Pattern Recognition ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Pattern Recognition
journal homepage: www.elsevier.com/locate/pr
Exact solution to median surface problem using 3D graph search and
application to parameter space exploration
Zhengwang Wu a, Xiaoyi Jiang b,c,d,n , Nanning Zheng a, Yuehu Liu a, Dachuan Cheng e
a
Institute of Artificial Intelligence and Robotics, Xi'an Jiaotong University, China
b Department of Mathematics and Computer Science, University of Münster, Einsteinstrasse 62, 48149 Münster, Germany
c European Institute for Molecular Imaging, University of Münster, Germany
d Cluster of Excellence EXC 1003, Cells in Motion, CiM, Münster, Germany
e Department of Biomedical Imaging and Radiological Science, China Medical University, Taiwan
article info
abstract
Article history:
Received 19 August 2013
Received in revised form 21
July 2014
Accepted 24 July 2014
In this paper we propose the generalized median surface problem and present an exact solution by applying a 3D graph search
algorithm. To our knowledge this is a novel class of median computation problem which has not been considered before in
the literature. In addition to the theoretic interest in median surface computation we also demonstrate its practical value by
means of the task of parameter space exploration without ground truth, which is an effective means of dealing with the
difficult parameter selection problem for image segmentation. We present a concrete application for artery boundary detection
in sonography (ultrasound imaging). It will be shown that the median computation can not only avoid the parameter training,
but also potentially achieve results comparable with trained parameters. The median-based approach can thus be a good
alternate, particularly in situations with no ground truth available.
Keywords:
Generalized median
Median surface
Graph search
Parameter handling
1. Introduction
Median computation is a useful concept in pattern recognition [1]. Given
an object set S in space U, the generalized median is defined by x AU which
minimizes the sum of distances to all objects in S. It can be considered as a
good representative of the given set. As another motivation, median
computation helps us to eliminate (smooth out) erroneous objects by
averaging over all objects. From a general point of view, the median concept
is motivated by well established results from consensus learning, e.g. multiple
classifier combination: by averaging the results of several classifiers a more
reliable classification can be achieved [2,3].
The median concept has been concretized to a lot of domains (see Section
2 for a brief overview). In particular, the 2D median contour problem has
been investigated [4]. In this work we go a step further from 2D to 3D and
consider the related 3D median surface problem.
Numerous median computation problems and related algo-rithms have
been developed for specific domains to integrate as much as possible domainspecific knowledge in order to obtain
n
Corresponding author at: Department of Mathematics and Computer Science, University of
Münster, Einsteinstrasse 62, 48149 Münster, Germany.
Tel.: þ 49 251 8333759; fax: þ 49 251 8333755.
E-mail address: xjiang@uni-muenster.de (X. Jiang).
& 2014 Elsevier Ltd. All rights reserved.
possibly exact solutions in an efficient way. For instance, the generalized
median string problem based on the edit distance is shown to be NP-hard, but
simplified histogram-based distances reduce the complexity to low-order
polynomial time only [5]. For 2D contours dynamic programming can be
used to determine the optimal median contour in a time linear to the image
size [4]. In this work we will show that for the class of the so-called terrainlike surfaces (to be formally defined in Section 3) and some extensions, a 3D
graph search algorithm can be applied to exactly and efficiently solve the
median surface problem.
The purpose of this paper is twofold. First, we study the median surface
problem and solve it by applying a 3D graph search algorithm. To our
knowledge this is a class of median computation problem which has not been
considered before in the literature. In addition to the theoretic interest in
median surface computation in its own right, we also demonstrate its practical
value motivated by recent works on exploring the parameter space of
segmentation algorithms without ground truth. Similar to multiple classifier
systems, ensemble techniques should be developed to achieve the best
possible segmentation result on a per-image basis.
The remainder of this paper is structured as follows. In the next section we
give a brief overview of the generalized median concept. In Section 3 the
median surface problem under consideration is defined, which is further
motivated by parameter exploration in the context of segmentation problems
in Section 4.
http://dx.doi.org/10.1016/j.patcog.2014.07.019 00313203/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Z. Wu, et al., Exact solution to median surface problem using 3D graph search and application to parameter space exploration,
Pattern Recognition (2014), http://dx.doi.org/10.1016/j.patcog.2014.07.019i
Z. Wu et al. / Pattern Recognition ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
The median surface problem will be exactly solved by applying a 3D graph
search algorithm (Section 5). Our definition of median surface computation is
very general and can cope with position-sensitive weighting of the input
surfaces. In Section 6 a scheme is presented for determining the weighting
factors. We report experimental results to demonstrate the usability of median
sur-face computation in the context of segmentation parameter exploration in
Section 7. Section 8 describes the extension from single surface to multiple
surfaces. Additional extensions of our work are discussed in Section 9.
Finally, some discussions in Section 10 conclude the paper.
This paper is an extended version of the conference paper [6]. The
generalized median concept is introduced with more details. We present more
algorithmic details of our approach, in particular the weighting scheme in
Section 6 and the multiple surface median computation in Section 8. The
experimental work in Section 7 has been reorganized and substantially
extended. Finally, various extensions are discussed in Section 9.
2. Generalized median concept
Assume that we are given a set S of objects in some representa-tion space
U and a distance function dðp; qÞ to measure the dissimilarity between any
two objects p; qAU. The generalized median p AU of S minimizes the sum of
distances to all objects from S, i.e.
generalized median of strings [11,23], graphs [14], and clusterings [24], all
being of NP-hard class.
3. Median surface problem
We start our discussion with terrain-like (height-field) as specified in
Definition 1 (defined in the same way as in [25]).
Definition 1. A terrain-like surface is a function: f : X _ Y-Z with X ¼ f1; 2;
…; Mg, Y ¼ f1; 2; …; Ng, and Z ¼ f1; 2; …; Lg. In order to guarantee
surface connectivity in 3D, an additional continuity constraint requires jf ðx þ
1; yÞ _ f ðx; yÞj r x and jf ðx; yþ 1Þ _ f ðx; yÞj r y for small positive
constants x and y.
It corresponds to a discrete version of the Monge patch in differential
geometry. For the sake of simplicity we will use the term “surface” only in the
following.
