Length-preserved Natural Boundary for Intrinsic Parameterization

Charlie C. L. Wang

Chinese University of Hong Kong

E-mail: cwang@acae.cuhk.edu.hk

Figure 1. Intrinsic parameterization with the natural boundary generated by [1] and our method – (a) the given

mesh surfaces for a polo-shirt, (b) the results of the front and the back pieces by the original intrinsic

parameterization approach in [1], and (c) the results of our length-preserved natural boundary working together

with [1].

Abstract This paper proposes a fast approach to generate

length-preserved natural boundary for intrinsic

parameterization. Given a triangular mesh with disk-

like topology, we compute an optimal boundary for its

parameterization. An optimal boundary is expected to

be length-preserved with the length of every triangular

edge on the boundary is invariant before and after

parameterization; also, the boundary is requested to

be natural where the inner angle is close to the angle

excess at every boundary vertex on the given mesh.

Computation of a length-preserved natural boundary

is formulated as a constrained non-linear optimization

problem, the procedure of solving which is in general

very time-consuming. Here, we speed up the

optimization by adopting the scheme of sequential

linearly constrained programming. It is shown at the

end of this paper that our length-preserved natural

boundary could greatly improve the speed and quality

of the original intrinsic parameterization.

1. Introduction Parameterization for mesh surfaces has become a

fundamental technique for numerous computer

graphics applications including texture mapping, shape

morphing, surface reconstruction and repairing, and

remeshing. The parameterization defines a

correspondence between the surface mesh M in 3D

and a 2D domain D , referred to as parameter space.

The correspondence is in general requested to be a

bijective mapping Ω satisfying DM →Ω : and

MD →Ω− :1 . However for practical applications, as

mentioned in [2], a weaker requirement of local

bijectivity is sufficient, where local bijectivity means

no flipped triangles is generated in the planar mesh.

Besides the bijectivity, the quality of a

parameterization in most applications depends heavily

on the distortion introduced in Ω . Ideally, a stretch-

free parameterization is desired, i.e., to let Ωisometric. Unfortunately, from differential geometry,

we know that only developable surfaces, which is a

small class among all freeform surfaces, can be

isometrically parameterized. Thus, all parameterization

techniques existed in literature are based on the spirit

of numerical optimization to achieve a local bijectivity

while minimizing distortion components, such as angle

deformation [1-3], length deformation [1, 4-6], or area

deformation [1, 6, 7]. An extensive survey of the state

of the art in parameterization research is given by

Floater and Hormann in [8].

As been linear, the approaches of Desbrun et al. [1]

and Lévy et al. [3] are the fastest parameterization in

literature. They can both give the parameterization

results with natural boundaries. Since the boundary

vertices are embedded into the linear equation system

during computing, no special constraints can be given

about the shearing (for shape) or stretching (for length)

on boundaries. However, the positions of boundary

vertices are more important than interior vertices

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because that the boundary vertices define the shape and

area of D . Therefore, it is always requested to have

the edges on the boundary D∂ of D with the same

length to ones on the boundary M∂ of M, and the

shape of D∂ at the meanwhile is desired to be similar

to M∂ . Considering that 2ℜ⊂∂D but 3ℜ⊂∂M , we

define angle-based metrics for the shape difference

between D∂ and M∂ .

The problem of computing a length-preserved

natural boundary is formulated as a constrained non-

linear optimization problem, where the difference

between D∂ and M∂ is minimized while the length of

D∂ is constrained. As known, the non-linear

optimization is very time-consuming; however, the

parameterization usually needs to be finished in an

interactive speed. In this paper, by adopting the

sequential linearly constrained programming [9], we

greatly simplify the optimization procedure – so that a

length-preserved natural boundary can be computed

speedily. Figure 1 demonstrates the functionality of our

approach comparing to the natural boundary generated

by the intrinsic parameterization approach of Desbrun

et al. [1].

The work presented in this paper has the following

major contributions:

• A novel approach is introduced to compute a

length-preserved natural boundary for the intrinsic

parameterization;

• Not only the boundary length but also the shape of

the parameterized domain is optimized

• The non-linear optimization for determining a

boundary is simplified by conducting the sequential

linearly constrained programming – so that the

computing can be finished in interactive speeds;

• By our method, after fixing all boundary vertices,

the linear system in the intrinsic parameterization

approach [1] can be separated into two sub-systems

(one is with only x-components while another having

only y-components involved) – the system size is

reduced, so the computation can be finished faster.

