This paper proposes the important issues in signal segmentation. The signal is disturbed by multiplicative noise where the number of segments is unknown. A Bayesian approach is proposed to estimate the parameter. The parameter includes the number of segments, the location of the segment, and the amplitude. The posterior distribution for the parameter does not have a simple equation so that the Bayes estimator is not easily determined. Reversible Jump Markov chain Monte Carlo (MCMC) method is adopted to overcome the problem. The Reversible Jump MCMC method creates a Markov chain whose distribution is close to the posterior distribution. The performance of the algorithm is shown by simulation data. The result of this simulation shows that the algorithm works well. As an application, the algorithm is used to segment a Synthetic Aperture Radar (SAR) signal. The advantage of this method is that the number of segments, the position of the segment change, and the amplitude are estimated simultaneously

Doisy, Michel

Suparman, Suparman

English

TELKOMNIKA (Telecommunication Computing Electronics and Control)

TELKOMNIKA, Vol.16, No.2, April 2018, pp. 673~680 
ISSN: 1693-6930, accredited A by DIKTI, Decree No: 58/DIKTI/Kep/2013 
DOI: 10.12928/TELKOMNIKA.v16i2.7510  673 
  
Received October 9, 2017; Revised February 17, 2018; Accepted February 28, 2018 
Bayesian Segmentation in Signal with Multiplicative 
Noise Using Reversible Jump MCMC 
 
 
Suparman*
1
, Michel Doisy
2
 
1
Department of Mathematics Education, Ahmad Dahlan University 
Jalan Prof. Dr. Soepomo, SH Yogyakarta, Indonesia, +62 274 563515 
2
Signal and Communications Group, ENSEEIHT 
2
Rue Camichel, Toulouse 31071, France 
*Corresponding author, e-mail: suparman@pmat.uad.ac.id 
 
 
Abstract 
This paper proposes the important issues in signal segmentation. The signal is disturbed by 
multiplicative noise where the number of segments is unknown. A Bayesian approach is proposed to 
estimate the parameter. The parameter includes the number of segments, the location of the segment, and 
the amplitude. The posterior distribution for the parameter does not have a simple equation so that the 
Bayes estimator is not easily determined. Reversib le Jump Markov chain Monte Carlo (MCMC) method is 
adopted to overcome the problem. The Reversib le Jump MCMC m ethod creates a Markov chain whose 
distribution is close to the posterior distribution. The performance of the algorithm is shown by simulation 
data. The result of this simulation shows that the algorithm works well. As an application, the algorithm is 
used to segment a Synthetic Aperture Radar (SAR) signal. The advantage of this method is that the 
number of segments, the position of the segment change, and the amplitude are estimated simultaneously. 
  
Keywords: Reversib le Jump MCMC, Bayesian, Multiplicative Noise, Signal Segmentation 
  
Copyright © 2018 Universitas Ahmad Dahlan. All rights reserved. 
 
 
1. Introduction 
Signal processing with additive noise has been investigated by several researches, for 
example Gustafsson et al. [1]. In many applications it is often found that signal is disturbed by 
multiplicative noise. Some researches have also discussed signal with multiplicative noise, such 
as Ullah et al. [2], Osoba and Kosko [3], Tian et al. [4], and Dong et al. [5]. Ullah et al. [2] used a 
variational approach to restore images with multiplicative noise. Osoba and Kosko [3] used the 
noisy Expectation-Maximization Algorithm for multiplicative noise injection. Tian et al. [4] used 
an adaptive fractional-order method to eliminate multiplicative noise. Dong et al. [5] proposed a 
method using a sparse analysis model for signal with multiplicative noise. In signal 
segmentation with multiplicative noise, generally the number of segments is unknown and must 
be estimated based on the data. This paper proposes the segmentation of signal with 
multiplicative noise when the number of segments is unknown. 
Let N be the many pixels contained in a line from the SAR image. The equation of the 
line can be expressed in the following form (Suparman et al. [6], Tourneret et al. [7]) : 
 
 
ttt
zry 
 
N,,2,1t    (1) 
 