This class of surfaces is very common in image analysis. In 3D
biomedical volume datasets, an important task is to detect such terrain-like
surfaces, possibly in an optimal manner. Besides, stacking 2D images along
an additional axis, e.g. time, also results in 3D volume datasets. Then, contour
detection in an image sequence can be recasted into a 3D surface detection
problem.
We assume a set S of K input surfaces fS1; S2; …; SK g and a distance
function dðÞ which measures the dissimilarity of two surfaces. Then, the
general median surface is defined by
K
p ¼ arg min ∑ dðp; qÞ:
p AU
¼ arg
min
s AUS
q AS
In general p is neither a member of S nor unique. A related concept is the socalled set median, which results from constraining the search to the given set
S:
p^ ¼ arg min ∑ dðp; qÞ:
p AS q AS
The set median may serve as an approximative solution for the generalized
median or an initialization for iterative generalized median computation. This
is justified by the fact ∑q ASdðp^ ; qÞ= ∑q ASdðp; qÞ r2_ 2=jSj if the
triangle inequality applies to the distance function dðp; qÞ (see [7] for a
proof).1 That is, the set median has a consensus error that is at most 2 _ 2=jSj
times the consensus error of the generalized median.
The median concept has been concretized to many domains including
vectors [8], tensors [9], rotations [10], strings [11], graphs [12–14],
clusterings [15,16], tree-like shapes [17], time series [18], ranking
(permutation) [19], images (for atlas construc-tion) [20], and image
segmentations [21,22]. In [4] the 2D median contour problem is investigated.
In this work we extend it from 2D to 3D for studying the related median
surface problem. We are not aware of the previous work on median
computation for the class of surfaces under consideration. Our work thus
enriches the generalized median related research.
There exist only very few general frameworks for median computation.
One such framework described in [14] is based on an embedding into the
vector space. The median vector is computed by means of the Weiszfeld
algorithm [8] and inversely transformed to the original space. Another general
framework [23] computes the weighted mean of pairs of objects in an
evolutionary scheme. Both frameworks are approximative only and therefore
suitable for those median problems with inherently high compu-tational
complexity. Indeed, they have been applied to computing
1
S
The proof in [7] is given for strings and edit distance. But the proof itself does not use any
string-specific property and thus holds for any distance function which satisfies the triangle
inequality.
∑
dsS
ð;
Þ
i
1
ðÞ
i¼ 1
where US represents the space (universe) of all potential solutions, i.e.
surfaces within the search volume X _ Y _ Z.
The distance function is defined by
MN
dðs; SiÞ ¼ ∑ ∑ wxy _ ρðsðx; yÞ; Siðx; yÞÞ
ð2Þ
x¼1y¼ 1
where ρ is a dissimilarity function for scalar values. Any function suitable for
a certain application, e.g. the Minkowski distance Lp, can be used for this
purpose. In particular, those from robust statistics [26] may help us to achieve
improved performance against outliers in the input surface data.
In the simplest case the weight wxy can be set to be constant for all ðx; yÞ
positions. However, the larger the wxy is, the more the influence of the
particular position and the input surface Si will be imposed on the final
median surface. Generally, a position-sensitive weighting gives us more
flexibility to incorporate problem-specific knowledge. For our segmentation
parameter exploration, we will fully utilize this flexibility (see Section 6).
Note that the case of constant weight corresponds to computing a mean
surface, i.e. averaging at each point but with guaranteed surface continuity. A
comparison of weighted and unweighted median computation will be
presented in Section 7.3.
The median surface problem defined above can be extended in various
ways. We can consider multiple surfaces simultaneously instead of one single
surface (see Section 8). In addition an extension to tube-like (cylindrical)
surfaces and closed surfaces substantially broadens the applicability of
median surface compu-tation in practice (see Section 9).
4. Motivation
One motivation of median surface computation is exploring segmentation
parameter space without ground truth, i.e., to get the best segmentation result
without the help of ground-truth for parameter training or selection.
Please cite this article as: Z. Wu, et al., Exact solution to median surface problem using 3D graph search and application to parameter space exploration,
Pattern Recognition (2014), http://dx.doi.org/10.1016/j.patcog.2014.07.019i
Z. Wu et al. / Pattern Recognition ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Segmentation algorithms mostly have some parameters and their optimal
values are not easy to find. Traditionally, a training image set with (manual)
ground truth segmentation is assumed to be available. Then, a subspace of the
parameter space is explored to find out the best parameter setting. For each
candidate of parameter setting in this subspace, a performance measure is
computed in the following way:
detection problem, which is solvable by a 3D graph search algorithm in loworder polynomial time.
First, we reformulate Eq. (1) as follows:
K
S
¼ arg
¼ arg
min
s AUS
min
¼ arg
min
s AUS
dsS
∑
ð;i
i¼1
K
M
min
s AUS
N
w
i¼1x¼1y¼1
M
N
∑ ∑
w
xy
x¼1y¼1
M
¼ arg
Þ
∑∑ ∑
s AUS
Segment each image in the training set based on this parameter setting.
Compute a performance measure by comparing the segmenta-tion result
and the corresponding ground truth.
Compute the average performance measure over all images of the training
set.
3
xy
N
S
ðð;
Þ;
xy
ið ;
ÞÞ
K
_∑
i¼ 1
∑ ∑
_ ρs x y
sx y S
ρz
ð ¼ ð ;
Þ;
xy
ið
;
ÞÞ
cxyz
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
c
xyz
x¼1y¼1
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}CðsÞ
The optimal parameter setting is selected as the one with the largest average
performance measure. Since fully exploring the subspace can be very costly,
several methods have been proposed to reduce the search space such as space
subsampling [27], genetic search [28], alternating scheme [29], and
orthogonal experiment design [30]. Although this general approach is
reasonable and has been successfully practiced in several applications, its
critical disadvantage is the need of ground truth segmentation. The manual
generation of ground truth is always painful and thus a main barrier of wide
use in many situations.
Recently, a novel approach is proposed by applying the concept of
generalized median for implicitly exploring the parameter space without the
need of ground truth segmentation. We assume a reasonable subspace of the
parameter space (i.e. a lower and a upper bound for each parameter). This
subspace is sampled into a finite number M of parameter settings. Then, the
segmentation procedure is run for all the M parameter settings and the
generalized median of the M segmentation results is computed. The rationale
here is that in line with the ensemble paradigm, the median result tends to be
a good one within the explored parameter subspace, as already successfully
demonstrated for 2D contour detection [4] and region segmentation [21,22].
