Informatica 40 (2016) 159–167 159 
A Mixed Noise Removal Method Based on Total Variation 
Dang N.H. Thanh  
Tula State University; 92 Lenin Ave., Tula, Russian Federation 
Hue College of Industry, 70 Nguyen Hue st., Hue, Vietnam 
E-mail: dnhthanh@hueic.edu.vn 
 
Dvoenko Sergey D. 
Tula State University; 92 Lenin Ave., Tula, Russian Federation 
E-mail: dsd@tsu.tula.ru 
 
Dinh Viet Sang 
Hanoi University of Science and Technology; 1 Dai Co Viet st., Hanoi, Vietnam 
E-mail: sangdv@soict.hust.edu.vn 
 
Keywords: total variation, ROF model, gaussian noise, poisson noise, image processing, biomedical image, euler-
lagrange equation 
Received: March 24, 2016 
Due to the technology limits, digital images always include some defects, such as noise. Noise reduces 
image quality and affects the result of image processing. While in most cases, noise has Gaussian distri-
bution, in biomedical images, noise is usually a combination of Poisson and Gaussian noises. This com-
bination is naturally considered as a superposition of Gaussian noise over Poisson noise. In this paper, 
we propose a method to remove such a type of mixed noise based on a novel approach: we consider the 
superposition of noises like a linear combination. We use the idea of the total variation of an image in-
tensity (brightness) function to remove this combination of noises. 
Povzetek: Članek predlaga izvirno kombinacijo Gaussovega in Poissonovega filtra za filtriranje šuma v 
slikah.
1 Introduction 
Image denoising has attracted a lot of attention in recent 
years. In order to suppress noise effectively, we need to 
know its type. There are many types of noises, for exam-
ple, Gaussian (digital images), Poisson (X-Ray images), 
Speckle (ultra sonograms) noises and so on. 
One of the most famous effective methods is the to-
tal variation model [2-4, 10, 12, 17, 18, 22, 26]. The first 
person who suggested it to solve the denoising problem 
is Rudin [17]. He used the total variation as a universal 
tool in image processing. His denoising model is well-
known as the ROF model [3, 17]. The ROF model is tar-
geted to efficiently remove Gaussian noise only.  
This model is often used to remove not only Gaussi-
an noise, but also other types of noise. For example, the 
ROF model suppresses Poisson noise not so effectively. 
Le T. [9] proposed another model, well-known as the 
modified ROF model to remove Poisson noise only.  
Gaussian and Poisson noises both are widespread in 
real situations, but their combination is important too, for 
example, in electronic microscopy images [7, 8]. In these 
images, both types of noises are combined as a superpo-
sition. In physical process, Poisson noise usually is added 
first, before Gaussian noise. Luisier F. with co-authors 
proposed the theoretically strong PURE-LET method 
[11] (Poisson-Gaussian unbiased risk estimate – linear 
expansion of thresholds) to remove this type of combina-
tion of noises.  
However, such kind of noises usually can be consid-
ered as dependent on the image acquisition systems. At 
the same time, in many papers devoted to the image de-
noising problem the idea of Poisson-Gaussian noise 
combination  is considered, even though such is not the 
case.   
From other side, many noise reduction approaches 
have been developed, particularly, wavelet-based trans-
forms, etc. It needs to draw attention, noise reduction 
approaches that have been developed based on wavelet 
transform are only for Gaussian or Poisson noise. 
In order to remove mixed noise, let us assume that 
the superposition of noises can be equivalent to some 
unknown linear combination of them. 
We can combine ROF and modified ROF models to 
suppress the linear combination of noises. The obtained 
model is supposed to remove the mixed noise better than 
ROF or modified ROF models separately. Additionally, 
it is simpler than PURE-LET, because we try to find only 
the proportion between Poisson and Gaussian noises in 
the mixed noise. 
In experiments, we use images and add noise into 
them. The image quality is compared with some other 
denoising methods such as ROF, modified ROF models, 
and PURE-LET method to remove the superposition of 
noises. In our paper [19], we proposed to remove the 
linear combination of Poisson and Gaussian noises and 
160 Informatica 40 (2016) 159–167 N.H.T. Dang et al.  
 
compared results with Wiener [1] and median [23] filters, 
and with Beltrami method [29]. Our method gives better 
results for Gaussian and Poisson noises separately, and 
for the combination of noises too. Hence, in this paper, 
we do not compare our approach with these methods.  
In order to compare image quality after restoration, 
we use criteria PSNR (Peak Signal-to-Noise Ratio), MSE 
(Mean Square Error), SSIM (Structure SIMilarity) [24, 
25]. The PSNR criterion is the most important, because it 
is always used to evaluate images and signals quality. 
In this paper, we try to represent and discuss only the 
case limited by the greyscale artificial and real images 
with artificial noise. According to it, we can use only 
criteria above based on the full-reference image quality 
evaluation approach.  
In the case of greyscale real images with unknown 
noises, we need to use the no-reference approach to eval-
uate the quality of denoising. In general, it is complicated 
theoretical problem to develop a criterion for it.  
Our investigation based on BRISQUE criterion [13] 
(Blind/Referenceless Image Spatial QUality Evaluator) 
in this case was discussed in paper [20].  
2 Combined denoising model  
Let in 2R  space a bounded domain 2R  be given. 
Let functions ( , ) Ru x y   and ( , ) R,v x y   respectively, 
be ideal (without noise) and observed (noisy) imag-
es, ( , )x y  .  For smooth function u , its total variation 
can be defined by [ ] | |TV u u dxdy

