FAST WAY TO CREATE SEAM BOUNDARY FOR SQUARE

PARAMETERIZATION WITH LOW-DISTORTION

Anuwat Dechvijankit, Hiroshi Nagahashi and Kota Aoki

Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, Tokyo, Japan

Keywords:

Mesh Parameterization, Seam Cutting.

Abstract:

In order to parameterize any three-dimensional surface like a closed boundary one, we need to convert its

polygonal mesh into a disk topology surface. For the quality of texturing or re-meshing that uses a parame-

terization technique, it is more effective if the distortion of two-dimensional manifold planar domain map is

as small as possible. We introduce a fast way to create seams along the three-dimensional surface for square

boundary of a two-dimensional planar domain map. This paper describes a faster and less error technique

than the existing seam cutting methods by simply observing the location of high distortion area in the planar

domain. The distortion rate of our proposed method is low as the original one but with less time consuming.

1 INTRODUCTION

Mesh parameterization is deﬁned as a mapping be-

tween a 3D manifold surface and a suitable target do-

main. To enable the mapping of a 3D surface into

a 2D planar domain, mesh parameterization requires

the 3D surface to be topologically equivalent to a disk

without any hole, which can be easily mapped. Con-

sequently, closed-boundary surfaces such as 3D mod-

els cannot be parameterized directly. Moreover, sur-

faces having any holes or being non-genus zero also

cannot be parameterized directly too. To solve the

above problem, we need to convert a 3D surface into

disk topology by cutting seams into the surface to

generate a boundary for parameterization. Parameter-

ization between these two domains causes distortion

errors such as stretch. Additionally, too short/long

seam boundary or random seam without any strategy

may give poor results of distortion easily.

We enhance the original method found in geome-

try images (Gu et al., 2002) by following the strategy

of cutting seams through high curvature areas (e.g.

ﬁngers, tails, ear) and connecting them together with

the shortest path. This strategy can improve distor-

tion error when the parameterization is made onto a

square planar domain. The difference from the orig-

inal one is that we bypass unnecessary time consum-

ing parameterization at the initial state and directly

do unit square parameterization when all appropriate

seams have been found at one time.

2 RELATED WORK

From a recent survey (Sheffer et al., 2006), there

are several methods for seam cutting that have been

proposed. (Gu et al., 2002), (Erickson and Har-

Peled, 2002) and (Ni et al., 2004) dealt with genus-

reducing, while (Lazarus et al., 2001) extracted

canonical schema and (Sheffer and Hart, 2002) found

high Gaussian curvature on the surface.

Dealing with high-genus models, (Erickson and

Har-Peled, 2002) has proposed a cutting method

which has some elegant theoretical guarantees but is

complex to implement. They ﬁnd the shortest loop

path connecting a vertex to the vertex itself by using

a front propagation technique, and then tests to see if

the considering loop path reduces the surface genus

or simply cut the surface into two pieces. The gener-

ation of minimal length cuts that convert a high genus

surface into a topological disk is a NP-hard problem.

The method used in (Erickson and Har-Peled, 2002) is

a brute force approach which consumes a lot of time.

The Seamster algorithm (Sheffer and Hart, 2002)

considers the differential geometry properties of the

surface which are independent of a particular param-

eterization technique. It ﬁrst ﬁnds regions of high

Gaussian curvature on the mesh and then uses a mini-

mal spanning tree of the mesh edges to connect them.

Visibility of edges is used as the weight of the short-

est path algorithms. When dealing with a high-genus

model, (Erickson and Har-Peled, 2002) is need to cre-

ate genus-reduce cutting ﬁrst.

185

Dechvijankit A., Nagahashi H. and Aoki K..

FAST WAY TO CREATE SEAM BOUNDARY FOR SQUARE PARAMETERIZATION WITH LOW-DISTORTION.

DOI: 10.5220/0003811701850188

In Proceedings of the International Conference on Computer Graphics Theory and Applications (GRAPP-2012), pages 185-188

ISBN: 978-989-8565-02-0

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

An practical approach is proposed by (Gu et al.,

2002), which traces a spanning graph of all the faces

in the mesh and then prunes this graph to obtain

a genus reducing cut. Then they parameterize the

surface by using circular shape-preserving (Floater,

1997). They ﬁnd high curvature position at the high-

est stretch on the mapping, and merge the boundary

edges and that position with the shortest path between

them. The process is terminated when the distortion

of square parameterization increases.