Segmentation of surfaces is one of the most important problems in
(biomedical) image analysis. But in most cases it involves para-meter
selection, which is however typically not easy to handle. Thus, median
surface computation can directly help us to alleviate the parameter problem in
3D surface segmentation as well.
As another situation of potential application we consider the segmentation
of 2D images along the time axis. Many algorithms from the literature, e.g.
[31], perform the segmentation indepen-dently on all images and thus cannot
guarantee a continuous segmentation over time, which is naturally desired
when working with sequences of images. If the parameter space exploration
technique described above is applied to the 3D volumes formed by stacking
all frame-wise segmentations along the time axis, we obtain a continuous
temporal segmentation without any extra effort as a nice spin-off of handling
the parameter problem.
It is important to emphasize that we do not intend to solve the
segmentation problem, but instead study the parameter explora-tion problem
for image segmentation based on some baseline segmentation algorithm. In
this sense our approach is on a meta-level above the segmentation level. The
median surface computa-tion is not only an interesting topic in its own right
but also of substantial practical value. This motivates us to find an efficient
way for exact median surface computation.
5. Exact computation by 3D graph search
¼ arg min CðsÞ
ð3Þ
s AUS
A candidate solution is a surface s AUS characterized by the z-value sðx; yÞ
for each position (x,y) on the grid X _ Y. Each point ðx; y; zÞ in the volume
X _ Y _ Z is assigned a cost cxyz, which is determined by its deviations (in zdirection) from the K input surfaces Siðx; yÞ and the position-specific weight
wxy. Then, the goodness of a candidate solution surface s is evaluated by the
measure C(s), i.e. summing up the costs of all positions. Therefore, the
median surface is simply the optimal surface with minimal cost from the
solution space US (consisting of all terrain-like surfaces within the volume X
_ Y _ Z). This discussion immediately leads to the following new
optimization problem. We first compute a cost cxyz for each point ðx; y; zÞ in
the volume X _ Y _ Z. Then, the median surface is determined by finding the
terrain-like surface within the volume with the minimal sum of costs.
It is important to mention that this optimization problem cannot be solved
by computing the optimal z-value for each of the M _ N positions (x,y)
independently (i.e. by enumerating all z-values from Z and minimizing cxyz).
Doing it in this simple way, we will encounter the trouble of generating a
discontinuous resultant surface in general. Only for some elementary cases
(e.g. constant weight wxy and ρ ¼ L22) the straightforward position-wise
optimization will deliver an optimal continuous resultant surface. But in
general, a global optimization approach is needed.
For the special case N¼ 1 (i.e. the y-axis vanishes), the 3D optimal
surface segmentation becomes a 2D optimal contour detection problem. This
reduced problem is much easier to solve, e.g. by a highly efficient dynamic
programming algorithm [4]. Unfortunately, there is no direct way of
extending the dynamic programming solution to the general 3D case (N 41).
Note that Eq. (3) can be viewed as a first order MRF minimum energy
problem, i.e.,
EðXÞ ¼ ∑ Gðv; XvÞþ ∑ Pðu; v; Xu; XvÞ
v AV
ðu;vÞ AE
but without the prior (the second term) where X is a labeling function.
Obtaining the exact solution of the first order MRF is a classical problem in
computer vision [32]. The work [33] solved the binary variable first order
MRF with specified priors. The work [34] discussed using the first order MRF
to solve the image segmentation problems and gave the exact solution when
Gðv; XvÞ is convex and Pðu; v; Xu; XvÞ is a linear function. In [35,36] a
general first order MRF energy function with convex priors and linear order
labels was solved with exact solution. In [37] an optimal net surface problem
is proposed, which can be viewed as a concrete case of [35,36], because the
base graph in [37] corre-sponds to the original graph in [35,36]. Later, in [25],
the problem of extracting the optimal surface from a 3D volume is proposed,
which is a special case of [37]. Fortunately, our problem can be viewed as the
optimal surface search as formulated in [25] and we
In this section we show that the median surface problem defined in Eq. (1)
can be transformed into an optimal 3D surface
Please cite this article as: Z. Wu, et al., Exact solution to median surface problem using 3D graph search and application to parameter space exploration,
Pattern Recognition (2014), http://dx.doi.org/10.1016/j.patcog.2014.07.019i
Z. Wu et al. / Pattern Recognition ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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thus use the algorithm in [25] to solve our optimal surface detection problem
in Eq. (3).
In the following we give a brief presentation of the most important steps
of this algorithm and the readers are referred to [25] for further details. A
node-weighted directed graph G ¼ ðV; E; WÞ is constructed as follows. For
each point ðx; y; zÞ in the volume X _ Y _ Z a corresponding node Vðx; y;
zÞ is defined in G, whose weight Wðx; y; zÞ is assigned according to
(
cxyz c
_
Wðx; y; zÞ ¼ cxyz;
xy;z _ 1
; z
1
ð4Þ
4
z
1
¼
where cxyz is the cost defined in Eq. (3). Two types of edges are defined for
G: E ¼ Ea [ Er . The set Ea of intraposition edges models the connections
within the same position (x,y). Each node Vðx; y; zÞ (z 41) has a direct edge
to the node V ðx; y; z _ 1Þ below it, i.e.,
Ea ¼ f oVðx; y; zÞ; Vðx; y; z _ 1Þ 4 j z 41g
The set Er of interposition edges models the connections of adjacent
positions. The node V ðx; y; zÞ is connected to the nodes Vðx þ 1; y; maxð0;
z _ xÞÞ at adjacent position ðx þ 1; yÞ, i.e. x below z; z Z x. Similar edges
are drawn to the other three adjacent positions ðx _ 1; yÞ; ðx; y þ 1Þ, and ðx;
y_ 1Þ. Overall, the interposi-tion edges are defined by
E
oV x; y; z ; V x
8 f oVðx; y; z
Þ; V xþ
r
ð
>f
¼
>
>
>
<f
>
>
ð
Þ
ð
Þ
oV x; y; z ; V x; y
f oV ðx; y ; zÞ; Vðx; y _ 1; maxð0; z _ y ÞÞ 4
ð _
ð
1; y; max 0; z
1; y; max
ð
ð
0; z_
_
1; max 0; z
þ
ð
xÞÞ 4
x
y
ÞÞ
_ ÞÞ
4
j
4
j
j
xA 1; …; M _ 1g; z AZg [
xAf 2; …; M ; z AZ
g[
f
g
y A 1; …; N
j y Af2; …; Ng; z AZg
f
assigned a cost cxyz, which is determined by its deviations (in z-direction)
from the K input surfaces Siðx; yÞ and the position-specific weight wxy.
