Abstract The 2-D maximum entropy method not only considers the distribution of the gray information, but also takes advantage of the spatial neighbor information with using the 2-D histogram of the image. As a global threshold method, it often gets ideal segmentation results even when the imageÕs signal noise ratio (SNR) is low. However, its time-consuming computation is often an obstacle in real time application systems. In this paper, the image thresholding approach based on the index of entropy maximization of the 2-D grayscale histogram is proposed to deal with infrared image. The threshold vector (t, s), where t is a threshold for pixel intensity and s is another threshold for the local average intensity of pixels, is obtained through a new optimization algorithm, namely, the particle swarm optimization (PSO) algorithm. PSO algorithm is realized successfully in the process of solving the 2-D maximum entropy problem. The experiments of segmenting the infrared images are illustrated to show that the proposed method can get ideal segmentation result with less computation cost

Chen Liangzhou

Deng Yong

Du Feng

Shi Wenkang

Zhu Zhenfu

CiteSeerX

Pattern Recognition Letters 26 (2005) 597–603
www.elsevier.com/locate/patrecInfrared image segmentation with 2-D maximum
entropy method based on particle swarm optimization (PSO)
Du Feng a,*, Shi Wenkang a, Chen Liangzhou a, Deng Yong a,*, Zhu Zhenfu b
a School of Electronics and Information Technology, Shanghai Jiao Tong University, No. 1954,
Huashan Road, Shanghai 200030, People’s Republic of China
b National Defence Key Laboratory of Target and Environment Feature, Beijing 100854, People’s Republic of China
Received 15 March 2004; received in revised form 22 October 2004
Available online 8 December 2004Abstract
The 2-D maximum entropy method not only considers the distribution of the gray information, but also takes
advantage of the spatial neighbor information with using the 2-D histogram of the image. As a global threshold
method, it often gets ideal segmentation results even when the images signal noise ratio (SNR) is low. However, its
time-consuming computation is often an obstacle in real time application systems. In this paper, the image thresholding
approach based on the index of entropy maximization of the 2-D grayscale histogram is proposed to deal with infrared
image. The threshold vector (t, s), where t is a threshold for pixel intensity and s is another threshold for the local aver-
age intensity of pixels, is obtained through a new optimization algorithm, namely, the particle swarm optimization
(PSO) algorithm. PSO algorithm is realized successfully in the process of solving the 2-D maximum entropy problem.
The experiments of segmenting the infrared images are illustrated to show that the proposed method can get ideal seg-
mentation result with less computation cost.
 2004 Elsevier B.V. All rights reserved.
Keywords: Entropy; Image segmentation; Particle swarm optimization; 2-D histogram1. Introduction
Image segmentation is a technique decompos-
ing an image into meaningful parts, or objects,0167-8655/$ - see front matter  2004 Elsevier B.V. All rights reserv
doi:10.1016/j.patrec.2004.11.002
* Corresponding authors. Tel./fax: +86 21 62932851.
E-mail addresses: dfengu@sjtu.edu.cn (D. Feng), dengyong@
sjtu.edu.cn (D. Yong).and infrared image segmentation plays a special
role in automatic target recognition (Pal and Pal,
1993; Boskovitz and Guterman, 2002). Image
thresholding is essentially a pixels classification
problem (Weszka, 1978). Its basic objective is to
classify the pixels of a given image into two classes:
those pertaining to an object and those pertaining
to the background. While one includes pixels withed.
L
t
sO
2
4
3
1
L
j
i
Fig. 1. The 2-D histogram plane.
598 D. Feng et al. / Pattern Recognition Letters 26 (2005) 597–603gray values that are below or equal to a certain
threshold, the other includes those with gray val-
ues above the threshold. Thresholding is a popular
tool for image segmentation for its simplicity,
especially in the fields where real time processing
is needed. Up to now, many threshold selection
techniques have been proposed (Portes de Albu-
querque et al., 2004).
In general, threshold selection techniques can
be broadly divided into two groups, namely, glo-
bal and local thresholding. A global technique
may be point-dependent or region-dependent.
The thresholding method is point-dependent
if the threshold value is determined solely from
the pixel gray tone as represented by gray-level
histogram and is independent of the gray tone
of the neighborhood of a pixel. On the other
hand, a method is called region-dependent if the
