CVGIP 2010 The 23th IPPR Conference on Computer Vision, Graphics, and Image Processing

1. 3-D Environment Model Construction and Adaptive
Foreground Detection for Multi-Camera Surveillance System
Yi-Yuan Chen1† , Hung-I Pai2† , Yung-Huang Huang∗ , Yung-Cheng Cheng∗ , Yong-Sheng Chen∗
Jian-Ren Chen† , Shang-Chih Hung† , Yueh-Hsun Hsieh† , Shen-Zheng Wang† , San-Lung Zhao†
†
Industrial Technology Research Institute, Taiwan 310, ROC
∗
Department of Computer and Information Science, National Chiao-Tung University, Taiwan 30010, ROC
E-mail:1 yiyuan@itri.org.tw, 2 HIpai@itri.org.tw
Abstract— Conventional surveillance systems usually use
multiple screens to display acquired video streams and may
cause trouble to keep track of targets due to the lack of spatial
relationship among the screens. This paper presents an effective
and efficient surveillance system that can integrate multiple
video contents into one single comprehensive view. To visualize
the monitored area, the proposed system uses planar patches to
approximate the 3-D model of the monitored environment and
displays the video contents of cameras by applying dynamic
texture mapping on the model. Moreover, a pixel-based shadow
detection scheme for surveillance system is proposed. After
an offline training phase, our method exploits the threshold
which determines whether a pixel is in a shadow part of the Fig. 1. A conventional surveillance system with multiple screens.
frame. The thresholds of pixels would be automatically adjusted
and updated according to received video streams. The moving
objects are extracted accurately with removing cast shadows and
then visualized through axis-aligned billboarding. The system
direction of cameras, and locations of billboards indicate
provides security guards a better situational awareness of the the positions of cameras, but the billboard contents will be
monitored site, including the activities of the tracking targets. hard to perceive if the angles between viewing direction and
Index Terms— Video surveillance system, planar patch mod- normal directions of billboards are too large. However, in
eling, axis-aligned billboarding, cast shadow removal rotating billboard method, when the billboard rotates and
faces to the viewpoint of user, neither camera orientations
I. I NTRODUCTION nor capturing areas will be preserved. In outdoor surveillance
Recently, video surveillance has experienced accelerated system, an aerial or satellite photograph can be used as
growth because of continuously decreasing price and better a reference map and some measurement equipments are
capability of cameras [1] and has become an important used to build the 3-D environment [3]–[5]. Neumann, et al.
research topic in the general field of security. Since the utilized an airborne LiDAR (Light Detection and Ranging)
monitored regions are often wide and the field of views sensor system to collect 3-D geometry samples of a specific
of cameras are limited, multiple cameras are required to environment [6]. In [3], image registration seams the video on
cover the whole area. In the conventional surveillance system, the 3-D model. Furthermore, video projection, such as video
security guards in the control center monitor the security flashlight or virtual projector, is another way to display video
area through a screen wall (Figure 1). It is difficult for the in the 3-D model [4], [7].
guards to keep track of targets because the spatial relationship However, the multi-camera surveillance system still has
between adjacent screens is not intuitively known. Also, it is many open problems to be solved, such as object tracking
tiresome to simultaneously gaze between many screens over across cameras and object re-identification. The detection of
a long period of time. Therefore, it is beneficial to develop a moving objects in video sequences is the first relevant step in
surveillance system that can integrate all the videos acquired the extraction of information in vision-based applications. In
by the monitoring cameras into a single comprehensive view. general, the quality of object segmentation is very important.
Many researches on integrated video surveillance systems The more accurate positions and shapes of objects are, the
are proposed in the literature. Video billboards and video more reliable identification and tracking will be. Cast shadow
on fixed planes project camera views including foreground detection is an issue for precise object segmentation or
objects onto individual vertical planes in a reference map to tracking. The characteristics of shadow are quite different
visualize the monitored area [2]. In fixed billboard method, in outdoor and indoor environment. The main difficulties in
billboards face to specified directions to indicate the capturing separating the shadow from an interesting object are due to
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2. the physical property of floor, directions of light sources and System configuration On-line monitoring
additive noise in indoor environment. Based on brightness
Manual operation video
and chromaticity, some works are proposed to decide thresh- streams
olds of these features to roughly detect the shadow from
objects [8]–[10]. However, current local threshold methods 2D 3D
patterns model Background modeling
couple blob-level processing with pixel-level detection. It
causes the performance of these methods to be limited due
to the averaging effect of considering a big image region. Registration with Segmentation and
corresponding points refinement
Two works to remove shadow are proposed to update
the threshold with time and detect cast shadow in different
Axis-aligned
scenes. Carmona et al [11] propose a method to detect Lookup billboarding
shadow by using the properties of shadow in Angle-Module tables
space. Blob-level knowledge is used to identify shadow, 3-D model
refection and ghost. This work also proposes a method to construction
update the thresholds to remove shadow in different positions
of the scene. However there are many undetermined param- Fig. 2. The flowchart and components of the proposed 3-D surveillance
system.
eters to update the thresholds and the optimal parameters are
hard to find in practice. Martel-Brisson et al [12] propose
a method, called GMSM, which initially uses Gaissian of
Mixture Model (GMM) to define the most stable Gaussian
distributions as the shadow and background distributions.
Since a background model is included in this method, more
computation is needed for object segmentation if a more com-
plex background model is included in the system. Besides,
because that each pixel has to be updated no matter how many
objects moving, it cost more computation in few objects.
In this paper, we develop a 3-D surveillance system based
on multiple cameras integration. We use planar patches to
build the 3-D environment model firstly and then visualize
videos by using dynamic texture mapping on the 3-D model.
To obtain the relationship between the camera contents and
the 3-D model, homography transformations are estimated
for every pair of image regions in the video contents and
the corresponding areas in the 3-D model. Before texture Fig. 3. Planar patch modeling for 3-D model construction. Red patches
mapping, patches are automatically divided into smaller ones (top-left), green patches (top-right), and blue patches (bottom-left) represent
with appropriate sizes according to the environment. Lookup the mapping textures in three cameras. The yellow point is the origin of
the 3-D model. The 3-D environment model (bottom-right) is composed of
tables for the homography transformations are also built for horizontal and vertical patches from these three cameras.
accelerating the coordinate mapping in the video visual-
ization processing. Furthermore, a novel method to detect
moving shadow is also proposed. It consists of two phases. quired from IP cameras deployed in the scene to the 3-D
The first phase is an off-line training phase which determines model by specifying corresponding points between the 3-D
the threshold of every pixel by judging whether the pixel is model and the 2-D images. Since the cameras are fixed, this
in the shadow part. In the second phase, the statistic data configuration procedure can be done only once beforehand.
of every pixel is updated with time, and the threshold is Then in the on-line monitoring stage, based on the 3-D
adjusted accordingly. By this way, a fixed parameters setting model, all videos will be integrated and visualized in a single
for detecting shadow can be avoided. The moving objects are view in which the foreground objects extracted from images
segmented accurately from the background and are displayed are displayed through billboards.
via axis-aligned billboarding for better 3-D visual effects.