Then, the goodness of a candidate solution surface s is measured by C(s)
in Eq. (3), i.e. summing up the costs of all (x,y) positions. Thus, the
median surface is simply the optimal surface with minimal cost in the
search volume.
Apply the 3D graph search algorithm in [25] to find the optimal surface
with minimal cost in the search volume.
The first step transforms the median surface computation problem into one of
the optimal surface detections, which is then solved by the 3D graph search
algorithm in the second step. Note that our approach can guarantee the true
median surface under the hard continuity constraint. The reason is that the
median surface should also satisfy the continuity constraint and the search
graph con-struction thus uses the continuity constraint.
It is worth mentioning that the number of input surfaces K does not pose
any problem (extra complexity) to our approach at all because the input
surfaces are only used to calculate the voxels' cost in the search volume (the
search volume itself is fixed). Afterwards, the complexity of median surface
computation does not depend on this number anymore.
6. Weighting scheme
1 ; z AZ
_ g
g[
>
>
:
Given the constructed directed graph G, a closed set C is a subset of nodes
such that all successors of any nodes in C are also contained in C. The cost of
a closed set is the total cost of all its nodes. For any feasible surface N, the
subset of nodes on or below N in G, namely C ¼ fVðx; y; zÞ j z rNðx; yÞg,
forms a closed set in G. Due to the node cost defined in (4), the costs of N and
C are clearly equal. In [25] it is shown that the original optimal surface
detection problem is equivalent to finding a minimum nonempty closed set in
G.
This problem is well studied in graph theory and can be solved by
computing a minimum s _ t cut in a related graph Gst (see [25] for the details
of constructing Gst from G). In our implementation we used the Boykov–
Kolmogorov algorithm [32] to compute the minimum s _ t cut. For a graph
with n nodes and m edges, the theoretical worst-case time complexity for this
algorithm is Oðn2mcÞ, where c is the cost of the minimum cut.
In summary our approach to median surface computation consists of the
following steps:
_ Given a set S of K surfaces fS1; S2; …; SK g, we construct a 3D
search volume X _ Y _ Z. Each point ðx; y; zÞ in this volume is
The median surface definition depends on the weight wxy. Generally, a
position-sensitive weighting gives us more flexibility to incorporate problemspecific knowledge. In this section we present a position-wise weighting
scheme along this line.
For any position (x,y) in the cost volume X _ Y _ Z, each ziðx; yÞ ASi
can be regarded as a sample of a random variable Fxy, which is assumed to
form a Gaussian distribution. As an illustra-tion, Fig. 1 shows the distribution
of the z-values from all input surfaces for a particular position (x,y) and the
estimated Gaussian density function. For the ziðx; yÞ ASi, the corresponding
wxy is then chosen to be the density at z ¼ sðx; yÞ. This weighting scheme
has a simple interpretation. If some z-value appears frequently in the input
surfaces, it is likely that the median surface should have a similar z-value at
this position. The density-based weighting reflects this expectation and
reinforces achieving the expected result.
The importance of position-sensitive weighting is further illu-strated in
Fig. 2. The distributions for two positions ðx1; y1Þ and ðx2; y2Þ are
considered. The green circle and the yellow triangle symbolize two different
input surfaces. At position ðx1; y1Þ the green surface should receive a high
weight since the correspond-ing z-value plays a strong majority role. In
contrast the yellow surface should be under-weighted. The situation at
position ðx2; y2Þ is exactly opposite. This illustrative example clearly
demonstrates that a constant weight is generally not a good choice. In our
experimental work we compare the constant weighting and the weighting
scheme presented above, and the results reported in Section 7 confirm the
superiority of position-sensitive weighting.
7. Experimental results
In this section we demonstrate a practical use of median surface
computation by applying the algorithm described above to segmentation
parameter exploration in sonography.
7.1. Sonographic data and experimental settings
Fig. 1. Weighting scheme: the distribution of z-value from all input surfaces for a particular
position (x,y) and its estimated Gaussian density function.
The task considered here is to alleviate the painful parameter selection
problem in the extraction of artery boundaries from
Please cite this article as: Z. Wu, et al., Exact solution to median surface problem using 3D graph search and application to parameter space exploration,
Pattern Recognition (2014), http://dx.doi.org/10.1016/j.patcog.2014.07.019i
Z. Wu et al. / Pattern Recognition ∎ (∎∎∎∎) ∎∎∎–∎∎∎
5
Fig. 2. Illustration of importance of position-sensitive weighting. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
sonographic videos. An artery has a near wall and a far wall, 2 as illustrated in
Fig. 3(b). Along with the time axis the 2D images can be regarded as a 3D
volume, see Fig. 3(a). Four sonographic videos from patients were used in our
experiments. For these videos, a ground truth of the arterial walls (golden
standard) was labeled manually. In total we thus have 8 test cases (4 videos,
near and far wall).
The dynamic programming approach from [38] was applied to detect the
two contours. This algorithm has two parameters. For both parameters a
reasonable interval ½0:1; 1:9& was identified and sampled into 10 values
(0.1, 0.3, …, 1.7, 1.9). In total we thus considered 100 different parameter
settings. For each video, these parameter settings were used to generate 100
near wall surfaces and 100 far wall surfaces. Then, a median surface was
computed from each of the two surface ensembles.
The diversity of quality of the 100 segmented surfaces in each test case is
documented by the statistics in Table 1. The quality is measured by
comparing each surface with the corresponding GT based on L1. The 100
input surfaces thus lead to 100 performance measures (average deviation per
(x,y) position). Each table item presents their value interval, mean, and
standard deviation. For all 8 test cases the corresponding ensemble shows
some diversity, indicating the varying suitability of the distinct parameter
settings. It is this diversity that promises potential by means of fusion (median
computation).