2. Methodology The methodology for computing length-preserved

natural boundary in parameterization is presented in

this section. The problem of determining the 2D

positions of boundary vertices on D∂ is converted into

a constrained optimization with inner angles on D∂ as

arguments. Before that, the metrics to measure the

difference between M∂ and D∂ are introduced.

Boundary Length Metrics

The first metrics measuring the difference between

D∂ and M∂ is the length of boundary edges. For a

triangular edge e on M∂ , letting )(0el and )(el

representing its corresponding length on M∂ and D∂ ,

the boundary length metrics is defined as

∑∂∈∀

−=ΠMee

L eleln

)()(1 0 , (1)

where en presents the number of triangle edges on

M∂ . The boundary length metrics is straightforward.

However, since 2ℜ∈∂D and 3ℜ∈∂M , it is not easy

to measure the shape similarity between M∂ and D∂ .

Boundary Morphological Metrics

Here, we conduct angles to measure the

morphological difference between D∂ and M∂ . For a

vertex v on M∂ , suppose that its angle excess on M is

)(vα and its inner turning angle on D is )(vθ (see the

illustration in Figure 2), the shape similarity between

D∂ and M∂ is detected with the help of the following

boundary morphological metrics and the above

boundary length metrics:

∑∂∈∀

−=ΠMvv

vvn

)()(1

θαθ . (2)

In θΠ , vn is the number of vertices on M∂ .

Optimization Functional

Based on the two boundary metrics, we can

compute an optimal natural boundary of a given

triangular mesh by the following optimization

functional:

0..minarg =ΠΠ∂∈∀

LDv

tsθ . (3)

However, as both θΠ and LΠ are complex in term of

the planar positions of boundary vertices, the procedure

of solving the optimization problem defined in Eq.(3)

is quite slow. Thus, the optimization functional needs

to be reformulated. By observation, we find that

solving the above optimization problem in angle space

could greatly simply the formulas. The following angle

distortion energy is proposed to be minimized

Figure 2. Illustration of the angle excess and the

inner turning angle of a vertex – (a) the given

triangular mesh M and (b) the boundary of its

corresponding parametrix domain D.

Ninth International Conference on Computer Aided Design and Computer Graphics (CAD/CG 2005)0-7695-2473-7/05 $20.00 © 2005 IEEE

∑ −=i

iiEJ 2)(2

1αθ , (4)

where iθ is the interior turning angle of a boundary

vertex Mvi ∂∈ , iα is the angle excess of iv on M,

and the index i of boundary vertices is in anti-

clockwise order (i.e., when walking along M∂ , left

side is always with the given surface M ).

From the closed-path theorem (ref. [9]), we know

that: for a simple non-self-intersecting 2D closed path,

if its path is anti-clockwise, the total turning is π2 . As

shown in Figure 3, considering D∂ , its total turning is

∑ −i

i )( θπ , the following constraint should thus be

satisfied in the angle space

πθπ 2

1

≡−∑=

n

i

in (5)

If the length of every boundary edge 1+ii vv is fixed to

as the length il on M∂ , the other two constraints are

added to ensure D∂ be a closed-path,

0cos

1

≡∑=

n

i

iil φ and 0sin

1

≡∑=

n

i

iil φ (6)

The relationship between iφ and iθ is derived as

follows. Referring to Figure 3, at the vertex iv we have

)(2 βφπθ −−= ii ,

and at the vertex 1−iv

πφβ −= −1i .

By these two formulas, we get

1−+−= iii φθπφ . (7)

Together with 11 θπφ −= (see Figure 3), the general

form of iφ could be written as

∑=

−=i

k

ki i

1

θπφ . (8)

In summary, the length-preserved natural boundary

is able to be determined by the following constrained

optimization problem in angle space.

0sin,0cos,2..

)(2

1minarg

111

2

≡≡≡−

⎭⎬⎫

⎩⎨⎧

−

∑∑∑

∑

===

n

i

ii

n

i

ii

n

i

i

iii

llnts

i

φφπθπ

αθθ

(9)

Next section will give the efficient numerical scheme

for solving Eq.(9).