A. Image registration
II. S YSTEM CONFIGURATION
For a point on a planar object, its coordinates on the plane
Figure 2 illustrates the flowchart of constructing the pro- can be mapped to 2-D image through homography citeho-
posed surveillance system. First, we construct lookup tables mography, which is a transformation between two planar
for the coordinate transformation from the 2-D images ac- coordinate systems. A homography matrix H represents the
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3. relationship between points on two planes:
sct = Hcs , (1)
where s is a scalar factor and cs and ct are a pair of corre-
sponding points in the source and target patches, respectively.
If there are at least four correspondences where no three
correspondences in each patch are collinear, we can estimate
H through the least-squares approach.
We regard cs as points of 3-D environment model and ct
as points of 2-D image and then calculate the matrix H to
map points from the 3-D model to the images. In the reverse
order, we can also map points from the images to the 3-D
model.
B. Planar patch modeling
Precise camera calibration is not an easy job [13]. In the Fig. 4. The comparison of rendering layouts between different numbers
and sizes of patches. A large distortion occurs if there are fewer patches for
virtual projector methods [4], [7], the texture image will be rendering (left). More patches make the rendering much better (right).
miss-aligned to the model if the camera calibration or the
3-D model reconstruction has large error. Alternatively, we
develop a method that approximates the 3-D environment where Iij is the intensity of the point obtained from homog-
model through multiple yet individual planar patches and ˜
raphy transformation, Iij is the intensity of the point obtained
then renders the image content of every patches to generate from texture mapping, i and j are the coordinates of row and
a synthesized and integrated view of the monitored scene. In column in the image, respectively, and m × n represents the
this way we can easily construct a surveillance system with dimension of the patch in the 2-D image. In order to have
3-D view of the environment. an reference scale to quantify the distortion amount, a peak
Mostly we can model the environment with two basic signal-to-noise ratio is calculated by
building components, horizontal planes and vertical planes.
The horizontal planes for hallways and floors are usually MAX2
I
surrounded by doors and walls, which are modeled as the PSNR = 10 log10 , (3)
MSE
vertical planes. Both two kinds of planes are further divided
into several patches according to the geometry of the scenes where MAXI is the maximum pixel value of the image.
(Figure 3). If the scene consists of simple structures, a few Typical values for the PSNR are between 30 and 50 dB and
large patches can well represent the scene with less rendering an acceptable value is considered to be about 20 dB to 25 dB
costs. On the other hand, more and smaller patches are in this work. We set a threshold T to determine the quality
required to accurately render a complex environment, at the of texture mapping by
expense of more computational costs.
In the proposed system, the 3-D rendering platform is PSNR ≥ T . (4)
developed on OpenGL and each patch is divided into tri-
angles before rendering. Since linear interpolation is used If the PSNR of the patch is lower than T , the procedure
to fill triangles with texture in OpenGL and not suitable divides it into smaller patches and repeats the process until
for the perspective projection, distortion will appear in the the PSNR values of every patches are greater than the given
rendering result. One can use a lot of triangles to reduce this threshold T .
kind of distortion, as shown in Figure 4, it will enlarge the
computational burden and therefore not feasible for real-time III. O N - LINE MONITORING
surveillance systems.
To make a compromise between visualization accuracy and The proposed system displays the videos on the 3-D model.
rendering cost, we propose a procedure that automatically However, the 3-D foreground objects such as pedestrians are
divides each patch into smaller ones and decides suitable projected to image frame and become 2-D objects. They will
sizes of patches for accurate rendering (Figure 4). We use the appear flattened on the floor or wall since the system displays
following mean-squared error method to estimate the amount them on planar patches. Furthermore, there might be ghosting
of distortion when rendering image patches: effects when 3-D objects are in the overlapping areas of
m−1 n−1 different camera views. We need to tackle this problem by
1 ˜
MSE = (Iij − Iij )2 , (2) separating and rendering 3-D foreground objects in addition
m×n i=0 j=0 to the background environment.
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4. our method such that the background doesn’t have to be
determined again.
In the indoor environment, we assume the color in eq.(7) is
similar between shadow and background in a pixel although
it is not evidently in sunshine in outdoor. Only the case of
indoor environment is considered in this paper.
Fig. 5. The tracking results obtained by using different shadow thresholds B. Collecting samples
while people stand on different positions of the floor. (a) Tr = 0.8 (b) Samples I(x, y, t) in some frames are collected to decide
Tr = 0.3. The threshold value Tθ = 6o is the same for both.
the shadow area, where t is the time. In [12] all samples are
collected including the classification of background, shadow
A. Underlying assumption and foreground by the pixel value changed with time. But
if a good background model has already built and some
Shadow is a type of foreground noise. It appears in any initial foreground objects were segmented, the background
zone of the camera scene. In [8], each pixel belongs to samples are not necessary. Only foreground and shadow
a shadow blob is detected by two properties. First, the samples If (x, y, t) were needed to consider. Besides, since
color vector of a pixel in shadow blob has similar direction background pixels are dropped from the samples list, this can
to that of the background pixel in the same position of save the computer and memory especially in a scene with
image. Second, the magnitude of the color vector in the T
few objects. Future, If θ (x, y, t) is obtained by dropping the
shadow is slightly less than the corresponding color vector of samples which not satisfy inequality eq.(7) from If (x, y, t).
background. Similar to [11], RGB or other color space can be Obviously, the samples data composed of more shadows
transformed into two dimensional space (called angle-module samples and less foreground samples. This also leads to that
space). The color vector of a pixel in position (x, y) of current the threshold r(x, y, t) can be derived more easily than the
frame, Ic (x, y), θ(x, y) is the angle between background threshold derived from samples of If (x, y, t).
vector Ib (x, y) and Ic (x, y), and the magnitude ratio r(x, y)
are defined as C. Deciding module ratio threshold
arccos(Ic (x, y) · Ib (x, y)) The initial threshold Tθ (x, y, 0) is set according to the
θ(x, y) = (5) experiment. In this case, Tθ (x, y, 0) = cos(6◦ ) is set as
|Ic (x, y)||Ib (x, y)| +
the initial value. After collecting enough samples, the ini-
|Ic (x, y)|
r(x, y) = (6) tial module ratio threshold Tr (x, y, 0) can be decided by
|Ib (x, y)| this method, Fast step minimum searching (FSMS). FSMS
where is a small number to avoid zero denominator. In [11], can fast separate the shadow from foreground distribution
the shadow of a pixels have to satisfy which collected samples are described above. The detail of
this method is described below. The whole distribution is
Tθ < cos θ(x, y) < 1 (7) separated by a window size w. The height of each window
Tr < r(x, y) < 1 (8) is the sum of the samples. Besides the background peak, two
peaks were found. The threshold Tr is used to search the
where Tθ is the angle threshold and Tr is the module peak which is closest to the average background value and
ratio threshold. According to the demonstration showed in smaller than the background value, the shadow threshold can
Figure 5, the best shadow thresholds are highly depends on be found by searching the minimum value or the value close
positions (pixels) in the scene, because of the complexity to zero.
of environment, the light sources and objects positions.