3 OUR PROPOSED METHOD

We have investigated the creation method of geome-

try image (Gu et al., 2002) from any kinds of meshes.

A goal of the creation method is to deliver the low-

est stretch of square parameterization. In practical

implementation, it takes a lot of computational time

during the square parameterizations to determine it-

eration’s termination. Since geometric-stretch param-

eterization (Sander et al., 2001; Sander et al., 2002)

has been used for stretch-minimizing method, it takes

a lot of computational time. Even though we change it

to a faster method, most of stretch-minimizing param-

eterization methods still require much solving time.

Besides the time consuming issue, we sometimes

have a poor result when dealing with a mesh contain-

ing holes as shown in ﬁgure 1(a). From our investiga-

tion, it proved that the hole-connected paths generated

in the phase of genus reduction cause unbalanced den-

sity of interior faces inside the boundary. This prob-

lem also affects to the iterated cut augmentation phase

that fails to detect actual high curvature area because

the highest stretch face will be located at boundary

area instead.

We propose an enhancement of seam cutting

method in geometry image (Gu et al., 2002), by im-

proving the hole-connected paths and time consuming

problems, which still maintains low stretch square pa-

rameterization result as original.

3.1 Hole-connected Paths

To avoid the problem of poor path connection among

holes, we detect a boundary of the surface ﬁrst in the

phase of genus reducing. If one or more boundaries

(holes) have been detected, we generate a cut-path by

connecting these holes with shortest-path edges to-

gether. After that, we consider the hole-connected

paths and existing holes as a single hole and then ap-

ply the genus reducing method in (Gu et al., 2002).

See ﬁgure 1(b) for better quality stretch at boundary

area after applying our approach.

(a) (b)

Figure 1: Comparison of stretch at bases of Stanford bunny.

(a) is the result using original genus reducing algorithms

directly; (b) is the result using our proposed algorithms by

connecting holes with the shortest path ﬁrst. Both cases

are parameterized after selecting the best corner points that

deliver lowest stretch already. The red lines indicate the

boundary edges of that mesh (cut-path).

3.2 Termination Condition

The methods of geometry images (Gu et al., 2002)

proposed that seam boundary should pass through

various high curvature areas in order to obtain low

stretch parameterization result. If we observe the

properties of non-natural boundaries in state-of-art

parameterizations, we can notice that too short or too

long boundary edges will affect distortion in the same

manner in terms of the location of the highest stretch

face in planar domain.

Our approach is to ignore the comparison of L

stretch of square stretch-minimizing parameterization

in each state as mentioned in geometry images. We

directly keep detecting high-curvature areas by us-

ing shape-preserving circular parameterization itera-

tively. The iteration stops when highest stretch face

locates very near to the boundary or boundary face

itself. Also, the iteration can terminate when L

stretch of shape-preserving circular parameterization

becomes lower than a speciﬁed threshold value.

The concept of this approach is to use the shape-

preserving circular parameterization as a prediction of

square parameterization. If we have too short bound-

ary edges, we will have over-pack of interior faces

that give high stretch values. Vice versa, if we have

too long boundary edges, we will have over-pack of

boundary faces that give high stretch at boundary

faces or interior faces nearby boundary.

The stretch of circular parameterization is also im-

portant too. Since we keep extending cutting-paths

during iterated cut augmentation, the number and

length of boundary edges are being increased from

the beginning. If the number and length of bound-

ary edges reach the best condition, then the circu-

lar parameterization should give the lowest stretch.

GRAPP 2012 - International Conference on Computer Graphics Theory and Applications

186

Table 1: Comparison of computational time and L

stretch between ours and the original method. We set low stretch threshold

value (for stop iteration) to 2.0. Unit square parameterization was computed by using the method (Yoshizawa et al., 2004)

with random boundary positions. In the “Reason that stopped iteration in our approach” ﬁeld, “A” means the highest stretch

triangle was boundary area or boundary face itself; “B” means the circular stretch is lower than threshold value.