After determining the optimal iθ s, their related

optimal iφ s are also obtained from Eq.(8). Therefore,

the optimal position of every boundary vertex iv is

given by T

iiiiii llqq )sin,cos(1 φφ+=+ (10)

with Tq )0,0(1 = – so that the optimal D∂ is obtained.

3. Numerical Implementation The constrained optimization problem is formulated

as an augmented objective function )(ΧJ using

Lagrange multipliers ),,( yx λλλθ :

∑∑∑

++

−−+==Χ

iiiy

iiix

iiEyxi

ll

nJJJ

φλφλ

θπλλλλθ θθ

sincos

))2((),,,()((11)

The objective function in Eq.(11) can be minimized

using the Newton’s method as follows.

while ε>Χ∇ )(J

solve )()(2 Χ−∇=Χ∇ JJ δ

δ+Χ←Χ

end

(12)

The size of Hessian matrix )(2 Χ∇ J is 3+vn where

vn is the number of boundary vertices. For very

complex models, vn could be a very large number that

make the procedure in (12) time-consuming.

Borrowing the idea of speeding up a non-linear

optimization in [2], we conduct the sequential linearly

constrained programming as [10] to minimize )(ΧJ .

The sequential linearly constrained programming

technique for solving constrained minimization

problems considers the constraints as linear at each

iteration – so that neglects the terms coming from the

2nd

derivatives of the constraints in the Hessian matrix

)(2 Χ∇ J . Therefore, )()(2 Χ−∇=Χ∇ JJ δ solved at

each step becomes

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

Λ

ΛΙ

λ

θ

λ

θ

δ

δ

B

BT

0 (13)

with [ ]ijJ θλ ∂∂∂=Λ 2 , JB θθ −∇= , and JB λλ −∇= .

It is now possible to dramatically reduce the

1θ

...

iθ2θ

1+iθ

nθ

1−nθ

1−iθ

...

y

x

2φ

1φ

iφ

1−iφ

1+iφ

β

β

il

1+il

2l

1l

nl1−nl 1θ

...

iθ2θ

1+iθ

nθ

1−nθ

1−iθ

...

y

x

2φ

1φ

iφ

1−iφ

1+iφ

β

β

il

1+il

2l

1l

nl1−nl

Figure 3. Closed-path constraints on planar boundary.

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dimensions of the matrix inverted at each iteration. By

Eq.(13), we can separately compute the step vector λδ

for the Lagrange multipliers, and express the step

vector θδ for the variables as a function of λδ :

λθλδ BBT −Λ=ΛΛ (14)

λθθ δδ TB Λ−= (15)

Linear equation system in Eq.(14) is 33× , so it can be

efficiently solved using Gaussian elimination [11] if

3)( =ΛΛTrank , or it can be solved by the singular

value decomposition (SVD) [11] if 3)( <ΛΛTrank .

The only left issue is how to efficiently evaluate Λ ,

θB , and λB . For λB , we can easily get

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−

−−

=−∇=

∑∑

∑

kkk

k kk

kk

l

l

n

JB

φ

φ

πθ

λλ

sin

cos

)2()(

. (16)

Before computing Λ and θB , let us first consider

about ik θφ ∂∂ – from Eq.(8), we have

⎩⎨⎧

≥−

<=

∂

∂

ik

ik

i

k

,1

,0

θ

φ. (17)

θθ Bbi∈∀ , the following expression can be derived

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂−++−−=

∂

∂−=

n

k i

kkkykxii

i

l

Jb

i

1

)cossin()(θ

φφλφλλαθ

θ

θ

θ

.

With Eq.(17), i

bθ is in general expressed as

∑=

−++−−=n

ik

kkykxii lbi

)cossin()( φλφλλαθ θθ . (18)

To efficiently compute i

bθ , during iterations, we

usually adopt the following recursion formula:

∑=

−++−−=n

k

kkykx lb

1

11 )cossin()(1

φλφλλαθ θθ (19)

iiyix

iiii

l

bbii

)cossin(

)()( 111

φλφλ

αθαθθθ

−−

−−−+= +++ (20)

Similarly, for [ ]iΛ=Λ , the column vector

( )

( )

( )⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂

∂∂

∂

−−∂

∂

=∂∂

∂=Λ

∑

∑

∑

kkk

i

kkk

i

kk

i

ij

i

l

l

n

J

φθ

φθ

θπθ

θλ

sin

cos

)2(

2

can be evaluated efficiently in the recursion form as

Tiiiiii ll )cos,sin,0(1 φφ−+Λ=Λ + (21)

