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
Copyright
c
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
2
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
2
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
2
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
2
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
2
stretch: 1.194532. (b) L
2
stretch: 1.316625. (c) L
2
stretch: 1.450614. (d) L
2
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
2
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).
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