with yt is the intensity of the measured SAR image, rt is SAR intensity, and zt is a multiplicative 
noise. In various SAR images, including agricultural images, the properties of rt and zt can be 
defined as follows (Oliver and Quegan [8]) : 
a. (
a
) 
SAR intensity rt is a step function. The equation can be written as :  
Kt
hr   1KK ntn   
with K = 0, 1, …, Kmax. Here, nK is the position of the height change of K
th
 step. (with 
agreement n0 = 0 and nK+1 = N) and hK is the height of K
th
 step, and K is the number of 
steps.  
b. (
b
Multiplicative noise z t is given in the form of a random variable that follows the gamma 
distribution with the mean L and variance 1/L, written z t ~ G(L, L), 
     ISSN: 1693-6930 
TELKOMNIKA Vol. 16, No. 2, April 2018 : 673 – 680 
674 
) 
   
t
1L
t
L
t
Lzexpz
)L(
L
zf 

   
N,,2,1t 
 
 
 
Here, L is the number of measurements. The value of L is known. 
Based on the data yt (t = 1, 2, ..., N), then the value of the parameter 
 
K, n
(K) 
= (n1, n2, …, nK+1) and h
(K+1)
 = (h0, h1, …, hK).  
 
will be estimated. To estimate the value of these parameters, a hierarchical Bayesian approach 
is used. 
 
 
2.    Research Method 
2.1. Hiearchical Bayesian 
The Bayesian approach [9] is a method for estimating parameter values
)h,n,K( )K()K(  
which is based on the information from the data yn (expressed in the 
probability distribution )y(f  ) and information from the parameter  (expressed in the prior 
distribution )( ). Due to a multiplicative noise )L,L(G~zt , then the probability distribution 
for yt can be written as : 
 
  




 
 
 
i
1iiK
0i
)n,n(L
i
h
)n,n,y(L
exphyf 1ii
 
  (2) 
with ab)b,a(  ,  
b
1an n
y)b,a,y( , and symbol “” means "proportional to". 
 To use the Bayesian approach, the prior distribution for the parameter  should be 
determined. Prior distribution for parameter  is taken the same as in Suparman et al. [6]. 
Suppose Kmax is the maximum number of steps. The K is assumed to follow a Binomial 
distribution with parameter . The prior distribution for K can be written as 
 
   KKKmaks maks)1(,KK   K = 0, 1, …, Kmax.  (3) 
    
For the value of K given, n
(K)
 is assumed to follow the following distribution :  
 
      K0i i1i)K( )1nn(Kn  
  (4) 
    
and h
(K)
 follows the inverse gamma distribution with parameters   dan  . Prior distribution for 
h
(K
 can be written as 
 
   ]
h
[exph,,Kh
i
1
i
K
0i
)1K( 



 
  (5) 
 
 The problem that arises is the presence of hyperparameter ),,(  in the above 
prior distributions. To simplify the problem, in Suparman et al. [6] value   is known. In this 
paper, as in Tourneret et al. [7] hyperparameter   is seen as a random variable with a given 
distribution, here   follows a uniform distribution at interval (0,1) and   follows Jeffrey 
distribution. The value   is taken relatively small. 
Suppose that )y,(  is a posterior distribution for  . By using Bayes's theorem, the 
posterior distribution )y,( 
 
can be expressed as the multiplication of the probability 
distribution for yi and prior distribution for ),(   :  
 
     yfy,    
 
  (6) 
 
TELKOMNIKA  ISSN: 1693-6930  
 
Bayesian Segmentation in Signal with Multiplicative Noise Using Reversible ... (Suparman) 
675 
The estimation of the parameter   will be based on the posterior distribution. Because 
the shape of the posterior distribution )y(
 
is very complex, it is difficult to estimate 
parameter value  . To overcome this problem, the Reversible Jump MCMC method is adopted. 
 