threshold value is determined from the local prop-
erty within a neighborhood of a pixel. A global
thresholding technique is one that segments the
entire image with a single threshold value,
whereas a local thresholding technique is one that
partitions a given image into subimages and
determines a threshold for each of these sub-
images. Though point-dependent global thres-
holding methods are comparatively simple, only
a small part of the independent information of
the image is used. Therefore, these techniques
are not always useful. In this paper we propose
a global threshold selection method to do infrared
image segmentation, which uses both gray-level
distribution and spatial information, namely, 2-
D maximum entropy thresholding (Abutaleb,
1989; Kapur et al., 1985). Whats more, taking
consideration of the complexity of its computa-
tion, we introduce a new heuristic optimization
algorithm, called the particle swarm optimization
(PSO) algorithm to search the result (Kennedy
and Eberhart, 1995). The work is organized as
follows. In Section 2, the proposed thresholding
method and its theoretical justifications are pre-
sented. In Section 3, the PSO algorithm and its
modification are described. In Section 4, it is
shown that how the two methods are extended to-
gether for segmenting infrared images. Finally,
some concluding remarks regarding the proposed
methods are given.2. 2-D maximum entropy thresholding
The motivation of application of the maximum
entropy method to solve threshold selection prob-
lem in our paper is that this method has been suc-
cessfully applied in many real systems such as
image reconstruction and target recognition (Zenzo
et al., 1998; Pal and Pal, 1989). The maximum en-
tropy principle states that, for a given amount of
information, the probability distribution which
best describes our knowledge is the one that max-
imizes the Shannon entropy subjected to the given
evidence as constraints (Shannon, 1948; Klir and
Folger, 1988). In this paper, we propose to use
the 2-D maximum entropy method to do image
segmentation. A lot of application examples have
proved that the performance of the 2-D maximum
entropy method is better than that of the 1-D max-
imum entropy method.
The 2-D maximum entropy method is based on
the 2-D histogram of the image. The 2-D histo-
gram is got as the following:
pij ¼
nij
N  N ð1Þ
where N · N denotes the image size, and nij de-
notes the number of a pixel whose grey value
equals i and local average value equals j. The 2-
D histogram plane can be described as Fig. 1:
where the area 1 and 2 denote objects and back-
ground respectively, and 3 and 4 denote edges
and noise. So a threshold vector (t, s), where t is
a threshold for pixel intensity and s is another
threshold for the local average of pixels, should
be determined to divide them. According to the
maximum entropy principle, the determined
threshold vector should make area 1 and area 2
have the maximum information.
D. Feng et al. / Pattern Recognition Letters 26 (2005) 597–603 599Suppose the area 1 and area 2 have different
probability distributions. According to the thres-
hold vector (t, s), we denote P1 and P2 as:
P 1 ¼
Xs1
i¼0
Xt1
j¼0
pij; P 2 ¼
XL1
i¼s
XL1
j¼s
pij: ð2Þ
Then the 2-D discrete entropy can be defined as:
H ¼ 
X
i
X
j
pij lg pij: ð3Þ
The 2-D entropy of area 1 can be got:
Hð1Þ ¼ 
Xs1
i¼0
Xt1
j¼0
ðpij=P 1Þ lgðpij=P 1Þ
¼ ð1=P 1Þ
Xs1
i¼0
Xt1
j¼0
ðpij lg pij  pij lg P 1Þ
¼ ð1=P 1Þ lg P 1
Xs1
i¼0
Xt1
j¼0
pij
 ð1=P 1Þ
Xs1
i¼0
Xt1
j¼0
pij lg pij
¼ lgðP 1Þ þ H 1=P 1: ð4Þ
The entropy of 2nd area can also be got:
Hð2Þ ¼ lgðP 2Þ þ H 2=P 2; ð5Þ
where H1 and H2 are described as,
H 1 ¼ 
Xs1
i¼0
Xt1
j¼0
pij lg pij;
H 2 ¼ 
XL1
i¼s
XL1
j¼t
pij lg pij: ð6Þ
Then the function of entropy is:
/ðs; tÞ ¼ Hð1Þ þ Hð2Þ; ð7Þ
where H(1) and H(2) are described in the formula
(4) and (5).
According to the maximum entropy principle,
the threshold vector (s*, t*) should be satisfied to
/ðs; tÞ ¼ maxf/ðs; tÞg: ð8Þ3. The particle swarm optimization (PSO)
algorithm
The particle swarm optimization (PSO) is a par-
allel evolutionary computation technique devel-
oped by Kennedy and Eberhart (1995) based onthe social behavior metaphor. Due to the limita-
tion of space, we only briefly introduce PSO in this
paper. For more information, we refer reader to
Kennedy et al. (2001), a standard textbook on
PSO, treating both the social and computational
paradigms. PSO differs from traditional optimiza-
tion methods in that a population of potential
solutions is used in the search. The direct fitness
information, instead of function derivatives or re-
lated knowledge, is used to guide the search. As
mentioned above, it is promising to solve 2-D
