310038

In Proceedings of the International Conference on Computer Graphics Theory and Applications (GRAPP-2011), pages 31-38

ISBN: 978-989-8425-45-4

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

automated method to build efficient and accurate

character skins from a set of examples. Using

motion capture and video cameras and providing a

reconstruction algorithm to solve the problems of

occlusion, hole-filling, deformation and noise-

removal, (Sand et al., 2003) examined how to

acquire deformable human geometry from

silhouettes. Taking the objective function to be

proximity of transformed vertices to the range data,

similarity between neighbouring transformations,

and proximity of sparse markers at corresponding

locations on the template and target surface, (Allen

et al., 2003) presented a new method by solving an

optimisation problem and explored its applications

in human body modeling. (Hyun et al., 2003)

represented each limb as a set of ellipsoids of

varying size, approximated these ellipsoids with a

swept ellipse, determined the difference between the

original and approximated limbs using a

displacement map, and proposed a new approach to

model and deform a human or virtual character’s

arms and legs. This work was extended to the

modeling and deformation of a whole human body,

and anatomical features are realised by a GPU-based

collision-detection procedure (Hyun et al., 2005).

(Seo and Thalmann, 2004) presented a set of

techniques to automatically generate a new human

body or modify an existing one by manipulating the

parameters provided. Using quasi-static linear

deformation model and finite element method to

calculate the deformation of chunks which represent

the internal structures of a virtual character, (Guo

and Wong, 2005) gave an approach to create skin

deformations. (Venkataraman et al., 2005)

introduced a combination of a kinematic and a

variational model to deal with the wrinkling of skin

by minimising a functional including energies for

stretching, bending and self-intersection. By

decomposing the facial meshes into the global shape

and 3D skin detail and recomposing the shape and

3D skin, (Lee and Soon, 2006) proposed a method to

reproduce the scanned model which allows to

simulate the exaggeration of the facial global shape,

retain the original skin detail and transfer 3D skin

from one to another. Using sweeps following a

simplified skeleton, (Lee et al., 2006) proposed a

new method to carry out realistic human hand

modeling and deformation which can achieve real-

time performance. (Yang and Zhang, 2006)

presented a new anatomy-based skin deformation

method which extracts major muscles automatically,

formulates muscle sliding around a joint and sliding

around a bone, develops a hybrid skin deformation

to combine the strengths of anatomy based and

smooth skinning, and is compatible with the current

animation workflow. By using the so-called curve

skeletons along with the joint-based skeletons,

(Yang et al., 2006) investigated an approach to deal

with the inherent non-linear relations between the

movement of skeletons and the caused skin shapes.

Motivated by the work of (Hyun et al., 2003,

2005) but without using standard ellipses and

displacement map, we will introduce trigonometric

series to approximate the cross-section curves of a

human body, and present a modeling method with a

small data size to build human models in this paper.

2 TRIGONOMETRIC SERIES

In the work of (Hyun et al., 2005), sweep surfaces

with an elliptic cross section have been used to

approximate human arms, legs, torso and neck, and

carry out human modeling and deformation. The

mathematical description of sweep surfaces has the

form of

⎥

⎦

⎤

⎢

⎣

⎡

⎥

⎦

⎤

⎢

⎣

⎡

⎥

⎦

⎤

⎢

⎣

⎡

)(

)sin()(

)cos()(

)()()(

)()()(),(

333231

232221

131211

vtb

vta

ururur

uvuvu

CERS

(1)

where

),( vuS is a sweep surface, )(uR and )(uC

stand for rotation and translation, respectively, and

)(v

is a standard ellipse of variable size.

Since the cross sections of most parts of a human

body are irregular curves. Using standard ellipses to

approximate these cross sections will bring in some

errors.

As indicated in Figure 1, we give some cross-

section curves of human torso from a human model

built with the polygon modeling approach and those

approximated by elliptic cross sections where the

curves in red indicate cross-section curves and those

in blue are from standard ellipses. It can be seen for

these images that there are noticeable differences

between the real cross-section curves and elliptic

ones.

In addition, for a certain value

of the

parametric variable

u , Eq. (1) can be written as the

following form

)()sin()()cos()(),(

00320310

00220210

00120110

uzvufvufvu

uyvufvufvu

uxvufvufvu

++=

(2)

The above equation indicates that at the plane

determined by a point

)(

and a unit normal

vector

[

]

urururuN )()()()(

0330230130

= , the cross-

GRAPP 2011 - International Conference on Computer Graphics Theory and Applications

section curve of the sweep surface is a simple curve

which is described with two trigonometric functions

)cos(v and )sin(v .

Cross-section curves of torso

Figure 1: Comparison between real human cross-section

curves and approximated ellipses.