For comparison purpose the best-performing one among the 100
parameter settings was determined by using all test videos and comparing
with the ground truth. This is the ultimate optimal segmentation (subject to
the explored parameter space) on a per-image basis, which may be
approached, but cannot be achieved by segmentation algorithms in general. In
all our tests the compar-ison between two surfaces, e.g. a segmented surface
and a ground truth, was done by computing the average L1 deviation in z per
(x,y) position.
7.2. Comparison with the ground truth
We compared our median result with the ground truth. The average L1
deviation in z is shown in Table 2. Contrasting with the ensemble statistics in
Table 1, the ensemble solution by means of median computation consistently
achieves a quality level between the minimum and the mean, in many cases
actually near the minimum.
In addition, the results from the best-performing parameter setting (BP)
were also compared with GT. Four examples of comparison from two
different frames are given in Fig. 4 for
2
More precisely, these are the intimas of the near and the far wall, see Section 8.2.
illustration purpose. While in (b) and (d) the median and BP results are
absolutely comparable, (a) and (c) illustrate superior median computation
compared to BP. As can be seen in Table 2, our median segmentation results
reach the quality of best-performing parameter setting very well. Using our
median surface algorithm thus can not only avoid the parameter training
(which is only possible with existing ground truth), but also potentially
achieve segmentation results similar to the best parameters. This fact is
clearly due to the ensemble nature of the median surface computation.
Due to the small number of videos we did not perform cross validation.
Instead, the best-performing parameter setting was determined by using all
available videos. This results in perfor-mance measures which usually cannot
be fully reached when the optimal parameters are learned using training data
and applied to unknown test data. Therefore, the comparison in Table 2 is
based on an overoptimistic estimation of training-based performance, making
the performance of median computation even more favorable.
7.3. Comparison with unweighted median computation
In Section 6 we have argued the importance of a position-sensitive
weighting and presented a weighting scheme. A test series was performed to
experimentally confirm the superior performance of weighting. Again, the
median surfaces are com-pared with GT based on L1. The average difference
reported in Table 3 shows an almost consistent improvement of the weighting
scheme over the unweighted median computation.
The weighting scheme presented in Section 6 is still rather simple. Note
that the performance evaluation in this work mainly follows the goal of
demonstrating the usability of the median computation and with regard to this
test series the potential of weighting. From an application point of view, it
certainly remains to improve the weighting scheme and to maximize the
benefit of position-sensitive weighting. But this is beyond the scope of our
current work.
7.4. Robustness study
We manually changed the population of input surfaces to simulate various
test situations. Three test scenarios were gener-ated by selecting 60 out of the
100 input surfaces:
50 best and 10 worst input surfaces in terms of GT comparison.
40 best and 20 worst input surfaces.
30 best and 30 worst input surfaces.
In particular, the last test scenario has a high percentage (50%) of parameter
settings, which are relatively non-optimal in terms of
Please cite this article as: Z. Wu, et al., Exact solution to median surface problem using 3D graph search and application to parameter space exploration,
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7.5. Computational time
The experiments were performed on an Intel Xeon x5560
2.8 GHz
with 16G memory. The program was written in C
(Boykov–Kolmogorov algorithm) and Matlab (other parts). The
computational time is listed in Table 5. Also shown is the
resolution of the subpart of a video, in which the near or the far
wall is searched for. The computational time increases with the
volume size (image size and number of images). Generally, the
graph search algorithm is quite fast.
7.6. Discussions
Our approach to deal with the parameter problem is in line
with the ensemble paradigm which is a powerful boosting
technique in machine learning and pattern recognition [2,3]: by
averaging the results of several classifiers a more reliable classification can be achieved. In our case an ensemble of K surfaces is
averaged to achieve an “optimal” segmentation. This principle has
been successfully validated for 2D contour detection [4] and image
Fig. 3. Sonographic video. (a) Along with the time axis the 2D images build a 3D
volume. (b) The near (blue) and the far (red) wall of the artery in a single image.
(For interpretation of the references to color in this figure caption, the reader is
referred to the web version of this paper.)
Table 1
Statistics of the 100 segmented surfaces: [minimum, maximum], mean, standard
segmentation [21,22] before. In the current work this ensemble
approach is demonstrated for parameter space exploration in the
context of 3D surface segmentation.
deviation (unit: pixels).
Video
1
2
3
4
#Images
86
86
111
73
Near wall
Far wall
[0.46, 1.00], 0.78, 0.17
[0.41, 1.35], 0.82, 0.33
[0.89, 2.82], 2.00, 0.73
[0.45, 1.76], 1.08, 0.56
[0.51, 0.71], 0.62, 0.05
[0.47, 0.75], 0.61, 0.08
[0.63, 0.96], 0.73, 0.09
[0.45, 1.45], 0.75, 0.22
Table 2
Comparison with the ground truth (unit: pixels).
Video
Comparison type
Near wall
Far wall
1
Median vs. GT
BP vs. GT
0.63
0.61
0.57
0.55
2
Median vs. GT
0.48
0.50
BP vs. GT
0.46
0.49
3
Median vs. GT
BP vs. GT
0.98
0.96
0.67
0.66
4
Median vs. GT
BP vs. GT
0.48
0.46
0.54
0.51
the studied parameter interval. The
performance is
given in
Table 4. In the first test scenario similar behavior as in Table 2
can be observed. The median computation basically achieves the
same performance as the best-performing parameter setting. Not
surprisingly, with the increasing percentage of non-optimal parameter settings, the performance of median computation slightly
drops. But even in the last test scenario the performance drop is
not dramatic, indicating the potential of robustness by using the
median surface approach in dealing with the segmentation parameter problem.
The robustness is mainly due to the use of L1 norm. In robust
statistics [26] it is known that L1 is robust and can tolerate up to
50% outliers. Any
other robust
dissimilarity function
with a
ρ
possibly high breakdown point
will make the median
surface
computation robust as well.
Since our approach is an ensemble one, the ensemble quality
certainly influences the final result. It is thus important to select a
reasonable parameter subspace for sampling. In practice, this is
mostly not difficult to achieve. Together with a robust dissimilarity
function it can be expected that the “bad” samples will be well
tolerated by the median surface computation and an overall good
quality of median surface can be reached.