Therefore, we propose a method to automatically adjust the D. Updating angle threshold
thresholds for detecting shadow for each pixel. The threshold When a pixel satisfies both conditions in inequality eq.(7,
for a pixel to be classified as shadow or not is determined by 8) at the same time, the pixel is classified as shadow. In other
the necessary samples (data) collected with time. Only one words, if the pixel Is (x, y) is actually a shadow pixel, and
c is classified as one of candidate of shadow by FSMS, the
parameter has to be manually initialized. It is Tθ (0), where
0 means the initial time. Then the method can update the property of the pixel is require to satisfy the below equation
thresholds automatically and fast. Our method is faster than at the same time
the similar idea, GMSM method [12], when a background
model has built up. There are two major advantages of
0 ≤ cos θ(x, y, t) < Tθ (x, y, t) (9)
the computation time for our method. First, only necessary
samples are collected. Second, compared with method [12], Tθ (x, y, t) can be decided by searching the minimum
any background or foreground results can combine with cos(θ) of pixels in Is which is obtained by FSMS. However
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5. Fig. 7. Orientation determination of the axis-aligned billboarding. L is
the location of the billboard, E is the location projected vertically from the
viewpoint to the floor, and v is the vector from L to E. The normal vector
(n) of the billboard is rotated according to the location of the viewpoint. Y
is the rotation axis and φ is the rotation angle.
Fig. 6. A flowchart to illustrate the whole method. The purple part is based
on pixel.
are always moving on the floor, the billboards can be aligned
to be perpendicular to the floor in the 3-D model. The 3-D
we propose another method to find out Tθ (x, y, t) more fast. location of the billboard is estimated by mapping the bottom-
The number of samples which are classified as shadow or middle point of the foreground bounding box in the 2-D
background at time t is ATr (x, y, t) by using FSMS. We
{b,s} image through the lookup tables. The ratio between the height
define a ratio R(Tr ) = ATr /A{b,s,f } where A{b,s,f } is all
{b,s} of the bounding box and the 3-D model determines the height
samples in position x, y, where b, s, f represent the back- of the billboard in the 3-D model. The relationship between
ground, shadow and foreground respectively. The threshold the direction of a billboard and the viewpoint is defined as
Tθ (x, y, t) can be updating to Tθ (x, y, t) by R(Tr ). The shown in Figure 7.
number of samples whose cos(θ(x, y)) values are larger than The following equations are used to calculate the rotation
the Tθ (x, y, t) is equal to A{b,s} and is required angle of the billboard:
R(Tθ (x, y, t)) = R(Tr ) (10) Y = (n × v) , (12)
Besides, we add a perturbation δTθ to the Tθ (x, y, t).
T
Since FSMS only finds out a threshold in If θ (x, y, t), if the φ = cos−1 (v · n) , (13)
initial threshold Tθ (x, y, 0) is set larger than true threshold,
the best updating threshold is equal to threshold Tθ not where v is the vector from the location of the billboard, L, to
smaller than threshold Tθ . Therefore the true angle threshold the location E projected vertically from the viewpoint to the
will never be found with time. To solve this problem, a per- floor, n is the normal vector of the billboard, Y is the rotation
turbation of the updating threshold is added to the updating axis, and φ is the estimated rotation angle. The normal vector
threshold of the billboard is parallel to the vector v and the billboard
is always facing toward the viewpoint of the operator.
Tθ (x, y, t) = Tθ (x, y, t) − δTθ (11)
F. Video content integration
Since the new threshold Tθ (x, y, t) has smaller value
to cover more samples, it can approach the true threshold If the fields of views of cameras are overlapped, objects in
with time. This perturbation can also make the method more these overlapping areas are seen by multiple cameras. In this
adaptable to the change of environment. Here is a flowchart case, there might be ghosting effects when we simultaneously
Figure 6 to illustrate the whole method. display videos from these cameras. To deal with this problem,
we use 3-D locations of moving objects to identify the cor-
E. Axis-aligned billboarding respondence of objects in different views. When the operator
In visualization, axis-aligned billboarding [14] constructs chooses a viewpoint, the rotation angles of the corresponding
billboards in the 3-D model for moving objects, such as billboards are estimated by the method presented above and
pedestrians, and the billboard always faces to the viewpoint of the system only render the billboard whose rotation angle is
the user. The billboard has three properties: location, height, the smallest among all of the corresponding billboards, as
and direction. By assuming that all the foreground objects shown in Figure 8.
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6. C1 C3
C2
C1
Fig. 8. Removal of the ghosting effects. When we render the foreground
object from one view, the object may appear in another view and thus
cause the ghosting effect (bottom-left). Static background images without Fig. 9. Determination of viewpoint switch. We divide the floor area
foreground objects are used to fill the area of the foreground objects (top). depending on the fields of view of the cameras and associate each area to one
Ghosting effects are removed and static background images can be update of the viewpoint close to a camera. The viewpoint is switched automatically
by background modeling. to the predefined viewpoint of the area containing more foreground objects.
G. Automatic change of viewpoint The experimental results shown in Figure 12 demonstrate
that the viewpoint can be able to be chosen arbitrarily in
The proposed surveillance system provides target tracking the system and operators can track targets with a closer
feature by determining and automatic switching the view- view or any viewing direction by moving the virtual camera.
points. Before rendering, several viewpoints are specified in Moreover, the moving objects are always facing the virtual
advance to be close to the locations of the cameras. During camera by billboarding and the operators can easily perceive
the viewpoint switching from one to another, the parameters the spatial information of the foreground objects from any
of the viewpoints are gradually changed from the starting viewpoint.
point to the destination point for smooth view transition.
The switching criterion is defined as the number of blobs V. C ONCLUSIONS
found in the specific areas. First, we divide the floor area into In this work we have developed an integrated video surveil-
several parts and associate them to each camera, as shown lance system that can provide a single comprehensive view
in Figure 9. When people move in the scene, the viewpoint for the monitored areas to facilitate tracking moving targets
is switched automatically to the predefined viewpoint of the through its interactive control and immersive visualization.
area containing more foreground objects. We also make the We utilize planar patches for 3-D environment model con-
billboard transparent by setting the alpha value of textures, so struction. The scenes from cameras are divided into several
the foreground objects appear with fitting shapes, as shown patches according to their structures and the numbers and
in Figure 10. sizes of patches are automatically determined for compromis-
ing between the rendering effects and efficiency. To integrate
IV. E XPERIMENT RESULTS video contents, homography transformations are estimated
for relationships between image regions of the video contents
We developed the proposed surveillance system on a PC
and the corresponding areas of the 3D model. Moreover,
with Intel Core Quad Q9550 processor, 2GB RAM, and one
the proposed method to remove moving cast shadow can
nVidia GeForce 9800GT graphic card. Three IP cameras with
automatically decide thresholds by on-line learning. In this
352 × 240 pixels resolution are connected to the PC through
way, the manual setting can be avoided. Compared with the
Internet. The frame rate of the system is about 25 frames per
work based on frames, our method increases the accuracy to
second.
remove shadow. In visualization, the foreground objects are
In the monitored area, automated doors and elevators are segmented accurately and displayed on billboards.
specified as background objects, albeit their image do change
when the doors open or close. These areas will be modeled in R EFERENCES
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7. Fig. 11. Dynamic background removal by ground mask. There is an
automated door in the scene (top-left) and it is visualized by a billboard (top-
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994

8. Morphing And Texturing Based On The Transformation Between Triangle Mesh And
Point
Wei-Chih Hsu Wu-Huang Cheng
Department of Computer and Communication Institute of Engineering Science and Technology,
Engineering, National Kaohsiung First University of National Kaohsiung First University of Science and
Science and Technology. Kaohsiung, Taiwan Technology. Kaohsiung, Taiwan
u9715901@nkfust.edu.tw
Abstract—This research proposes a methodology of [1] has proposed a method to represent multi scale surface.