  , where 
( , )x yu u u   is a gradient, /xu u x   , /yu u y   , 
2 2| | x yu u u   . In this paper, we consider that the func-
tion u  always has limited total variation [ ]TV u   . 
According to [2, 3, 17, 18], an image smoothness is 
characterized by the total variation of an image intensity 
function. The total variation of the noisy image is always 
greater than the total variation of the corresponding 
smooth image. In order to solve the problem 
[ ] minTV u  , we need to use the following condition 
2( )v u dxdy const

  . 
Hence, we obtain the ROF model to remove Gaussi-
an noise in the image [17, 18]: 
2argmin | | ( )
2u
u u dxdy v u dxdy
 
 
    
 
 

, 
where 0  is Lagrange multiplier. This is a solution of 
the unconstrained optimization problem. 
In order to remove Poisson noise, Le T. built another 
model based on ROF model [9] as the optimization prob-
lem [ ] minTV u   with the following constraint 
ln( ( | )) ( ln( ))p v u dxdy u v u dxdy const
 
    . 
This model resulted in the following unconstrained 
optimization problem 
* argmin | | ( ln( ))
u
u u dxdy u v u dxdy
 
 
    
 
  , 
where 0  is a coefficient of regularization. This is the 
known modified ROF model to remove Poisson noise. 
In order to build a model for removing the mixed 
Poisson-Gaussian noise, we also solve the same optimi-
zation problem [ ] minTV u  , but with a different con-
straint as follows. 
This constraint is very similar to constraints above. 
We consider, the noise variance is unchangeable (Pois-
son noise is not changed and Gaussian noise only de-
pends on noise variance): 
      ln( ( | ))p v u dxdy const

 ,   (1) 
where ( | )p v u  is a conditional probability of the real 
image v  with the ideal image u  given. 
The probability density function of Gaussian noise is 
2
1 2
( )
( | ) exp / ( 2 )
2
v u
p v u
 
  
 
 

, 
and the probability distribution of Poisson noise is 
2 ( | ) exp( ) / !
vp v u u u v  . 
We have to notice that intensity functions of images 
u  and v  are integer (for example, for 8-bits greyscale 
image the range of intensity is from 0 to 255). 
In order to combine Gaussian and Poisson noises, we 
consider the following linear combination 
1 1 2 2ln( ( | )) ln( ( | )) ln( ( | ))p v u p v u p v u   , 
1 0 , 2 0 , 1 2 1   . 
According to (1), we obtain the denoising problem as 
a constrained optimization problem 
*
21
22
arg min | |
( ) ( ln( )) ,
2
u
u u dxdy
v u u v u dxdy


  



      
  



 

 
where   is a constant value. We transform this problem 
into unconstrained optimization problem by using La-
grange functional 
21
2
( , ) | | ( )
2
L u u dxdy v u dxdy
 

    

 

 

2 ( ln( ))u v u dxdy


  

   
to find the solution as 
 
* *
,
( , ) arg min ( , )
u
u L u

   (2) 
where 0  is Lagrange multiplier. 
If 1 0  and 2   , we obtain the modified ROF 
model to remove Poisson noise. If 2 0  and 
2
1 /    , we obtain the ROF model to remove 
Gaussian noise. If 1 20, 0   , we obtain our model to 
remove mixed Poisson-Gaussian noise. 
 
A Mixed Noise Removal Method Based on Total Variation Informatica 40 (2016) 159–167 161 
3 Discrete denoising model  
The problem (2) can be solved by using Lagrange multi-
pliers method [5, 16, 28].  
We use Euler-Lagrange equation [28]. Let a function 
( , )f x y  be defined in a limited domain  2R  and be 
second-order continuously differentiated by x  and y , 
where ( , )x y  . Let ( , , , , )x yF x y f f f  be a convex 
functional, where /xf f x   , /yf f y   . Then the 
solution of the following optimization problem 
( , , , , ) minx yF x y f f f dxdy

  
satisfies the following Euler-Lagrange equation  
( , , , , ) ( , , , , )
xf x y f x y
F x y f f f F x y f f f
x

 

 
( , , , , ) 0
yf x y
F x y f f f
y



, 
where /fF F f   , /xf xF F f   , /yf yF F f   . 
We use the above result to solve the obtained model. 
Then the solution of the problem (2) satisfies the follow-
ing Euler-Lagrange equation 
1
22
( ) (1 )
v
v u
u
    



 
2 2 2 2
0,
yx
x y x y
uu
x yu u u u
   
     
     
   
   