In Section 4 we have discussed several parameter exploration
methods [27–30]. However, their common critical disadvantage is
the need of ground truth segmentation. The manual generation of
ground truth is always painful and thus a main barrier of wide use
in many situations. Our intention in the application part of this
work is to overcome the difficulty with parameters, to some extent
at least, without assuming the ground truth. Due to the very
different nature of our unsupervised ensemble approach and other
supervised methods we do not perform direct comparison with
them. However, we have shown that our median segmentation
results reach
the quality of best-performing
parameter
setting
very well. Since even ground truth based parameter learning
cannot achieve the performance of best-performing parameter
setting, this result is an indirect proof of the power of our
approach compared to the existing supervised methods.
In our tests the involved dynamic programming approach from
[38] has two parameters only. In the case of more parameters the
number of sampled parameter settings certainly increases. One
can either fix some non-critical parameters and sample a small
number of remaining critical parameters, or sample the (noncritical)
parameters relatively sparsely. Another solution
is to
compute the K input surfaces in a parallel manner. It is important
to emphasize that this is a general issue related to many, if not all,
parameter learning methods from the literature.
The application of our median surface computation intends to
alleviate the parameter problem. It is therefore important to make
sure that our approach does not introduce new parameter(s). This
is the case since all steps
described are parameter-free. For
instance, the weighting scheme described in Section 6 is based
on function fitting. Note that the continuity values x and
y in
Definition 1 are not parameters introduced by our median computation algorithm. Indeed, any baseline surface detection algorithm, which
delivers the input for our median
surface
computation, needs these values as well. They mean the degree
of surface continuity a user wishes to achieve and thus have to be
specified by the user, most
likely in an
application-dependent
manner.
8. Multiple surfaces median computation
Due to the imperfections of medical imaging techniques,
insufficient or even deficient image-derived information may be
Please cite this article as: Z. Wu, et al., Exact solution to median surface problem using 3D graph search and application to parameter space exploration,
Pattern Recognition (2014), http://dx.doi.org/10.1016/j.patcog.2014.07.019i
Z. Wu et al. / Pattern Recognition ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Fig. 4. Examples of comparison of GT, BP, and median.
Table 3
Comparison weighted vs. unweighted (unit: pixels).
Video
Variant
Near wall
Far wall
1
Weighted
Unweighted
0.63
0.67
0.57
0.57
2
Weighted
Unweighted
0.48
0.54
0.50
0.53
3
Weighted
0.98
0.67
Unweighted
1.08
0.68
Weighted
Unweighted
0.48
0.52
0.54
0.53
coupled terrain-like
surfaces contain two
Definition 2. Two
f
i : X _ Y-Z
terrain-like
single surfaces:
with X ¼ f1; 2; …; Mg,
Y ¼ f1; 2; …; Ng, and Z ¼ f1; 2; …; Lg, i¼ 1,2. The pre-defined coupling constraints δl and δu specify, respectively, the minimum and
maximum distances between the two non-intersecting surfaces,
i.e. δl rjf 1ðx; yÞ _ f 2ðx; yÞj rδu.
4
This definition can be easily extended to the case of K (42)
surfaces. Denoting the K surfaces by f 1; f 2; …; f K , we need to define
for each pair of neighboring surfaces fi and fj the related constraints δlij and δuij.
The 3D graph search algorithm [25] has an extension for K
i ¼ 1; …; K, are combined to build an overall search graph:
WiÞ;
available. This insufficiency can be alleviated by using clues from
other mutually
related boundaries
or surfaces. Indeed, codetection of multiple coupled surfaces frequently yields superior
performance compared to the common single-surface detection
approaches [31,25,38–40]. Thus, it is essential to extend the single
terrain-like surface median problem to the case
of multiple
coupled surfaces.
8.1. Algorithmic extension
We start with the definition of multiple surfaces (defined in the
same way as in [25]).
individual graphs Gi ¼ ðV i; Ei;
coupled surfaces. In this case the
G ¼ ðV1 [ ⋯ [ VK ; E1 [ ⋯ [ EK [ Es; W1 [ ⋯ [ WK Þ
The extra intersurface edges in Es
model the relations between
surfaces. As done in [25], we specify the construction of Es for
K¼ 2. The ideas can be easily generalized to handling more than
two surfaces. Supposing f1 is above f2, then
oV1 x; y; z ; V2 x; y; z _δu 4 z Zδu
gδ [
ð
Þ
ð
Þ j
Es 8 f oV2
l
x; y; z ; V1
x; y; z δ
4 z oL
¼> f
Þ
_ Þ j
_ g[
ð
ð
l
< oV1
>f
:
0; 0; δl ; V 2 0; 0; 0 4
ð
Þ
ð
Þ
g
Note that the formulation above is given for non-intersecting
surfaces. The needed extension for intersecting surfaces can be
found in [25], also the details of computing the K optimal surfaces.
Please cite this article as: Z. Wu, et al., Exact solution to median surface problem using 3D graph search and application to parameter space exploration,
Pattern Recognition (2014), http://dx.doi.org/10.1016/j.patcog.2014.07.019i
Z. Wu et al. / Pattern Recognition ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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The K-surface median surface computation, however, may suffer from
high demand of computation. Compared to the original single-surface
algorithm, the worst-case time complexity of the
Boykov–Kolmogorov algorithm (Oðn2mcÞ) increases at least by a factor of
K3 (because of K _ n nodes and K _ m plus extra intersur-face edges in the
enlarged graph).
Table 4
Performance of three populations of input surfaces (unit: pixels).