transforming triangle mesh object into point-based object and M. Müller et al. The [2] has developed a method for
the applications. Considering the cost and program functions, modeling and animation to show that point-based has
the experiments of this paper adopt C++ instead of 3D flexible property.
computer graphic software to create the point cloud from Morphing can base on geometric, shape, or other features.
meshes. The method employs mesh bounded area and planar Mesh-based morphing sometimes involves geometry, mesh
dilation to construct the point cloud of triangle mesh. Two structure, and other feature analysis. The [3] has
point-based applications are addressed in this research. 3D demonstrated a method to edit free form surface based on
model generation can use point-based object morphing to
geometric. The method applies complex computing to deal
simplify computing structure. Another application for texture
mapping is using the relation of 2D image pixel and 3D planar.
with topology, curve face property, and triangulation. The [4]
The experiment results illustrate some properties of point- not only has divided objects into components, but also used
based modeling. Flexibility and scalability are the biggest components in local-level and global-level morphing. The [5]
advantages among the properties of point-based modeling. The has adopted two model morphing with mesh comparison and
main idea of this research is to detect more sophisticated merging to generate new model. The methods involved
methods of 3D object modeling from point-based object. complicate data structure and computing. This research has
illustrated simple and less feature analysis to create new
Keywords-point-based modeling; triangle mesh; texturing; model by using regular point to morph two or more objects.
morphing Texturing is essential in rendering 3D model. In virtual
reality, the goal of texture mapping is try to be as similar to
I. INTRODUCTION the real object as possible. In special effect, exaggeration
texturing is more suitable for demand. The [6] has built a
In recent computer graphic related researches, form•Z, mesh atlas for texturing. The texture atlases' coordinates,
Maya, 3DS, Max, Blender, Lightwave, Modo, solidThinking considered with triangle mesh structure, were mapped to 3D
and other 3D computer graphics software are frequently model. The [7] has used the conformal equivalence of
adopted tools. For example, Maya is a very popular software, triangle meshes to find the flat mesh for texture mapping.
and it includes many powerful and efficient functions for This method is more comprehensible and easy to implement.
producing results. The diverse functions of software can The rest arrangements are described as followings:
increase the working efficiency, but the methodology design Transforming triangle mesh into point set for modeling are
must follow the specific rules and the cost is usually high. addressed in Section II and III, and that introduce point-
Using C++ as the research tool has many advantages, based morphing for model creating. The point-based texture
especially in data information collection. Powerful functions mapping is addressed in Section IV, and followed by the
can be created by C language instructions, parameters and conclusion of Section V.
C++ oriented object. More complete the data of 3D be
abstracted, more unlimited analysis can be produced. II. TRANSFORMING TRIANGLE MESH INTO POINT SET
The polygon mesh is widely used to represent 3D models
In order to implement the advantages of point-based
and has some drawbacks in modeling. Unsmooth surface of
model, transforming triangle mesh into point is the first step.
combined meshes is one of them. Estimating vertices of
The point set can be estimated by using three normal bound
objects and constructing each vertex set of mesh are the
lines of triangle mesh. The normal denoted by n can be
factors of modeling inefficiency. Point-based modeling is the
calculated by three triangle vertices. The point in the triangle
solution to conquer some disadvantages of mesh modeling.
area is denoted by B in , A denotes the triangle mesh area, the
Point-based modeling is based on point primitives. No
structure of each point to another is needed. To simplify the 3D space planar can be presented by p with coordinate
point based data can employ marching cube and Delaunay ( x, y , z ) , vi =1, 2,3 denotes three triangle vertices of triangle
triangulation to transform point-based model into polygon
mesh. Mark Pauly has published many related researches mesh, v denotes the mean of three triangle vertices. The
about point-based in international journals as followings: the formula that presents the triangle area is described below.
995

9. A = { p ( x, y , z ) | pn T − v i n T = 0 , i ∈ (1,2,3), p ∈ Bin } The experiments use some objects file which is the wave
front file format (.obj) from NTU 3D model database ver.1
Bin = { p( x, y, z ) | f (i , j ) ( p) × f (i , j ) (v) > 0} of National Taiwan University. The process of transforming
f (i , j ) ( p) = r × a − b + s triangle mesh into point-based is shown in Figure 1. It is
clear to see that some areas with uncompleted the whole
b j − bi point set shown in red rectangle of figure 1. The planar
r= , s = bi - r × ai dilation process is employed to refine fail areas.
a j − ai
Planar dilation process uses 26-connected planar to refine
i, j = 1,2,3 a , b = x, y , z i < j a<b the spots leaved in the area. The first half portion of Figure 2
shows 26 positions of connected planar. If any planar and its
26 neighbor positions are the object planar is the condition.
The main purpose to estimate the object planar is to verify
the condition is true. The result in second half portion of
Figure 2 reveals the efficiency of planar dilation process.
III. POINT-BASED MORPHING FOR MODEL CREATING
The more flexible in objects combining is one of property
of point-based. No matter what the shape or category of the
objects, the method of this study can put them into morphing
process to create new objects.
The morphing process includes 3 steps. Step one is to
Figure 1. The process of transforming triangle mesh into point-based equalize the objects. Step two is to calculate each normal
point of objects in morphing process. Step three is to
estimate each point of target object by using the same normal
point of two objects with the formula as described below.
n −1
ot = p r1o1 + p r 2 o2 + ⋅ ⋅ ⋅ + (1 − ∑ p ri )o n
i =1
n
0 ≤ p r1 , p r 2 ,⋅ ⋅ ⋅, p r ( n −1) ≤ 1 , ∑ p ri = 1
i =1
ot presents each target object point of morphing, and oi is
the object for morphing process. p ri donates the object
effect weight in morphing process, and i indicates the
number of object. The new model appearance generated
from morphing is depended on which objects were chosen
and the value of each object weight as well. The research
experiments use two objects, therefore i = 1 or 2, n = 2 .
The results are shown in Figure 3. First row is a simple
flat board and a character morphing. The second row shows
the object selecting free in point-based modeling, because
two totally different objects can be put into morphing and
produced the satisfactory results. The models the created by
objects morphing with different weights can be seen in figure
4.
IV. POINT-BASED TEXTURE MAPPING
Texturing mapping is a very plain in this research method.
It uses a texture metric to map the 3D model to the 2D image
pixel by using the concept of 2D image transformation into
3D. Assuming 3D spaces is divided into α × β blocks, α is
the number of row, and β is the number of column. Hence
the length, the width, and the height of 3D space
is h × h × h ; afterwards the ( X , Y ) and ( x. y, z ) will
denote the image coordination and 3D model respectively.
Figure 2. Planar dilation process. The texture of each block is assigned by texture cube, and it
996

10. is made by 2D image as shown in the middle image of first confirmed by the scalability and flexibility of proposed
raw in figure 5. The process can be expressed by a formula methodologies.
as below.
At T = c T REFERENCES
h h h [1] MARK PAULY, “Point-Based Multiscale Surface
t = [ x mod , y mod , z mod ] , c = [ X,Y ]
α β β Representation,” ACM Transactions on Graphics, Vol. 25, No.