T

kkk

kkk ll )cos,sin,1(1 ∑∑ −−=Λ φφ . (22)

Using the above formulas, the procedure of

computing optimal iθ s is finally as

while ε>Χ∇ )(J

solve Eq.(14)

compute θδ by Eq.(15)

λδλλ +← and θδθθ +←

End

(23)

In our tests, this procedure usually stops after 8-10

iterations. The optimal planar positions of boundary

vertex Dvi ∂∈ are then computed by Eq.(10). Finally,

the planar position of interior vertices can be

determined by the parameterization approach in [1].

4. Results and Comparisons We test all the examples by our implementation of

the proposed algorithm and the intrinsic

parameterization [1] using C++ (plus OpenGL) on a

PC with PIV 2.99GHz CPU + 1GB RAM. The sparse

linear system BAX = in the intrinsic parameterization

is solved by PCBCG (ref. [11]) with the terminal

conditions setting to the iteration number greater than 4

10 or the error 610/ −≤− BBAX . By our proposed

approach, as the boundary vertices are fixed, the linear

equation system in [1] can be separated into sub-

systems relating to x- and y-coordinates and solved,

which also speeds up the computation because of the

half reduction of system dimension. The time

comparison of our approach and [1] is given in the

computation statistic table (see Table 1), where fn

represents the number of triangles on the given mesh.

Since the natural boundary mode in [1] has no

control on the boundary length, the resultant

parameterization usually shrinks significantly – we

simply scale the parameterization by equaling the total

length of D∂ and M∂ . Checkboard texture is adopted

to evaluate the visual quality of a parameterization on

the results. Besides, the comparisons between the

intrinsic parameterization [1] and ours are also

quantitatively given based on the boundary

morphological metrics θΠ (by Eq.(2)), the edge

distortion error EΠ

∑∑∑

−=Πji

ji

ji

ji

Evv

vv

qq

qq, (24)

and the area distortion error AΠ

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∑ ∑∑ −=Π ΩΩ )(/)()(/)( iiiiA TATATATA (25)

with )(LA and )(LΩA representing the area of a

triangle before and after parameterization.

The first example of our tests includes the front and

the back piece of a polo-shirt, which has been

previously shown in Figure 1 – we conduct checkboard

in Figure 4 to demonstrate the differences between the

results from [1] and our method. It is easy to find that

distortions happen on the results from [1] (Figure 4a)

but not on ours (Figure 4b). Our second example is

lady pants shown in Figure 5, where the results from

[1] also show larger distortions comparing to ours. The

statistics shown in Table 1 also prove this – θΠ , EΠ ,

and AΠ on the results from our method are much less

than the results from [1]. Also, by our method, the

parameterization can be computed faster. Example III

is the mannequin model that is usually used as

benchmark in the papers of surface parameterization.

The results from [1] and ours are similar (see Figure 6),

but ours is much faster. Our last example shown in

Figure 7 is a cube that is symmetrically cut into a disk-

like topology – the parameterization result from ours is

symmetric while the one from [1] is not.

5. Conclusion and Discussion This paper presents a fast approach to compute

length-preserved natural boundary for the

parameterization of a given triangular mesh. The

problem is formulated as a constrained non-linear

optimization problem in angle space, and is speedily

solved by adopting the sequential linearly constrained

programming. The tests show that our length-preserved

natural boundary could greatly improve the speed and

quality of the original intrinsic parameterization

scheme.

Although the constraints derived from the closed-

path theorem have avoided the local-self-intersection

(e.g., the case shown in Figure 8a), the method to

prevent the global-self-intersection (e.g., Figure 8b) is

still under investigation. We are planning to solve the

global-self-intersection problem by borrowing ideas

from [12] in our future research.

6. References [1] M. Desbrun, M. Meyer, and P. Alliez, “Intrinsic

parameterizations of surface meshes,” Proceedings of

EUROGRAPHICS 2002, pp.209-218, 2002.

[2] A. Sheffer, B. Lévy, M. Mogilnitsky, and A.

Bogomyakov, “ABF++: fast and robust angle based

flattening,” ACM Transactions on Graphics, vol.24,

no.2, pp.311-330, 2005.