2.2. Reversible Jump MCMC Method 
Suppose that ),(M   is a Markov chain. The MCMC method is a sampling method. 
This sampling method makes a homogeneous Markov chain 
m21
M,,M,M   which satisfies 
periodic and irreducible properties such that 
m21
M,,M,M   can be considered as a random 
variable that follows the distribution  y,
 
[10]. The chain Markov 
m21
M,,M,M   can be 
used to estimate the parameter M. To realize it, Gibbs algorithm is adopted which consists of 
two stages: 
a. Simulate the distribution )y,(   
b. Simulate the distribution )y,(   
The Gibbs algorithm is used to simulate the distribution )y,(  . Hybrid algorithm is 
used to simulate the distribution )y,(  . This hybrid algorithm combines the Reversible 
Jump MCMC algorithm [11] to simulate parameter )y,n,K( )K(   and algoritma Gibbs to 
simulate parameter )y,h( )K(  . Reversible Jump MCMC algorithm is an extension of the 
Metropolis-Hastings algorithm. 
 
2.2.1.Distribution Simulation )y,(   
Suppose that )y,( 
 
is a conditional distribution of parameter   given  and y. The 
distribution )y,(  can be expressed as 
 
 
)
h
1
(exp)1()y,(
i
K
0i
)1K(KKK maks  

 
 
 
This distribution is the multiplication of distribution )1KK,1K(B
max
  and 
)h/1,1)1K((G
i
K
0i  . The Gibbs algorithm is used to simulate it. 
 
2.2.2. Distribution Simulation )y,(   
Suppose that )y,(  is a conditional distribution of   given )y,( . This conditional 
distribution can be expressed as 
  
)y,(  
 
 







  




 K
0i i1i
1K
1K2
2N
KKKK
K
1)nn(
)(C
1
)1(C maksmaks  
  
]
h
)n,n,y,L,(
[exph
i
1ii1)n,n,L,(
i
K
0i
1ii 


 
 
 
where )b,a(L  
and ).b,a,y(L)b,a,y,L,(   
The conditional distribution )y,(   is integrated against )K(h , it will be obtained 
  
)y,n,K( )K(  
 
 







  




 K
0i i1i
1K
1K2
2N
KKKK
K
1)nn(
)(C
1
)1(C maksmaks  
  
)n,n,y,L,(
))n,n,L,((
1ii
1iiK
0i






 
 
 
 
     ISSN: 1693-6930 
TELKOMNIKA Vol. 16, No. 2, April 2018 : 673 – 680 
676 
On the other hand, we have 
  
)y,,n,Kh( )K()K(  
 
]
h
)n,n,y,L,(
[exph
i
1ii1)n,n,L,(
i
K
0i
1ii 


   
  ))n,n,y,L,(,n,n,L,((IG
1ii1ii
K
0i  
 The distribution )y,( 
 
as the multiplication of the distribution )y,n,K(
)K( 
 
and 
distribution )y,,n,Kh( )K()K(  , that is : 
 
 )y,(  = )y,n,K(
)K(  )y,,n,Kh( )K()K( 
 
 
 
To simulate the distribution )y,(  , a Gibb algorithm is used. This Gibb algorithm consists of 
two stages : 
a. Stage 1 : Simulate the distribution )y,,n,Kh( )K()K(   
b. Stage 2 : Simulate the distribution )y,n,K(
)K(   
To simulate the distribution )y,,n,Kh( )K()K(  , the Gibbs algorithm is used. On the other 
hand, distribution )y,n,K(
)K(   is not explicitly so that the MCMC Reversible Jump algorithm 
is used to simulate it. 
 
 
3. Results and Analysis 
As an application, this method is applied to segment simulation multiplicative and real 
multiplicative signals. As in [12], a simulation study was undertaken to confirm the performance 
of the Reversible Jump MCMC algorithm whether it works well or not. The Case studies are 
given to provide examples of application of research to solve problems in everyday life.To 
segment the multiplicative signals of simulation and real multiplicative signals, the Reversible 
Jump MCMC algorithm is implemented as much as 25 thousand iterations with a 5 thousand 
burn-in period. 
 