2, pp. 177–193, April 2006.
⎡α 0 0⎤
A=⎢ β (h − z ) ⎥ [2] M. Müller1, R. Keiser1, A. Nealen2, M. Pauly3, M. Gross1
⎢ 0 0 ⎥ and M. Alexa2, “Point Based Animation of Elastic, Plastic
⎣ y ⎦ and Melting Objects,” Eurographics/ACM SIGGRAPH
A denotes the texture transforming metric, t denotes the 3D Symposium on Computer Animation, pp. 141-151, 2004.
model current position, c denotes the image pixel content in [3] Theodoris Athanasiadis, Ioannis Fudos, Christophoros Nikou,
the current position. “Feature-based 3D Morphing based on Geometrically
The experiment results are shown in the second row of Constrained Sphere Mapping Optimization,” SAC’10 Sierre,
figure 5 and 6. The setting results α = β = 2 are shown in Switzerland, pp. 1258-1265, March 22-26, 2010.
second row of figure 5. The setting results α = β = 4 create [4] Yonghong Zhao, Hong-Yang Ong, Tiow-Seng Tan and
Yongguan Xiao, “Interactive Control of Component-based
the images are shown in the first row of figure 6. The last
Morphing,” Eurographics/SIGGRAPH Symposium on
row images of figure 6 indicate the proposed texture Computer Animation , pp. 340-385, 2003.
mapping method can be applied into any point-based model.
[5] Kosuke Kaneko, Yoshihiro Okada and Koichi Niijima, “3D
V. CONCLUSION Model Generation by Morphing,” IEEE Computer Graphics,
Imaging and Visualisation, 2006.
In sum, the research focuses on point-based modeling
[6] Boris Springborn, Peter Schröder, Ulrich Pinkall, “Conformal
applications by using C++ instead of convenient facilities or
Equivalence of Triangle Meshes,” ACM Transactions on
other computer graphic software. The methodologies that Graphics, Vol. 27, No. 3, Article 77, August 2008.
developed by point-based include the simple data structure
properties and less complex computing. Moreover, the [7] NATHAN A. CARR and JOHN C. HART, “Meshed Atlases
for Real-Time Procedural Solid Texturing,” ACM
methods can be compiled with two applications morphing
Transactions on Graphics, Vol. 21,No. 2, pp. 106–131, April
and texture mapping. The experiment results have been
2002.
Figure 3. The results of point-based modeling using different objects morphing.
997

11. Figure 4. The models created by objects morphing with different weights.
Figure 5. The process of 3D model texturing with 2D image shown in first row and the results shown in second row.
998

12. Figure 6. The results of point-based texture mapping with α = β = 4 and different objects.
999

13. LAYERED LAYOUTS OF DIRECTED GRAPHS USING A GENETIC
ALGORITHM
Chun-Cheng Lin1,∗, Yi-Ting Lin2 , Hsu-Chun Yen2,† , Chia-Chen Yu3
1
Dept. of Computer Science,
Taipei Municipal University of Education, Taipei, Taiwan 100, ROC
2
Dept. of Electrical Engineering,
National Taiwan University, Taipei, Taiwan 106, ROC
3
Emerging Smart Technology Institute,
Institute for Information Industry, Taipei, Taiwan, ROC
ABSTRACT charts, maps, posters, scheduler, UML diagrams,
etc. It is important that a graph be drawn “clear”,
By layered layouts of graphs (in which nodes are
such that users can understand and get information
distributed over several layers and all edges are di-
from the graph easily. This paper focuses on lay-
rected downward as much as possible), users can
ered layouts of directed graphs, in which nodes are
easily understand the hierarchical relation of di-
distributed on several layers and in general edges
rected graphs. The well-known method for generat-
should point downward as shown in Figure 1(b).
ing layered layouts proposed by Sugiyama includes
By this layout, users can easily trace each edge from
four steps, each of which is associated with an NP-
top to bottom and understand the priority or order
hard optimization problem. It is observed that the
information of these nodes clearly.
four optimization problems are not independent, in
the sense that each respective aesthetic criterion
may contradict each other. That is, it is impossi-
ble to obtain an optimal solution to satisfy all aes-
thetic criteria at the same time. Hence, the choice
for each criterion becomes a very important prob-
lem. In this paper, we propose a genetic algorithm
to model the first three steps of the Sugiyama’s al-
gorithm, in hope of simultaneously considering the Figure 1: The layered layout of a directed graph.
first three aesthetic criteria. Our experimental re-
sults show that this proposed algorithm could make Specifically, we use the following criteria to es-
layered layouts satisfy human’s aesthetic viewpoint. timate the quality of a directed graph layout: to
minimize the total length of all edges; to mini-
Keywords: Visualization, genetic algorithm, mize the number of edge crossings; to minimize the
graph drawing. number of edges pointing upward; to draw edges
as straight as possible. Sugiyama [9] proposed a
1. INTRODUCTION classical algorithm for producing layered layouts
of directed graphs, consisting of four steps: cycle
Drawings of directed graphs have many applica- removal, layer assignment, crossing reduction, and
tions in our daily lives, including manuals, flow assignment of horizontal coordinates, each of which
∗ Research supported in part by National Science Council
addresses a problem of achieving one of the above
under grant NSC 98-2218-E-151-004-MY3
criteria, respectively. Unfortunately, the first three
† Research supported in part by National Science Council problems have been proven to be NP-hard when the
under grant NSC 97-2221-E-002-094-MY3 width of the layout is restricted. There has been
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14. a great deal of work with respect to each step of is quite different between drawing layered layouts
Sugiyama’s algorithm in the literature. of acyclic and cyclic directed graphs. In acyclic
Drawing layered layouts by four independent graphs, one would not need to solve problems on
steps could be executed efficiently, but it may not cyclic removal. If the algorithm does not restrict
always obtain nice layouts because preceding steps the layer by a fixed width, one also would not need
may restrain the results of subsequent steps. For to solve the limited layer assignment problem. Note
example, four nodes assigned at two levels after the that the unlimited-width layer assignment is not an
layer assignment step lead to an edge crossing in NP-hard problem, because the layers of nodes can
Figure 2(a), so that the edge crossing cannot be be assigned by a topological logic ordering. The
removed during the subsequent crossing reduction algorithm in [10] only focuses on minimizing the
step, which only moves each node’s relative posi- number of edge crossings and making the edges as
tion on each layer, but in fact the edge crossing straight as possible. Although it also combined
can be removed as drawn in Figure 2(b). Namely, three steps of Sugiyama’s algorithm, but it only
the crossing reduction step is restricted by the layer contained one NP-hard problem. Oppositely, our
assignment step. Such a negative effect exists ex- algorithm combined three NP-hard problems, in-
clusively not only for these two particular steps but cluding cycle removal, limited-width layer assign-
also for every other preceding/subsequent step pair. ment, and crossing reduction.
In addition, our algorithm has the following ad-
vantages. More customized restrictions on layered
layouts are allowed to be added in our algorithm.
For example, some nodes should be placed to the
(a) (b) left of some other nodes, the maximal layer number
should be less than or equal to a certain number,
Figure 2: Different layouts of the same graph. etc. Moreover, the weighting ratio of each optimal
criterion can be adjusted for different applications.