In order to describe the cross-section curves of a

human body more accurately, we propose to use

trigonometric series to represent cross-section

curves. With the application of more terms of

trigonometric series, the cross-section curves of a

human body can be approximated very accurately.

In the previous work (You et al., 2004),

trigonometric series has been applied to describe

blending surfaces. By degenerating the two

dimensional problems to one dimensional ones, the

trigonometric series proposed by (You et al., 2004)

can be modified to represent cross-section curves of

a human body. Assuming that a cross-section curve

is perpendicular to one of x-y, x-z and y-z plane, with

the centre

[]

ccc

zyx=C , and taking the one

perpendicular to x-y plane as an example, the

mathematical equation of the cross-section curve can

be written in the following trigonometric series

)cossin(

)sincos(

120

=−

++=−

∑

−

jvbjvbbyy

jvajvaaxx

(3)

where

a and

b (j=0,1,2,3,…,2J) are unknown

constants.

If there is a cross-section curve represented by a

number of discrete points

),,(

cii

zyx

(i=1,2,3,…,I),

the centre of the cross-section curve can be

determined by the average value of each component

x and y. That is

∑

(4)

Then, we use curve fitting and the least squares

algorithm to determine the unknown constants in Eq.

(3). To this aim, we calculate the squares sum of the

errors between the curve and the trigonometric series

at the points

),,(

cii

zyx

(i=1,2,3,…,I) for x and y

position components, respectively

[

212

)cossin(

)sincos(

⎥

⎦

⎤

+−

−−=

⎥

⎦

⎤

+−

−−=

∑

−

ijij

ciy

ijij

cix

jvbjvb

byyE

jvajva

axxE

(5)

The errors in Eq. (5) are minimized by setting the

derivatives of the square sums with respect to the

unknown constants to zero

)2,,3,2,1,0(

∂

(6)

which leads to the following linear algebraic

equations

),,3,2,1(

0sinsinsincos

sinsin)(

0cossincoscos

coscos)(

0sin

cos)(

kvjvakvjv

akvakvxx

kvjvakvjv

akvakvxx

ajvaaxx

jii

iic

jii

iic

=−

−−−

=−

−−−

−−−−

∑∑

∑∑∑∑

∑∑

∑∑∑∑

∑∑∑∑∑∑

−

====

−

(7)

and

),,3,2,1(

0coscoscossin

coscos)(

0sincossinsin

sinsin)(

0cos

sin)(

kvjvbkvjv

bkvbkvyy

kvjvbkvjv

bkvbkvyy

bjvbbyy

jii

iic

jii

iic

=−

−−−

=−

−−−

−−−−

∑∑

∑∑∑∑

∑∑

∑∑∑∑

∑∑∑∑∑∑

−

====

−

(8)

TRIGONOMETRIC CURVE-BASED HUMAN MODELING

Solving equations (7) and (8), respectively, we

determine all unknown constants and obtain the

mathematical representation of the cross-section

curve.

For the curve which is not perpendicular to any

coordinate planes, we must carry out the coordinate

transformation and find the mathematical description

of the curve in the local coordinate system where

one of the coordinate axes is perpendicular to the

plane containing the curve.

For some representative curves taken from

human leg and torso, we use the trigonometric series

(3) and standard ellipse

)(v

to regenerate them

and the obtained results were depicted in Figure 2

where J is defined in Eq. (3), the curve in red is the

original one, the one in green is from the algorithm

of the trigonometric series, and that in blue is

created with the equation of the standard ellipse.

J=2 J=3 J=4

a) Cross-section curve of human leg

J=3 J=7 J=10

b) Cross-section curve of human torso

Figure 2: Curve generation with different approaches.

It can be seen from these curves that the

algorithm of the trigonometric series can

approximate the cross-section curves of human body

quite well and the approximation can be greatly

improved by increasing the terms of the

trigonometric series. Since most cross-section curves

of a human body have the similar complexity to

those in this figure, the proposed approach can use

few terms to achieve high accuracy of curve

modeling.

Depending on different shapes of cross-section

curves, different terms in Eq. (3) will be used to

achieve the required accuracy. Too few terms in Eq.

(3) will cause too large errors. Too many terms will

increase the computational cost and slow down the

human modeling process. Therefore, suitable terms

should be used. In the situation of interactive

modeling, it is required to automatically find out

which value of the term J is the best. In order to

tackle this issue, here we propose the following

strategy.

First, we define an average error

and a

maximum error

below. The former is used to

measure the global approximation of the curve from

the trigonometric series to the original curve and the

latter quantifies the maximum difference between

the curve from the trigonometric series and the

original curve.