Model genus number of time (seconds) Stopped iteration L

stretch

open/closed vertices/faces our original reason in our approach our original

Mannequin head 0-open (689/1355) 0.078 0.156 AB(1.84/2.0) 1.226 1.226

Hand 0-closed (1002/2000) 0.34 0.982 A (2.10/2.0) 1.485 1.485

Max-Planck 0-closed (49132/98260) 98 495 A (2.26/2.0) 1.257 1.257

Bunny 0-closed (35947/69451) 26 36 A (2.19/2.0) 1.409 2336

Bunny(ﬁlled hole) 0-closed (34823/69642) 51 252 B (1.88/2.0) 1.196 1.196

Cow 4-closed (16612/33244) 20.55 21.53 A (29.7/2.0) 93009 153106

Dragon 1-closed (50000/100000) 139 152 A (98.7/2.0) 10.94 21.68

If the shape-preserving circular parameterization can

achieve a low stretch result, then stretch-minimizing

square parameterization method with same boundary

edges most likely delivers low stretch result too. In

other words, if we stop iteration by only detecting

the highest stretch at boundary area without consid-

ering stretch from circular parameterization, it might

give worse square parameterization result due to the

boundary constraint or too long boundary edges. Also

if it gives a better result, the margin is slightly small.

4 RESULTS

We apply our approach to various models, both open

and closed meshes, genus zero and non-genus zero

models. Our test system was on an Intel Core 2 Quad

2.67GHz with 4GB RAM computer.

We compare computational time between our

and original approaches for ﬁnding seam-cutting

path. The original approach returns seam-cutting

path and square parameterization result, but our ap-

proach does not give square parameterization result.

Therefore, we also do square parameterization af-

ter our approach is ﬁnished. To reduce calculation

time, we use method (Yoshizawa et al., 2004) as

stretch-minimizing square parameterization in both

approaches instead of geometric-stretch method. Ta-

ble 1 shows the result of the computational time.

In case of high genus models, the unit square pa-

rameterization results are poor because genus-reduce

cutting paths are not the shortest loop as we ex-

pected. Same as hole-connected paths problem, the

long boundary edges while covering small areas in

3D domain might cause unbalanced density of interior

faces inside the boundary in square planar domain. If

we manage to create the shortest genus-reduce cutting

of these high genus models, square parameterization

might give a better result. Even though we got poor

genus-reduce cutting paths, our algorithm still could

detect high curvature area more than the original one.

See ﬁgure 2 as the results of cow model.

If seam-cutting edges pass through every high cur-

vature areas, it is not true that it will give the best re-

sult. We extended seam-cutting edges to another high

curvature area after our method was ﬁnished. The

results show that they do not give a signiﬁcant bet-

ter result because the boundary edges are already too

long since the beginning. A worse result was found in

bunny model and a small margin difference was found

in armadillo model. (see ﬁgures 3)

(a) Original approach. (b) Our approach.

Figure 2: Show cow models with seam cutting paths. (a)

shows the result from original approach that enables to de-

tect high curvature area only 2 areas. (b) shows our result

that enables to detect high curvature area 6 areas.

5 CONCLUSIONS

We proposed an enhanced method of seam cutting

method (Gu et al., 2002) in order to calculate appro-

priate seam boundary over a three-dimensional sur-

face, which becomes a square boundary in planar do-

main. We try to convert any closed or open surface

into the disk topological patch. By observing the

highest stretch area and its stretch value in shape-

preserving circular parameterization result, we use

this information to predict the unit square parame-

terization result. Also the seam cutting path of the

FAST WAY TO CREATE SEAM BOUNDARY FOR SQUARE PARAMETERIZATION WITH LOW-DISTORTION

187

(a) L

stretch: 1.194532. (b) L

stretch: 1.316625. (c) L

stretch: 1.450614. (d) L

stretch: 1.437329.