Even if one could obtain the optimal solution for According to our experimental results, our genetic
each step, those “optimal solutions” may not be the algorithm may effectively adjust the ratio between
real optimal solution, because those locally optimal edge crossings number and total edge length. That
solutions are restricted by their respective preced- is, our algorithm may make layered layouts more
ing steps. Since we cannot obtain the optimal solu- appealing to human’s aesthetic viewpoint.
tion satisfying all criteria at the same time, we have
to make a choice in a trade-off among all criteria.
For the above reasons, the basic idea of our 2. PRELIMINARIES
method for drawing layered layouts is to combine
the first three steps together to avoid the restric-
tions due to criterion trade-offs. Then we use the The frameworks of three different algorithms for
genetic algorithm to implement our idea. In the layered layouts of directed graphs (i.e., Sugiyama’s
literature, there has existed some work on produc- algorithm, the cyclic leveling algorithm, and our
ing layered layouts of directed graphs using ge- algorithm) are illustrated in Figure 2(a)–2(c), re-
netic algorithm, e.g., using genetic algorithm to re- spectively. See Figure 2. Sugiyama’s algorithm
duce edge crossings in bipartite graphs [7] or entire consists of four steps, as mentioned previously; the
acyclic layered layouts [6], modifying nodes in a other two algorithms are based on Sugiyama’s algo-
subgraph of the original graph on a layered graph rithm, in which the cyclic leveling algorithm com-
layout [2], drawing common layouts of directed or bines the first two steps, while our genetic algo-
undirected graphs [3] [11], and drawing layered lay- rithm combines the first three steps. Furthermore,
outs of acyclic directed graphs [10]. a barycenter algorithm is applied to the crossing re-
Note that the algorithm for drawing layered lay- duction step of the cyclic leveling and our genetic
outs of acyclic directed graphs in [10] also com- algorithms, and the priority layout method is ap-
bined three steps of Sugiyama’s algorithm, but it plied to the x-coordinate assignment step.
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15. Sugiyama’s Algorithm Cyclic Leveling Genetic Algorithm
Cycle Removel Cycle Removel Cycle Removel edge-node crossing
Layer Assignment Layer Assignment Layer Assignment
edge crossing
Crossing Reduction (a) An edge crossing. (b) An edge-node crossing
Crossing Reduction Crossing Reduction Barycenter Algorithm
x-Coordinte Assignment x-Coordinte Assignment Priority Layout Method
x-Coordinte Assignment
Figure 4: Two kinds of crossings.
(a) Sugiyama (b) Cyclic Leveling (c) Our
we reverse as few edges as possible such that the
Figure 3: Comparison among different algorithms. input graph becomes acyclic. This problem can
be stated as the maximum acyclic subgraph prob-
2.1. Basic Definitions lem, which is NP-hard. (2) Layer assignment: Each
node is assigned to a layer so that the total vertical
A directed graph is denoted by G(V, E), where V is length of all edges is minimized. If an edge spans
the set of nodes and E is the set of edges. An edge across at least two layers, then dummy nodes should
e is denoted by e = (v1 , v2 ) ∈ E, where v1 , v2 ∈ V ; be introduced to each crossed layer. If the maxi-
edge e is directed from v1 to v2 . A so-called layered mum width is bounded greater or equal to three,
layout is defined by the following conditions: (1) the problem of finding a layered layout with min-
Let the number of layers in this layout denoted by imum height is NP-compete. (3) Crossings reduc-
n, where n ∈ N and n ≥ 2. Moreover, the n-layer tion: The relative positions of nodes on each layer
layout is denoted by G(V, E, n). (2) V is parti- are reordered to reduce edges crossings. Even if we
tioned into n subsets: V = V1 ∪ V2 ∪ V3 ∪ · · · ∪ Vn , restrict the problem to bipartite (two-layer) graphs,
where Vi ∩ Vj = ∅, ∀i ̸= j; nodes in Vk are assigned it is also an NP-hard problem. (4) x-coordinate as-
to layer k, 1 ≤ k ≤ n. (3) A sequence ordering, signment: The x-coordinates of nodes and dummy
σi , of Vi is given for each i ( σi = v1 v2 v3 · · · v|Vi | nodes are modified, such that all the edges on the
with x(v1 ) < x(v2 ) < · · · < x(v|Vi | )). The n- original graph structure are as straight as possi-
layer layout is denoted by G(V, E, n, σ), where σ = ble. This step includes two objective functions: to
(σ1 , σ2 , · · · , σn ) with y(σ1 ) < y(σ2 ) < · · · < y(σn ). make all edges as close to vertical lines as possible;
An n-layer layout is called “proper” when it fur- to make all edge-paths as straight as possible.
ther satisfies the following condition: E is parti-
tioned into n − 1 subsets: E = E1 ∪ E2 ∪ E3 ∪ 2.3. Cyclic Leveling Algorithm
· · · ∪ En−1 , where Ei ∩ Ej = ∅, ∀i ̸= j, and
Ek ⊂ Vk × Vk+1 , 1 ≤ k ≤ n − 1. The cyclic leveling algorithm (CLA) [1] combines
the first two steps of Sugiyama’s algorithm, i.e., it
An edge crossing (assuming that the layout is
focuses on minimizing the number of edges point-
proper) is defined as follows. Consider two edges
ing upward and total vertical length of all edges.
e1 = (v11 , v12 ), e2 = (v21 , v22 ) Ei, in which v11
It introduces a number called span that represents
and v21 are the j1 -th and the j2 -th nodes in σi ,
the number of edges pointing upward and the total
respectively; v12 and v22 are the k1 -th and the k2 -
vertical length of all edges at the same time.
th nodes in σi+1 , respectively. If either j1 < j2 &
k1 > k2 or j1 > j2 & k1 < k2 , there is an edge The span number is defined as follows. Consider
crossing between e1 and e2 (see Figure 4(a)). a directed graph G = (V, E). Given k ∈ N, define a
layer assignment function ϕ : V → {1, 2, · · · , k}.
An edge-node crossing is defined as follows. Con-
Let span(u, v) = ϕ(v) − ϕ(u), if ϕ(u) < ϕ(v);
sider an edge e = (v1 , v2 ), where v1 , v2 ∈ V i; v1
span(u, v) = ϕ(v) − ϕ(u) + k, otherwise. For each
and v2 are the j-th and the k-th nodes in σi , re-
edge e = (u, v) ∈ E, denote span(e) = span(u, v)
∑
spectively. W.l.o.g., assuming that j > k, there are
and span(G) = e∈E span(e). In brief, span
(k − j − 1) edge-node crossings (see Figure 4(b)).
means the sum of vertical length of all edges and
the penalty of edges pointing upward or horizontal,
2.2. Sugiyama’s Algorithm
provided maximum height of this layout is given.
Sugiyama’s algorithm [9] consists of four steps: (1) The main idea of the CLA is: if a node causes
Cycle removal: If the input directed graph is cyclic, a high increase in span, then the layer position of
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16. the node would be determined later. In the algo- then priority(v) = B − (|k − m/2|), in which B is a
rithm, the distance function is defined to decide big given number, and m is the number of nodes in
which nodes should be assigned first and is ap- layer k; if down procedures (resp., up procedures),
plied. There are four such functions as follows, then priority(v) = connected nodes of node v on
but only one can be chosen to be applied to all layer p − 1 (resp., p + 1).
the nodes: (1) Minimum Increase in Span Moreover, the x-coordinate position of each node
= minϕ(v)∈{1,··· ,k} span(E(v, V ′ )); (2) Minimum v is defined as the average x-coordinate position of
Average Increase in Span (MST MIN AVG) connected nodes of node v on layer k − 1 (resp.,
= minϕ(v)∈{1,··· ,k} span(E(v, V ′ ))/E(v, V ′ ); (3) k + 1), if down procedures (resp., up procedures).