8.2. Experimental results
Comparison type
Near wall
Far wall
Median vs. GT
Bp vs. GT
0.57
0.61
0.55
0.55
40 best þ 20 worst
Median vs. GT
BP vs. GT
0.56
0.61
0.55
0.55
30 best þ 30 worst
Median vs. GT
BP vs. GT
0.68
0.61
0.58
0.55
Median vs. GT
BP vs. GT
0.44
0.46
0.49
0.49
40 best þ 20 worst
Median vs. GT
BP vs. GT
0.44
0.46
0.49
0.49
30 best þ 30 worst
Median vs. GT
BP vs. GT
0.51
0.46
0.50
0.49
Video 1
50 best þ 10 worst
Video 2
50 best þ 10 worst
9. Extensions
The 3D graph search algorithm in [25] designed for terrain-like
surface detection has several variants of extensions which further
broaden the usability of median surface computation. The exten-
Video 3
50 best þ 10 worst
Median vs. GT
BP vs. GT
0.97
0.96
0.66
0.66
40 best þ 20 worst
Median vs. GT
0.96
0.66
BP vs. GT
0.96
0.66
Median vs. GT
BP vs. GT
1.10
0.96
0.67
0.66
Median vs. GT
BP vs. GT
0.47
0.46
0.51
0.51
40 best þ 20 worst
Median vs. GT
BP vs. GT
0.47
0.46
0.51
0.51
30 best þ 30 worst
Median vs. GT
BP vs. GT
0.48
0.46
0.53
0.51
30 best þ 30 worst
Video 4
50 best þ 10 worst
sion to the case of multiple surfaces in already discussed in Section
8. In this section we briefly discuss three additional extensions and
another issue related to computational cost.
9.1. Tube-like surfaces
Sometimes, the desired surface is required to be wraparound
along some direction (say, the x- or y-axis). For instance, this is a
common situation in medical image analysis to detect tube-like
(cylindrical) surfaces. In this case a tubular surface is first unfolded
into a terrain-like surface using cylindrical coordinate transform
[41]. Then, the boundary of the tubular surface in the original
image corresponds to a terrain-like surface to be detected in the
unfolded image and the 3D graph search algorithm [25] can thus
be applied. Moreover, the first and last rows along the unfolding
plane should satisfy the continuity constraints as well, which is
guaranteed by a modified graph construction. Median surface
computation is directly applicable in this case.
Table 5
Computational time (seconds).
Video
#images
Near wall
Resolution
The experimental work for the single-surface case reported in Section 7 is
restricted to the intima of near and far walls only. In fact the another so-called
adventitia is also of interest, see Fig. 5 for an illustration. Using the same
sonographic data and experi-mental settings as described in Section 7.1, we
have performed experiments for multiple surface median computation (K¼
2). For the near wall, a pair of median intima and median adventitia are
computed simultaneously and the same is also done for the far wall. Similar
to Table 2, Table 6 lists a comparison with the GT and the best-performing
parameter setting. Again, our median seg-mentation results reach the quality
of best-performing parameter setting very well. The corresponding
computational time is reported in Table 7. As expected, it is much higher than
that for the single surface case.
Far wall
Time
Resolution
time
1
2
3
86
86
111
45 _ 121
47 _ 163
72 _ 188
3.2
6.3
21.8
49 _ 121
46 _ 163
56 _ 188
4.0
6.8
19.1
4
73
40 _ 126
2.4
45 _ 126
7.5
9.2. Closed surfaces
Besides the tube-like surfaces, closed surfaces are also fre-quently
encountered in practical situations, but cannot be directly transformed to a
general volume like the tube-like surfaces. Fortunately, as we mentioned in
Section 5, the optimization
Fig. 5. Illustration of intima and adventitia for near wall and far wall in artery ultrasound image (compare Fig. 3).
Please cite this article as: Z. Wu, et al., Exact solution to median surface problem using 3D graph search and application to parameter space exploration,
Pattern Recognition (2014), http://dx.doi.org/10.1016/j.patcog.2014.07.019i
Z. Wu et al. / Pattern Recognition ∎ (∎∎∎∎) ∎∎∎–∎∎∎
10. Discussions and conclusion
Table 6
Comparison with the ground truth (unit: pixels) in multiple surface setting.
Video
Comparison type
Near wall
Far wall
Intima
Adventitia
Intima
Adventitia
1
Median vs. GT
BP vs. GT
0.63
0.61
0.51
0.48
0.57
0.55
0.47
0.47
2
Median vs. GT
BP vs. GT
0.48
0.46
0.80
0.80
0.50
0.49
0.81
0.80
3
Median vs. GT
BP vs. GT
0.98
0.96
1.43
1.46
0.67
0.66
0.76
0.75
4
Median vs. GT
BP vs. GT
0.48
0.46
0.74
0.69
0.54
0.51
0.53
0.50
Table 7
Computational time (seconds) in multiple surface setting.
video
#images
Near wall
Far wall
Resolution
Time
Resolution
9
Time
1
2
3
86
86
111
45 _ 121
47 _ 163
72 _ 188
93.44
172.79
405.76
49 _ 121
46 _ 163
56 _ 188
96.62
164.89
379.08
4
73
40 _ 126
74.36
45 _ 126
71.59
problem can be regarded as a general MRF problem and both [36] and [37]
provide ways for an exact computation. Some other closed surface works
[42,43] are based on dedicated base graph generation and the algorithm in
[37]. Therefore, the median surface computation is also applicable for closed
surfaces.
In this paper we have formulated the generalized median surface problem
and presented its exact solution by means of a 3D graph search algorithm.
The median concept has been con-cretized for many domains. This work can
be understood as an extension of the simpler case of 2D median contour
computation by means of dynamic programming [4] to 3D median surface.
Since dynamic programming cannot be directly extended to 3D in an efficient
manner, we resort to the 3D graph search algorithm [25]. In addition to
terrain-like surfaces including single and multiple coupled surfaces, the
median computation can also be applied to deal with tubular surfaces and
closed surfaces (of both types).
This work is motivated by the task of parameter space explora-tion
without ground truth, which is an effective means of dealing with the difficult
parameter problem and has been successfully applied to domains like 2D
contour detection [4] and region segmentation [21,22]. Our median surface
computation algorithm thus provides a useful tool for parameter exploration
in 3D surface segmentation or 2D contour segmentation in a temporal
context. A concrete application has been demonstrated on artery boundary
detection in sonography, which confirmed the findings from the previous
studies. That is, the median computation can not only avoid the parameter
training, but also potentially achieve results comparable with trained
parameters. Parameter training is only possible with the existing ground truth,
which is not always available. The median-based approach can thus be a good
alter-nate in case of no ground truth.
One question that may be of interest to the readers is why we do not
directly apply the graph search to solve the 3D surface segmentation problem.
Our motivation in this work is not the segmentation itself, but exploring
segmentation parameter space without ground truth. In this sense our
approach is on a meta-level above the segmentation level and can be based on
any baseline segmentation algorithm.