Figure 3: Models with seam-cutting paths and check-board texture mapping using square parameterization results. (a) shows

the results using our approach which terminated cut augmentation process when stretch of circle parameterization becomes

lower than threshold (2.0). (b) shows the results that manually extend seam-cutting paths to the highest stretch face in (a).

(c) shows the results using our approach which terminated cut augmentation process when the highest stretch face locate at

boundary area. (d) shows the results that manually extend seam-cutting paths to another high-curvature area (right-side ear).

All cases are parameterized after selecting best corner points that deliver lowest stretch already. (a) has a better result than

(b), (d) has slightly a better result than (c); judging from L

stretch value.

original method was depended on a starting face, and

the obtained seam boundary that connected holes to-

gether might cause extremely under and over stretch

around the boundary area. Hence, we avoided the

problem by connecting them with the shortest path

ﬁrst before further analysis.

The issue that we are still interested in is how

to ﬁnd the perfect seam boundary length. When the

highest stretch face is at boundary area, we think that

we have over-pack faces at boundary area situation

because boundary edges are too long already. We are

also interested in how to assign positions in boundary

that deliver the lowest stretch. Different positions give

different distortion. Additionally, our method is still

time consuming because of a brute force algorithm

that checks almost possible positions.

ACKNOWLEDGEMENTS

We would like to gratefully thank Shin Yoshizawa

for helpful advice and discussion including C++ code

of (Yoshizawa et al., 2004), Hugues Hoppe for ﬁlled

holes bunny and hand model data, Christian Rau

who developed OpenGI library and all reviewers,

The models are courtesy of the Stanford Univer-

sity (bunny, dragon and armadillo), the University of

Washington (mannequin head), MPI f

ur Informatik

(Max-Planck), AIM@SHAPE(cow).

REFERENCES

Erickson, J. and Har-Peled, S. (2002). Optimally cutting a

surface into a disk. In Proceedings of the eighteenth

annual symposium on Computational geometry, SCG

’02, pages 244–253, New York, NY, USA. ACM.

Floater, M. S. (1997). Parametrization and smooth ap-

proximation of surface triangulations. Comput. Aided

Geom. Des., 14:231–250.

Gu, X., Gortler, S. J., and Hoppe, H. (2002). Geometry

images. ACM Trans. Graph., 21:355–361.

Lazarus, F., Pocchiola, M., Vegter, G., and Verroust, A.

(2001). Computing a canonical polygonal schema of

an orientable triangulated surface. In Proceedings of

the seventeenth annual symposium on Computational

geometry, SCG ’01, pages 80–89. ACM.

Ni, X., Garland, M., and Hart, J. C. (2004). Fair morse

functions for extracting the topological structure of a

surface mesh. In ACM SIGGRAPH 2004 Papers, SIG-

GRAPH ’04, pages 613–622, New York, NY, USA.

ACM.

Sander, P. V., Gortler, S. J., Snyder, J., and Hoppe,

H. (2002). Signal-specialized parametrization. In

Proceedings of the 13th Eurographics workshop on

Rendering, EGRW ’02, pages 87–98, Aire-la-Ville,

Switzerland, Switzerland. Eurographics Association.

Sander, P. V., Snyder, J., Gortler, S. J., and Hoppe, H.

(2001). Texture mapping progressive meshes. In Pro-

ceedings of the 28th annual conference on Computer

graphics and interactive techniques, SIGGRAPH ’01,

pages 409–416, New York, NY, USA. ACM.

Sheffer, A. and Hart, J. C. (2002). Seamster: inconspicuous

low-distortion texture seam layout. In Proceedings of

the conference on Visualization ’02, VIS ’02, pages

291–298, Washington, DC, USA. IEEE Computer So-

ciety.

Sheffer, A., Praun, E., and Rose, K. (2006). Mesh param-

eterization methods and their applications. In Foun-

dations and Trends in Computer Graphics and Vision,

page 2006. Now Publishers.

Yoshizawa, S., Belyaev, A., and Seidel, H.-P. (2004). A

fast and simple stretch-minimizing mesh parameteri-

zation. In SMI ’04: Proceedings of the Shape Model-

ing International 2004, pages 200–208, Washington,

DC, USA. IEEE Computer Society.

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188