Maximum Increase in Span = 1/δM IN (v);
(4) Maximum Average Increase in Span =
2.6. Genetic Algorithm
1/δM IN AV G (v). From the experimental results
in [1], using “MST MIN AVG” as the distance The Genetic algorithm (GA) [5] is a stochastic
function yields the best result. Therefore, our global search method that has proved to be success-
algorithm will be compared with the CLA using ful for many kinds of optimization problems. GA
MST MIN AVG in the experimental section. is categorized as a global search heuristic. It works
with a population of candidate solutions and tries
2.4. Barycenter Algorithm to optimize the answer by using three basic princi-
The barycenter algorithm is a heuristic for solv- ples, including selection, crossover, and mutation.
ing the edge crossing problem between two lay- For more details on GA, readers are referred to [5].
ers. The main idea is to order nodes on each
layer by its barycentric ordering. Assuming that 3. OUR METHOD
node u is located on the layer i (u ∈ Vi ), the
The major issue for drawing layered layouts of di-
barycentric∑ value of node u is defined as bary(u) =
rected graphs is that the result of the preceding step
(1/|N (u)|) v∈N (u) π(v), where N (u) is the set
may restrict that of the subsequent step on the first
consisting of u’s connected nodes on u’s below or
three steps of Sugiyama’s algorithm. To solve it, we
above layer (Vi−1 or Vi+1 ); π(v) is the order of v
design a GA that combines the first three steps of
in σi−1 or σi+1 . The process in this algorithm is
Sugiyama’s algorithm. Figure 5 is the flow chart
reordering the relative positions of all nodes accord-
of our GA. That is, our method consists of a GA
ing to the ordering: layer 2 to layer n and then layer
and an x-coordinate assignment step. Note that
n − 1 to layer 1 by barycentric values.
the barycenter algorithm and the priority method
are also used in our method, in which the former is
2.5. Priority Layout Method
used in our GA to reduce the edge crossing, while
The priority layout method solves the x-coordinate the latter is applied to the x-coordinate assignment
assignment problem. Its idea is similar to the step of our method.
barycenter algorithm. It assigns the x-coordinate
position of each node layer to layer according to the Initialization
priority value of each node.
At first, these nodes’ x-coordinate positions in
each layer are given by xi = x0 + k, where x0 is
k
Assign dummy nodes
i
a given integer, and xk is the k-th element of σi . Draw the best Chromosome Terminate? Barycenter
Next, nodes’ x-coordinate positions are adjusted
Fine tune Selection
according to the order from layer 2 to layer n, layer
n − 1 to layer 1, and layer n/2 to layer n. The im- Mutation Remove dummy nodes
provements of the positions of nodes from layer 2 to
Crossover
layer n are called down procedures, while those from
layer n−1 to layer 1 are called up procedures. Based
on the above, the priority value of a k-th node v on Figure 5: The flow chart of our genetic algorithm.
layer p is defined as: if node v is a dummy node,
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17. 3.1. Definitions 4. MAIN COMPONENTS OF OUR GA
For arranging nodes on layers, if the relative hori- Initialization: For each chromosome, we ran-
√ √
zontal positions of nodes are determined, then the domly assign nodes to a ⌈ |V |⌉ × ⌈ |V |⌉ grid.
exact x-coordinate positions of nodes are also de- Selection: To evaluate the fitness value of each
termined according to the priority layout method. chromosome, we have to compute the number of
Hence, in the following, we only consider the rela- edge crossings, which however cannot be computed
tive horizontal positions of nodes, and each node is at this step, because the routing of each edge is
arranged on a grid. We use GA to model the lay- not determined yet. Hence, some dummy nodes
ered layout problem, so define some basic elements: should be introduced to determine the routing of
Population: A population (generation) includes edges. In general, these dummy nodes are placed
many chromosomes, and the number of chromo- on the best relative position with the optimal edge
somes depends on setting of initial population size. crossings between two adjacent layers. Neverthe-
Chromosome: One chromosome represents one less, permuting these nodes on each layer for the
graph layout, where the absolute position of each fewest edge crossings is an NP-hard problem [4].
(dummy) node on the grid is recorded. Since the Hence, the barycenter algorithm (which is also used
adjacencies of nodes and the directions of edges by the CLA) is applied to reducing edge crossings
will not be altered after our GA, we do not need on each chromosome before selection. Next, the
record the information on chromosomes. On this selection step is implemented by the truncation se-
grid, one row represents one layer; a column rep- lection, which duplicates the best (selection rate ×
resents the order of nodes on the same layer, and population size) chromosomes (1/selection rate)
these nodes on the same layer are always placed times to fill the entire population. In addition, we
successively. The best-chromosome window reserves use a best-chromosome window to reserve some of
the best several chromosomes during all antecedent the best chromosomes in the previous generations
generations; the best-chromosome window size ra- as shown in Figure 6.
tio is the ratio of the best-chromosome window size
Best-Chromosome Window
to the population size. Best-Chromosome Window
Fitness Function: The ‘fitness’ value in our def- duplicate
inition is abused to be defined as the penalty for
the bad quality of chromosome. That is, larger ‘fit- Parent Population Child Population Child Population
ness’ value implies worse chromosome. Hence, our
GA aims to find the chromosome with minimal ‘fit-
ness’ value. Some aesthetical criteria to determine Figure 6: The selection process of our GA.
the quality of chromosomes (layouts) are given as
follows (noticing that these criteria are referred Crossover: Some main steps of our crossover pro-
∑7
from [8] and [9]): f itness value = i=1 Ci × Fi cess are detailed as follows: (1) Two ordered par-
where Ci are constants, 1 ≤ i ≤ 7, ∀i; F1 is the to- ent chromosomes are called the 1st and 2nd parent
tal edge vertical length; F2 is the number of edges chromosome. W.l.o.g., we only introduce how to
pointing upward; F3 is the number of edges point- generate the first child chromosome from the two
ing horizontally; F4 is the number of edge crossing; parent chromosomes, and the other child is similar.
F5 is the number of edge-node crossing; F6 is the (2) Remove all dummy nodes from these two par-
degree of layout height over limited height; F7 is ent chromosomes. (3) Choose a half of the nodes
the degree of layout width over limited width. from each layer of the 1st parent chromosome and
In order to experimentally compare our GA place them on the same relative layers of child chro-
with the CLA in [1], the fitness function of our mosome in the same horizontal ordering. (4) The
GA is tailored to satisfy the CLA as follows: information on the relative positions of the remain-
f itness value = span + weight × edge crossing + ing nodes all depends on the 2nd chromosomes.