[3] B. Lévy, S. Petitjean, N. Ray, and J. Maillot,

“Least squares conformal maps for automatic texture

atlas generation”, ACM Transactions on Graphics,

vol.21, no.3, pp.362-371, 2002.

[4] P. Sander, S. Gortler, J. Snyder, and H. Hoppe,

“Signal-specialized parametrization,” Eurographics

Workshop on Rendering 2002, pp.87-100.

[5] P. Sander, J. Snyder, S. Gortler, and H. Hoppe,

“Texture mapping progressive meshes,” Proceedings

of ACM SIGGRAPH 2001, pp.409-416.

[6] S. Yoshizawa, A.G. Belyaev, and H.-P. Seidel, “A

fast and simple stretch-minimizing mesh

parameterization,” Proceedings of International

Conference on Shape Modeling and Applications

2004, pp.200-208.

Figure 4. The checkboard texture for Example I

shown in Figure 1 – (a) texture mapping by the

parameterization results of Desbrun et al. [1] and (b)

texture mapping from our method’s results.

Figure 7. Example V – parameterize a cube (a) after

symmetrically cutting, the result from [1] is

asymmetric (b) while our result (c) is symmetric.

(a) (b)(a) (b)

Figure 8. Self-intersection – (a) local vs. (b) global.

Ninth International Conference on Computer Aided Design and Computer Graphics (CAD/CG 2005)0-7695-2473-7/05 $20.00 © 2005 IEEE

[7] P. Degener, J. Meseth and R. Klein, “An adaptable

surface parameterization method,” Proceedings of 12th

International Meshing Roundtable, pp.201-213, 2003.

[8] M.S. Floater and K. Hormann, “Surface

parameterization: a tutorial and Survey,” Advances in

Multiresolution for Geometric Modelling, N.A.

Dodgson, M.S. Floater, and M.A. Sabin (eds.), pp.157-

186, Springer-Verlag, Heidelberg, 2004.

[9] M.E. Mortenson, Geometric Modeling (2nd

Edition), pp.282-310, Wiley: New York, 1997.

[10] J. Nocedal, and S.J. Wright, Numerical

optimization, Springer-Verlag, 1999.

[11] W.H. Press, B.P. Flannery, S.A. Teukolsky, and

W.T. Vetterling, Numerical Recipes in C: the Art of

Scientific Computing (2nd ed.), Cambridge: Cambridge

University Press, 1995.

[12] A. Sheffer and E. de Sturler, “Parameterization of

faceted surfaces for meshing using angle based

flattening,” Engineering with Computers, vol.17, no.3,

pp.326-337, 2001.

Table 1 Computational Statistics

Intrinsic Parameterization [1] Our method together with Intrinsic Parameterization Example fn

Figure Time θΠ EΠ AΠ Figure Time* θΠ EΠ AΠ

I (front) 3.62k 1b, 4a 0.704 0.020 0.094 0.186 1c, 4b 0.157 0.006 0.038 0.067

I (back) 4.08k 1b, 4a 0.906 0.034 0.189 0.353 1c, 4b 0.187 0.021 0.029 0.053

II (front) 2.61k 5b 0.547 0.010 0.113 0.233 5c 0.078 0.005 0.051 0.072

II (back) 2.24k 5b 0.359 0.010 0.175 0.331 5c 0.078 0.006 0.029 0.038

III 44.4k 6b 65.6 0.064 1.037 1.457 6c 18.0 0.064 1.006 1.420

IV 0.8k 7b 0.062 0.039 0.299 0.633 7c 0.016 0.039 0.273 0.485

*The time reported here includes both the computation of natural boundary and the followed intrinsic parameterization.

Figure 5: Example II – mesh parameterization for lady pants, where (a) the given 3D mesh surface patches, (b) the

results of the front and the back pieces by the intrinsic parameterization in [1], and (c) the results from our

boundary followed by the intrinsic parameterization. The parameterizations are colored using the 3D normal map.

Figure 6: Example III – parameterization for the refined mannequin model with (a) the 3D model, (b) the

parameterization result of [1], and (c) the result from our natural boundary + intrinsic parameterization.

Ninth International Conference on Computer Aided Design and Computer Graphics (CAD/CG 2005)0-7695-2473-7/05 $20.00 © 2005 IEEE