3.1. Multiplicative Signal Simulation 
Figure 1 is a simulated multiplicative signal created according to the Equation (1) with  
N = 250 and L = 5. The value K = 3, vector value n
(3)
 = (75, 125, 200) and vector value  
h
(4)
 = (1,7,3,5). 
 
 
 
 
Figure 1. Multiplicative Signal Simulation 
 
TELKOMNIKA  ISSN: 1693-6930  
 
Bayesian Segmentation in Signal with Multiplicative Noise Using Reversible ... (Suparman) 
677 
Based on the simulation multiplicative signal in Figure 1, the parameter is estimated. 
The number of segments K, vector n
(K)
 and vector h
(K)
 are estimated by using the Reversible 
Jump MCMC algorithm. Estimator of K, n
(K)
 and h
(K)
 are 
 
,3Kˆ  ),196,125,75(n )Kˆ(   and )1.5,1.3,3.7,9.0(h )Kˆ( 
 
 
The histogram for K is given in Figure 2.  
 
 
 
Figure 2. Histogram for K 
 
 
The signal segmentation generated by the algorithm is presented in Figure 3. 
 
 
 
 
Figure 3. Result of Multiplicative Segmentation of Simulated Signal 
 
 
If the true parameter value of K, n(K), h(K) is compared against the estimated value of 
parameter K, n
(K)
, h
(K)
 obtained by algorithm, it appears that the algorithm can work well. 
 
3.2. Real Multiplicative Signal 
Now, algorithm is used to segment a line on the real image. The real image used is 480 
x 640 as shown in Figure 4. The image taken using the Nokia 3220 mobile phone is a natural 
scene around the Imogiri Tomb, Yogyakarta Indonesia. 
     ISSN: 1693-6930 
TELKOMNIKA Vol. 16, No. 2, April 2018 : 673 – 680 
678 
 
 
Figure 4. Images of the Natural Scene around the Imogiri Tomb, Yogyakarta 
Indonesia 
 
 
The 198
th
 column of the real image is presented in Figure 5. The 198
th
 line will be called a real 
signal. 
 
 
 
 
Figure 5. The 198
th
 Column of the Real Image 
 
 
The Reversible Jump MCMC algorithm is implemented in this real signal. An estimation for the 
value K, n
(K)
 dan h
(K)
 is given as follows :  
 
,3Kˆ  )394,213,145(n )Kˆ(  and ).139,107,217,151(h
)Kˆ( 
 
 
The histogram for K is given in Figure 6. Because the mode of the histogram is 3, the estimator 
of the number of segments is 4. 
TELKOMNIKA  ISSN: 1693-6930  
 
Bayesian Segmentation in Signal with Multiplicative Noise Using Reversible ... (Suparman) 
679 
 
 
Figure 6. Histogram for K 
 
 
The results of its segment are presented in Figure 7. 
 
 
 
 
Figure 7. Results of Real Multiplicative Signal Segmentation 
 
 
4. Conclusion 
The above description was a theoretical study of Reversible Jump MCMC algorithm and 
their applications for segmenting signal models with multiplicative noise. From the simulation 
results showed that Reversible Jump MCMC algorithm can segment the signal well.  
As an algorithm implementation, real signal was drawn from columns in a natural scene 
around the Imogiri Tomb, Yogyakarta Indonesia. If the Reversible Jump MCMC algorithm is 
implemented on each row or column of the image it will generate segmentation of the image.  
The advantage of this method is that the number of segments, the position of the segment 
change, and the amplitude are estimated simultaneously. The development of a Reversible 
Jump MCMC algorithm to segment images directly will be an interesting research topic.  
 
 
 
     ISSN: 1693-6930 
TELKOMNIKA Vol. 16, No. 2, April 2018 : 673 – 680 
680 
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TELKOMNIKA (Telecommunication Computing Electronics and Control). 2014;12(1): 87-96 


Bayesian Segmentation in Signal with Multiplicative Noise Using Reversible Jump MCMC

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Bayesian Segmentation in Signal with Multiplicative Noise Using Reversible Jump MCMC

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