C6 × F6 + C7 × F7 where we will adjust the weight Specifically, we choose a node adjacent to the small-
of edge crossing number in our experiment to rep- est number of unplaced nodes until all nodes are
resent the major issue which we want to discuss. placed. If there are many candidate nodes, we ran-
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18. domly choose one. The layer of the chosen node is Note that the x-coordinate assignment problem
equal to base layer plus relative layer, where base (step 4) is solved by the priority layout method
layer is the average of its placed connected nodes’ in our experiment. In fact, this step would not
layers in the child chromosome and relative layer is affect the span number or edge crossing number. In
the relative layer position of its placed connected addition, the second step of Sugiyama’s algorithm
nodes’ layers in the 2nd parent chromosome. (5) (layer assignment) is an NP-hard problem when the
The layers of this new child chromosome are mod- width of the layered layout is restricted. Hence,
ified such that layers start from layer 1. we will respectively investigate the cases when the
Mutation: In the mutated chromosome, a node width of the layered layout is limited or not.
is chosen randomly, and then the position of the
chosen node is altered randomly. 5.1. Experimental Environment
Termination: If the difference of average fitness
All experiments run on a 2.0 GHz dual core lap-
values between successive generations in the latest
top with 2GB memory under Java 6.0 platform
ten generations is ≤ 1% of the average fitness value
from Sun Microsystems, Inc. The parameters of
of these ten generations, then our GA algorithm
our GA are given as follows: Population size:
stops. Then, the best chromosome from the latest
100; Max Generation: 100; Selection Rate: 0.7;
population is chosen, and its corresponding graph
Best-Chromosome Window Size Ratio: 0.2; Mutate
layout (including dummy nodes at barycenter po-
Probability: 0.2; C6 : 500; C7 : 500; f itness value =
sitions) is drawn.
span + weight × edgecrossing + C6 × F6 + C7 × F7 .
Fine Tune: Before the selection step or after the
termination step, we could tune better chromo- 5.2. Unlimited Layout Width
somes according to the fitness function. For ex-
ample, we remove all layers which contain only Because it is necessary to limit the layout width
dummy nodes but no normal nodes, called dummy and height for L M algorithm, we set both limits
layers. Such a process does not necessarily worsen for width and height to be 30. It implies that there
the edge crossing number but it would improve are at most 30 nodes (dummy nodes excluded) on
the span number. In addition, some unnecessary each layer and at most 30 layers in each layout. If
dummy nodes on each edge can also be removed we let the maximal node number to be 30 in our
after the termination step, in which the so-called experiment, then the range for node distribution
unnecessary dummy node is a dummy node that is equivalently unlimited. In our experiments, we
is removed without causing new edge crossings or consider a graph with 30 nodes under three differ-
worsening the fitness value. ent densities (2%, 5%, 10%), in which the density
is the ratio of edge number to all possible edges,
5. EXPERIMENTAL RESULTS i.e. density = edge number/(|V |(|V | − 1)/2). Let
the weight ratio of edge crossing to span be de-
To evaluate the performance of our algorithm, our noted by α. In our experiments, we consider five
algorithm is experimentally compared with the different α values 1, 3, 5, 7, 9. The statistics for
CLA (combing the first two steps of Sugiyama’s the experimental results is given in Table 1.
algorithm) using MST MIN AVG as the distance Consider an example of a 30-node graph with
function [1], as mentioned in the previous sections. 5% density. The layered layout by the LM B algo-
For convenience, the CLA using MST MIN AVG rithm, our algorithm under α = 1 and α = 9 are
distance function is called as the L M algorithm shown in Figure 7, Figure 8(a) and Figure 8(b), re-
(Leveling with MST MIN AVG). The L M algo- spectively. Obviously, our algorithm performs bet-
rithm (for step 1 + step 2) and barycenter algo- ter than the LM B.
rithm (for step 3) can replace the first three steps
in Sugiyama’s algorithm. In order to be compared
5.3. Limited Layout Width
with our GA (for step 1 + step 2 + step 3), we con-
sider the algorithm combining the L M algorithm The input graph used in this subsection is the same
and barycenter algorithm, which is called LM B al- as the previous subsection (i.e., a 30-node graph).
gorithm through the rest of this paper. The limited width is set to be 5, which is smaller
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19. Table 1: The result after redrawing random graphs
with 30 nodes and unlimited layout width.
method measure density =2%density=5%density=10%
span 30.00 226.70 798.64
LM B crossing 4.45 57.90 367.00
running time 61.2ms 151.4ms 376.8ms
α =1 span 30.27 253.93 977.56
crossing 0.65 38.96 301.75
α =3 span 31.05 277.65 1338.84
crossing 0.67 32.00 272.80
our α =5 span 30.78 305.62 1280.51
GA crossing 0.67 29.89 218.45
α =7 span 32.24 329.82 1359.46
crossing 0.75 26.18 202.53 (a) α = 1 (b) α = 9
α =9 span 31.65 351.36 1444.27
crossing 0.53 24.89 200.62 (span: 188, crossing: 30)(span: 238, crossing: 14)
running time 3.73s 17.32s 108.04s
Figure 8: Layered layouts by our GA.
Table 2: The result after redrawing random graphs
with 30 nodes and limited layout width 5.
method measure density =2%density=5%density=10%
span 28.82 271.55 808.36
LM B crossing 5.64 59.09 383.82
running time 73.0ms 147.6ms 456.2ms
Figure 7: Layered layout by LM B (span:262, α =1 span 32.29 271.45 1019.56
crossing:38). crossing 0.96 39.36 292.69
α =3 span 31.76 294.09 1153.60
crossing 0.80 33.16 232.76
our α =5 span 31.82 322.69 1282.24
GA crossing 0.82 30.62 202.31
than the square root of the node number (30), be- α =7 span 32.20 351.00 1369.73
cause we hope the results under limit and unlimited crossing 0.69 27.16 198.20
α =9 span 33.55 380.20 1420.31
conditions have obvious differences. The statistics crossing 0.89 24.95 189.25
for the experimental results under the same settings running time 3.731s 3.71s 18.07s
in the previous subsection is given in Table 2.
Consider an example of a 30-node graph with 5%
density. The layered layout for this graph by the
our GA may produce simultaneously span and edge
LM B algorithm, our algorithm under α = 1 and
crossing numbers both smaller than that by LM B.
α = 9 are shown in Figure 9, Figure 10(a) and Fig-
ure 10(b), respectively. Obviously, our algorithm Moreover, we discovered that under any condi-
also performs better than the LM B. tions the edge crossing number gets smaller and
the span number gets greater when increasing the
weight of edge crossing. It implies that we may ef-
5.4. Discussion
fectively adjust the weight between edge crossings
Due to page limitation, only the case of 30-node and spans. That is, we could reduce the edge cross-
graphs is included in this paper. In fact, we con- ing by increasing the span number.
ducted many experiments for various graphs. Be- Under limited width condition, because the re-
sides those results, those tables and figures show sults of L M are restricted, its span number should
that under any conditions (node number, edge den- be larger than that under unlimited condition.
sity, and limited width or not) the crossing number However, there are some unusual situations in our
by our GA is smaller than that by LM B. How- GA. Although the results of our GA are also re-
ever, the span number by our GA is not neces- stricted under limited width condition, its span
sarily larger than that by LM B. When the layout number is smaller than that under unlimited width
width is limited and the node number is sufficiently condition. Our reason is that the limited width
small (about 20 from our experimental evaluation), condition may reduce the possible dimension. In
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