maximum entropy problem by adopting PSO.
PSO is initialized with a group of random parti-
cles (solutions) and then searches for optima by
updating generations. Particles profit from the dis-
coveries and previous experience of other particles
during the exploration and search for higher objec-
tive function values. Let i indicate a particles index
in the swarm. Each of m particles fly through the n-
dimensional search space Rn with a velocity vi,
which is dynamically adjusted according to its
own previous best solution si and the previous best
solution ŝ of the entire swarm. The velocity updates
are calculated as a linear combination of position
and velocity vectors. The particles interact and
move according to the following equations
vðtþ1Þi ¼ wv
ðtÞ
i þ c1r
ðtÞ
1 ðs
ðtÞ
i  p
ðtÞ
i Þ þ c2r
ðtÞ
2 ðŝðtÞ  p
ðtÞ
i Þ;
ð9Þ
pðtþ1Þi ¼ v
ðtþ1Þ
i þ p
ðtÞ
i ; ð10Þ
where rðtÞ1 ; r
ðtÞ
2 
 UNIFð0; 1Þ are random numbers
between zero and one. c1, c2 are learning factors,
usually c1 = c2 = 2. And w is an inertia weight. It
is possible to clamp the velocity vectors by specify-
ing upper and lower bounds on vi, to avoid too
rapid movement of particles in the search space.
Then we can use the standard procedure to find
the optimum. The searching is a repeat process,
and the stop criteria are that the maximum itera-
tion number is reached or the minimum error con-
dition is satisfied.
The standard procedure is described as below:
(1) Set the iteration number t to zero. Initialize
randomly the swarm S of m particles (popula-
tion number) such that the position p0i of each
particle to meet the prescribed conditions.
Fig. 2. The first original infrared image.
600 D. Feng et al. / Pattern Recognition Letters 26 (2005) 597–603(2) Evaluate the fitness of each particle (F ðpðtÞi Þ,
object function).
(3) Compare the personal best of each particle to
its current fitness, and set sðtÞi to the better per-
formance, i.e.
sðtÞi ¼
sðt1Þi if f ðp
ðtÞ
i Þ 6 f ðs
ðt1Þ
i Þ;
pðtÞi if f ðp
ðtÞ
i Þ > f ðs
ðt1Þ
i Þ:
(
ð11Þ
(4) Set the global best ŝðtÞ to the position of the
particle with the best fitness within the swarm,
i.e.
ŝðtÞ 2 fsðtÞ1 ; s
ðtÞ
2 ; . . . ; s
ðtÞ
m g j F ðŝ
ðtÞÞ
¼ maxfF ðsðtÞ1 Þ; F ðs
ðtÞ
2 Þ; . . . ; F ðsðtÞm Þg: ð12Þ
(5) Change the velocity vector for each particle
according to Eq. (9).
(6) Move each particle to its new position, accord-
ing to Eq. (10).
(7) Let t = t + 1.
(8) Go to step 2, and repeat until meets the stop
criteria.
It can be easily seen that there are two key steps
when applying PSO to optimization problems: the
representation of the solution and the fitness func-
tion. One of the desirable merits of PSO is that
PSO takes real numbers as particles. It is not like
genetic algorithm (GA) (Davis, 1991), where
transformation of binary encoding and special
genetic operators are needed. The complete appli-
cation of PSO, as well as the method to do
image segmentation, is discussed in the following
section.Fig. 3. Segmented result of the first one.4. Image segmentation based on the proposed
method
Considering the 2-D maximum entropy method
and PSO together, we set the threshold vector (t, s)
as the particle, and /(s, t) as the fitness to guide the
search. After getting the 2-D histogram of the
image, we adopt the standard PSO procedure to
search the optimum result of (s*, t*) which can
produce the maximum fitness. Then, the infrared
image can be segmented according to the value
of (s*, t*).Experiments on image segmentation are used to
illustrate the validation of the proposed algorithm.
The first original image used contains 290 · 200
pixels, which is shown as Fig. 2, and Fig. 3 is its
segmentation result of the proposed method. The
optimum answer is (143,132) with the greatest en-
tropy of 10.6285. As we can see, the segmentation
result is ideal. Because there is a lot of papers have
demonstrated the 2-D histogram entropy method
is better than the 1-D histogram entropy method,
we pay more attention on the advantage of its
computation speed. If the maximum entropy of
2-D histogram is obtained by exhaustive search
method, it needs to do 256 · 256 times computa-
tion of entropy function (8). However, with the
method of PSO, some nice results can be shown
in the following tests. When the population num-
ber of particles is initiated with 15 and max itera-
Fig. 5. Segmented result of the second one.
Table 1
The process of the convergence
Iteration number Threshold Object value
1th (55,58) 9.8942
2th (114,107) 10.2274
3th (111,108) 10.2427
4–10th (120,119) 10.2796
11–30th (123,122) 10.2822
D. Feng et al. / Pattern Recognition Letters 26 (2005) 597–603 601tion number is set as 30, the results can be shown
as follows.