∑

(9)

and

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

E ,,,,max

(10)

where

()( )

)cossin(

)sincos(

120

iji

jciyi

iji

jcixi

cicii

yixii

jvbjvbbyyd

jvajvaaxxd

yyxxd

ddd

+−−−=

−+−=

∑

−

(11)

Then, we set different errors to determine the

required terms in Eq. (3). For example, we take the

average error and maximum error not more than 1%

and 5%, respectively. That is

≤

(12)

A linear interpolation operation is employed to

find out the suitable terms efficiently. Initially, we

take J=3 and J=10, and calculate the average errors

and

10a

and the maximum errors

and

10m

where the subscripts 3 and 10 stand for the

values of J. If both

and

have met Eq. (12),

or one of

10a

and

10m

does not satisfy Eq. (12), a

linear extrapolation operation is used to find a

smaller

for the former and a larger

for the

latter. Otherwise, a linear interpolation is applied to

calculate the

between J=3 and J=10. The

obtained

is usually not an integer. We round off

it to the nearest integer. With

, we calculate the

average errors

E and the maximum errors

and examine whether they have met Eq. (12). If both

E and

E have met Eq. (12) and

, the

linear extrapolation operation occurs with

and

GRAPP 2011 - International Conference on Computer Graphics Theory and Applications

J=3. If one of

E and

E does not satisfy Eq.

(12) and

, the linear extrapolation operation is

performed with J=10 and

. If both

E and

have met Eq. (12) and

103

<< J

, the linear

interpolation operation is conducted between J=3

and

. If one of

E and

E does not satisfy Eq.

(12) and

103

<< J

, the linear interpolation

operation is carried out between

and J=10.

Since there are two quantities

and

which

can be employed for the interpolation, we must

determine which one should be used. Obviously, if

only one of

and

does not meet Eq. (12), this

error is used for the interpolation. If both of them do

not satisfy Eq. (12), we always use the average error

for the interpolation since the average error is a

global measurement of the difference between the

original curve and that from the trigonometric series.

3 HUMAN PARTS DEFINED

WITH TRIGONOMETRIC

SERIES

In order to build a human model, we draw some

cross-section curves which define the human

profiles. Then we approximate the original cross

sections with the trigonometric series. After that, we

construct surface patches from the curves generated

using the trigonometric series with the following

treatment.

As mentioned by (Tokuyama, 2000), among

three surface interpolation methods, i. e.,

interpolating through distinct point data, skinning

over a family of curves and interpolating the surface

simultaneously over two families of intersection

curves, the skinning method is generally considered

to be the most frequently used technique for surface

construction. Here we use this skinning method to

construct surface patches of human parts.

If a surface patch will be constructed from K

curves determined with the trigonometric series

))(),(),(( vcvcvc

zkykxk

(k=0,1,2,3,…,K-1) where

)(vc

, )(vc

and )(vc

are determined by Eq. (3),

we use the following equation to describe the

surface to be constructed

),,(

)(),(

zyxt

vfuvut

∑

−

(13)

where

)(vf

(m=0,1,2,3,…,K-1) are unknown

functions.

Uniformly dividing the region u=0 to u=1 into

K-1 equal intervals which gives the interval length to

)1/(1 −= Kdu , we have

duku

×=

(k=0,1,2,3,…,K-1). The unknown functions

)(vf

(m=0,1,2,3,…,K-1) can be determined by

solving the following linear algebraic equations

),,;1,,3,2,1,0(

)()(

zyxtKk

vfuvc

ktk

=−=

∑

−

(14)

Expanding Eq. (14) and rewriting it into the form

of matrix, we obtain the following mathematical

expression

[

]

{

}

{

}

)()()( vvuR

ttkmt

CF =

(15)

where

[

]

)(

kmt

is a

× square matrix with the

elements

kkm

uu = (k=0,1,2,…,K-1; m=0,1,2,…,K-

1),

{

}

[

]

tKtttt

vfvfvfvfv )()()()()(

1210 −

= "F and

{

}

[

]

tKtttt

vcvcvcvcv )()()()()(

1210 −

= "C are two

vectors with K elements.

Using

[

]

)(

−

kmt

uR to indicate the inverse matrix

[

]

)(

kmt

and left multiplying both sides of Eq.

(15) by this inverse matrix, we obtain the unknown

functions with the following equation

{

}

[

]

{}

)()()(

vuv

tkmtt

CRF

−

(16)

At the boundary curves where two different

surface patches are to be connected together, we

must consider the continuity between the two

surface patches. For an existing surface patch

indicated by Eq. (13), different order continuities

such as the boundary tangents and boundary

curvature etc. at its boundaries can be determined

from the different orders of partial derivatives

vut

∂

∂ ),(

(n=1,2,3,…) of the surface patch with

respect to the parametric variable u. By introducing

these partial derivatives into the above operation,

different order continuities between two connected

surface patches can be obtained.