9.3. Smoothness constraint
Smoothness constraint is doubtless desirable for surface extrac-tion
problems. Although it is quite natural for the variational methods such as
active contours and level set approaches, impos-ing smoothness constraint to
graph search based algorithms is not trivial, especially for problems of
boundary or surface extraction. In [44,45], the graph search algorithm from
[37] is augmented by smoothness constraints by means of arc penalty. For our
median surface computation these extended versions can be used if
smoothness should be integrated into the detection.
9.4. Computational cost
As discussed before, the K-surface median surface computation may
suffer from high demand of computation. Even for the single-surface scenario
the computation demand may significantly increase when dealing with videos.
A large number T of frames will cause both space and time problems. For
typical image sizes in medical image analysis, for instance, a large T may
result in memory overflow in practice. Even if sufficient memory can be
allocated, the computation time can become too long to be useful for real
applications.
Both situations (K 41 and large T) call for a practical solution. To reduce
the complexity one can partition a video into blocks and process each block
by the algorithm from [25]. A separate handling of the blocks, however,
cannot guarantee the consistency between two adjacent blocks. Therefore,
suitable methods are needed to take care of smooth transition from one block
to another. We are currently investigating various block-wise approximate
variants.
Even if it is not our intention to directly apply the graph search algorithm
for 3D segmentation, we also tested this option by applying the same
preprocessing as in the baseline segmentation method used in this work [38].
For video 1, see Table 2, the performance measure (average L1 deviation in z)
is 0.57 (near wall) and 0.53 (far wall), which is comparable with our
ensemble solution (0.63 and 0.57, respectively). In contrast, video 3 leads to
1.74 (near all) and 1.44 (far wall), which is substantially worse than our
ensemble solution (0.98 and 0.67, respectively). Overall, the achieved
segmentation performance is not always satisfactory and can neither beat the
baseline segmentation algorithm nor our ensemble approach. The reason for
this observation may be the strong basic assumption of graph search that the
desired surface must be the minimum-cost surface. Generally, this assumption
poses substantial challenges to feature computation in the pre-processing step.
In practice, high-level properties are thus typically adopted as constraints to
relax this assumption which may not be easy to realize in the framework of
graph search. This motivates further work on improved preprocessing that
may enhance the performance of the graph search algorithm for 3D
segmentation, but this is a research topic different from the main motivation
of the current work, i.e. exploring segmentation parameter space without
ground truth.
Additional future work will be to study alternative weighting schemes, for
example, weighting schemes that adopt more high-level and prior properties
such as curvature. Furthermore, we will extend our ensemble approach to
more applications as motivated in Section 4. As an example, the median result
could also serve for selecting the best preprocessing and feature from various
potential options.
Please cite this article as: Z. Wu, et al., Exact solution to median surface problem using 3D graph search and application to parameter space exploration,
Pattern Recognition (2014), http://dx.doi.org/10.1016/j.patcog.2014.07.019i
Z. Wu et al. / Pattern Recognition ∎ (∎∎∎∎) ∎∎∎–∎∎∎
10
Conflict of interest
None declared.
Acknowledgments
This work is supported by the State Key Program of National Nature
Science Foundation of China, Grant no. 61231018. Xiaoyi Jiang was
supported by the Deutsche Forschungsgemeinschaft (DFG): SFB 656 MoBil
(project B3) and EXC 1003 Cells in Motion
– Cluster of Excellence.
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Zhengwang Wu received the B.S. degree in Computer Science from Xi'an Jiaotong University, Xi'an, China, in 2006. He is currently a Ph.D. student in Institute of Artificial Intelligence and
Robotics, Xi'an Jiaotong University. His research interests include medical image processing, structural pattern recognition, and combinatory optimization.
Xiaoyi Jiang studied Computer Science at Peking University and received his Ph.D. and Venia Docendi (Habilitation) degree from University of Bern, Switzerland. He was an associate professor at
Technical University of Berlin. Since 2002 he is a full professor of Computer Science at University of Münster, Germany. Currently, he is Editor-in-Chief of International Journal of Pattern
Recognition and Artificial Intelligence and also serves on the advisory board and editorial board of several journals including Pattern Recognition, IEEE Transactions on Cybernetics, and Chinese
Science Bulletin. He is a senior member of IEEE and fellow of IAPR.
Please cite this article as: Z. Wu, et al., Exact solution to median surface problem using 3D graph search and application to parameter space exploration,
Pattern Recognition (2014), http://dx.doi.org/10.1016/j.patcog.2014.07.019i
Z. Wu et al. / Pattern Recognition ∎ (∎∎∎∎) ∎∎∎–∎∎∎
11
Nanning Zheng graduated in 1975 from the Department of Electrical Engineering, Xi'an Jiaotong University, Xi'an, China, received the M.E. degree in information and control engineering from
Xi'an Jiaotong University in 1981, and the Ph.D. degree in electrical engineering from Keio University, Japan, in 1985. He is currently a full professor and the Director of the Institute of Artificial
Intelligence and Robotics at Xi'an Jiaotong University. His research interests include computer vision, pattern recognition, computational intelligence, image processing, and hardware implementation
of intelligent systems. He is a member of the Chinese Academy Engineering and IEEE fellow.
Yuehu Liu received the B.S. and M.E. degrees in computer science at Xi'an Jiaotong University, China, in 1984 and 1989, respectively, and the Ph.D. degree in electrical engineering from Keio
University, Japan, in 2000. He is currently a professor of the Institute of Artificial Intelligence and Robotics at Xi'an Jiaotong University. His research interests include computer vision, pattern
recognition, computational intelligence, image processing.
Da-Chuan Cheng studied Electrical Engineering (B.S.) at Chinese Culture University, Taiwan, Biomedical Engineering (M.S. and Ph.D.) at National Cheng Kung University, Taiwan. He is an
associate professor at the Department of Biomedical Imaging and Radiological Science of the China Medical University, Taiwan. His research interests include medical image processing and
optimization.
Please cite this article as: Z. Wu, et al., Exact solution to median surface problem using 3D graph search and application to parameter space exploration,
Pattern Recognition (2014), http://dx.doi.org/10.1016/j.patcog.2014.07.019i