From the 2nd iteration to the 6th iteration, the
fitness is convergent to a local extremum of
10.28991 with (94,94), and from the 7th iteration
to the 9th iteration, the object function is conver-
gent to another local extremum of 10.577085 with
(142,141). And when it reaches the 10th iteration,
the global maximum value of 10.6285 is got and
the threshold is (143,142). In other words, only
with the cost of doing 15 · 10 = 150 times compu-
tation of entropy function, the optimum answer
can be got. Comparing with exhaustive searching
method, the speed is improved 256 · 256/
150  436 times. What is more, when the popula-
tion number of particles is initiated with 30, the
best answer can be captured at the 4th iteration.
It means the algorithm only does 30 · 4 = 120
times computation of entropy function. It is rea-
sonable that the computation can be further re-
duced by trying more proper parameters.
We also do another image segmentation with
Fig. 4, which contains 352 · 288 pixels.
And the result with the proposed method is
shown like Fig. 5. Through exhaustive search,
the optimum threshold is (123,122) with the max-
imum value of 10.2882. Again, we focused on the
advantage of its computation speed. Table 1 de-
scribes the process of the algorithms convergence,
where all the constants are set as the same as the
first experiments. The optimum answer is got onlyFig. 4. The second original infrared image.with cost of doing 15 · 11 = 165 times computa-
tion of entropy function. Comparing with exhaus-
tive searching method, the speed is improved
256 · 256/165  397 times.
It is well known that evaluating segmentation
results and comparing segmentation algorithms
are not simple tasks (Yasnoff et al., 1977; Hoover
et al., 1996; Yu et al., 1994). However, one of the
most widely used criteria for performance evalua-
tion is whether the system can outline the desired
or important regions in the image. In addition,
Haralick and Shapiro (1985) pointed out that
good segmentation results should present simple,
uniform and homogeneous regions, with simple,
not ragged and spatially accurate boundaries. We
believe our results satisfy these requirements. Be-
cause sometimes there are more background pixels
than object ones, it does not make sense the direct
measure of the performance since both types of
errors have not the same relevance. A balanced
error measure to quantitatively evaluate our exper-
imental results is proposed:
Fig. 6. Manual segmented result of the first one.
Fig. 7. Manual segmented result of the second one.
602 D. Feng et al. / Pattern Recognition Letters 26 (2005) 597–603Error rate ¼ 1
2
misdetections
object pixels
þ 1
2
false alarms
background pixels
; ð13Þ
where the object pixels and background pixels can
be got by manual method.
According to (13), the error rates of Figs. 3 and
5 are 4.77% and 3.79%, respectively. We think the
results obtained by our method can be regarded as
reasonably good and applicable in subsequent pro-
cessing. The manual segmented results of Figs. 2
and 4 are shown in Figs. 6 and 7, respectively.5. Conclusion
2-D histogram entropy is a good method to do
infrared image segmentation except its complexcomputation. With the help of PSO method, the
maximum threshold can be obtained easily with
high efficiency. The image segmentation method
of 2-D maximum entropy method based on PSO
is a simple and effective method to do infrared im-
age segmentation. Especially, it is full of impor-
tance in the system where real time processing is
needed. What is more, although there are a lot of
examples of applying PSO to the continuous opti-
mization problem, there is relative less in the dis-
crete fields. The maximum entropy problem is an
example of discrete problem. Owing to the success
of the proposed method in this paper, we can see
the capability of PSO to deal with discrete prob-
lems is also promising.Acknowledgments
The authors would like to express their thanks
for the anonymous referees for their valuable com-
ments and suggestions, which help us to make
much improvements in the present version of the
manuscript. This work was supported by National
Natural Science Foundation of China grant no.
30400067, Shanghai Nature Science Foundation
grant no. 03ZR14065 and National Defence Key
Laboratory grant no. 51476040103JW13.References
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Infrared image segmentation with 2-D maximum entropy method based on particle swarm optimization (PSO

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Infrared image segmentation with 2-D maximum entropy method based on particle swarm optimization (PSO

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