For example, if we intend to connect two surface

patches with the tangent continuity at u=1 of the

existing surface patch

),( vut , we obtain the

mathematical expressions of the first partial

derivative

∑

−

⎥

⎦

⎤

⎢

⎣

⎡

∂

)(

),(

vfm

vut

of the existing

TRIGONOMETRIC CURVE-BASED HUMAN MODELING

surface patch

),( vut

and

)(

),(

vut

⎥

⎦

⎤

⎢

⎣

⎡

∂

of the

unknown surface patch

),( vut with respect to the

parametric variable u from Eq. (13), respectively.

Then the following tangential continuity constraint

is added to Eq. (14).

),,(

)()(

zyxt

vfmvf

tmt

∑

−

(17)

Since one more linear algebraic equation is

introduced, the unknown functions

)(vf

in Eq.

(13) should be increased from

)(

tK −

)(vf

and all K in Eq. (13) and thereafter will be replaced

by (K+1).

Figure 3: Human torso, legs and arm created from the

trigonometric series.

If both opposite edges of a surface patch ),( vut

will be connected to two separate surface patches

),( vut and ),( vut , the continuity at the boundary

curve u=0 of the exiting surface patch

),( vut should

also be considered. Similar to the above treatment,

the following boundary condition for the tangential

continuity will be added to Eq. (14)

),,(

)()(

zyxt

vfvmf

∑

−

(18)

and the unknown functions

)(vf

in Eq. (13) will be

increased to

)(

tK +

and all K in Eq. (13) and

thereafter will be replaced by (K+2).

With the above constructed surface function, we

create human torso, legs and arms from the

trigonometric series as demonstrated in Figure 3.

4 ASSEMBLY OF HUMAN PARTS

BY SURFACE BLENDING

After successful construction of human parts

including human torso, arms, and legs, the

remaining task is how to assemble them together

which maintains smooth transition between different

human parts and ensures the surface in the transition

region has the realistic appearance of the human

part.

There are two approaches which can be used to

achieve this aim: one is surface blending (Zhang and

You, 2004) and the other is local extrusion

deformation (You and Zhang, 2003). Here we take

the connection between human torso and right arm

as an example to explain the operation of surface

blending.

Firstly, we determine the boundary curves and

boundary tangents of the surface which will

smoothly connect human right arm to torso together.

According to the position function of the surface

patch at the leftmost end of the right arm, we can

determine the right boundary curve and boundary

tangent of the transition surface. In order to

determine the left boundary curve and boundary

tangent of the transition surface, we create a cross-

sectional curve on the human torso near the left end

of the right arm and calculate the tangent of the

human torso at this curve. Then, we construct the

transition surface with these boundary curves and

boundary tangents. We can change the size of the

boundary tangents to change the shape of the

transition surface and to make the transition surface

look more realistically. If the manipulation of the

boundary tangents does not reach the required

realism, we add one or more cross-section curves

between the two boundary curves and create the

transition surface from two boundary curves, two

boundary tangents and the in-between curves. With

such a method, we connect two different parts

together. The top image of Figure 4 gives two

human parts: right arm and torso, and the bottom

image shows these two human parts have been

smoothly connected together. With the same

treatment, we also connected the left arm to the

torso.

For the connection between the human torso and

legs, we first use a surface to close the bottom

opening of the human torso. Then we determine

boundary curves and tangents of the torso

corresponding the right and left legs, respectively.

And finally, the above method is employed to

produce the transition surfaces between the three

human parts.

GRAPP 2011 - International Conference on Computer Graphics Theory and Applications

Figure 4: Connection of different human parts.

In Figure 5, we gave a human model built with

our proposed approach.

The approach presented in this paper greatly

lowers the data size of geometric modeling. For the

human body given in the following figure, the

modeling with the polygon approach uses 6457

vertices with total 14371 design parameters to create

the model. With our method, only 6510 design

parameters are required to build it.

Figure 5: Human model assembled from different human

parts.

5 CONCLUSIONS

Human modeling can be performed with NURBS,

polygon and subdivision. These approaches are

effective and popular. However, they involve a lot of

design parameters. In order to reduce design

parameters, we present a new modeling method to

build human models.

The method starts from cross-sectional curves of

a human body. With introduction of trigonometric

series, these cross-sectional curves are approximated

very accurately. Then human parts are constructed

from the curves generated with the trigonometric

series. We examined how to construct human parts

from these trigonometric curves and demonstrated

that surface patches produced with our proposed

approach can maintain the required continuities

between two adjacent patches. Finally, we discussed

the assembly issue of human models.

Unlike the sweep-based human modeling, our

method can represent cross-sectional curves of

human body very accurately. Due to this reason, the

treatment to improve the modeling realism using

some approaches such as displacement map is not

required.

Our proposed approach greatly reduces design

parameters of human modeling. It is especially

suitable for reconstruction of human models from

scanned point clouds.

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