(3) 
where 1/  . We rewrite (3) in the form 
1
22
( ) (1 )
v
v u
u
   



2 2
2 2 3/2
2
0
( )
xx y x y xy x yy
x y
u u u u u u u
u u
 



, 
  (4) 
2
2xx
u
u
x



, 
2
2yy
u
u
y



, 
xy yx
u u
u u
x y y x
     
     
      . 
In order to obtain the discrete form of the model (4), we 
add an artificial time parameter and consider the function 
( , , )u u x y t  in the following diffusion equation 
1
22
( ) (1 )t
u v
u v u
t u

     




 
2 2
2 2 3/2
2
( )
xx y x y xy x yy
x y
u u u u u u u
u u
 

 . 
(5) 
Then the discrete form of the equation (5) is 
11
2
( )k k ki j i j i j i ju u v u


   




2 (1 )
i j k
i jk
i j
v
u

  


  , 
(6) 
2
2 2 3/2
( )( ( ))
(( ( )) ( ( )) )
k k
xx ij y ijk
i j k k
x ij y ij
u u
u u
 
 
  
  
2
2 2 3/2
2 ( ) ( ) ( ) ( ( )) ( )
(( ( )) ( ( )) )
k k k k k
x ij y ij xy ij x ij yy ij
k k
x ij y ij
u u u u u
u u
      
  
, 
1, 1,
( )
2
k k
i j i jk
x ij
u u
u
x
 
 

, 
, 1 , 1
( )
2
k k
i j i jk
y ij
u u
u
y
 
 

,
1, 1,
2
2
( )
( )
k k k
i j ij i jk
xx ij
u u u
u
x
  
 

, 
, 1 , 1
2
2
( )
( )
k k k
i j ij i jk
yy ij
u u u
u
y
  
 

,
1, 1 1, 1 1, 1 1, 1
( )
4
k k k k
i j i j i j i jk
xy ij
u u u u
u
x y
         
 
 
, 
1 1 2 20 1 1, , 0 1 , 1 ,
; ; ; ;k k k k k k k kj j N j N j i i i N i Nu u u u u u u u      
1 21,..., ; 1,..., ;i N j N 
0,1,..., ; 1; 0 1k K x y       , 
where K  is enough great number, 500K  . 
4 Optimal model parameters 
In practice, parameters 1 2, , ,     in procedure (6) are 
usually unknown. We have to change 1 2, ,    into 
1 2, ,
k k k    to evaluate them on every iteration step k .  
4.1 Optimal parameters λ1 and λ2 
Let ( , )u   be a solution of problem (2). Then we obtain 
the following condition ( , ) / 0L u u   . This condition 
give us optimal 1  and 2 : 
1
2
(1 )
1
( ) (1 )
v
dxdy
u
v
v u dxdy dxdy
u

 


  

 


, 2 11   . 
The discrete form for 0,1,...,k K  is 
1 2
1 2
1 1
1
2
1 1
(1 )
( 1 )
N N
ij
k
i j ijk
kN N
ij ij ij
k
i j ij
v
u
v u v
u
 
 



 




, 2 11
k k   . 
4.2 Optimal parameter μ 
In order to find the optimal  , we multiply (3) by 
( )v u  and integrate by parts over domain  . Finally, 
we obtain the formula to find the optimal 
2
21
22
2 2
2 2
( )
( ( ) )
( )
x x y y
x y
x y
v u
v u dxdy
u
u v u v
u u dxdy
u u



  


 






 . 
The discrete form is 
1 2
1 2
2
21
22
1 1
1 1
( )
( ( ) )
kN N k
ij ijk k
ij ij k
i j ijk
N N
k
ij
i j
v u
v u
u 
 

  








, 
162 Informatica 40 (2016) 159–167 N.H.T. Dang et al.  
 
2 2( ( )) ( ( ))k k kij x ij y iju u    
2 2
( ) ( ) ( ) ( )
( ( )) ( ( ))
k k
x ij x ij y ij y ij
k k
x ij y ij
u v u v
u u
   
  
, 
1, 1,
( )
2
k k
i j i jk
x ij
u u
u
x
 
 

, 
, 1 , 1
( )
2
k k
i j i jk
y ij
u u
u
y
 
 

, 
1, 1,
( )
2
k k
i j i jk
x ij
v v
v v
x
 
 

, 
, 1 , 1
( )
2
k k
i j i jk
y ij
v v
v
y
 
 

, 
1 1 2 20 1 1, , 0 1 , 1 ,
; ; ; ;k k k k k k k kj j N j N j i i i N i Nu u u u u u u u      
1 1 2 20 1 1, 0 1 , 1 ,
; ; ; ;j j N j N j i i i N i Nv v v v v v v v      
1 21,..., ; 1,..., ;i N j N  0,1,..., ; 1k K x y     . 
4.3 Optimal parameter σ 
The parameter   is calculated at the first step of the 
iteration process. We use the method of Immerker [6]: 
1 2
1 11 2
/ 2
| * |
6( 2)( 2)
N N
ij
i j
u
N N  
 
 


 , 
with the mask 
1 2 1
2 4 2
1 2 1

 

 
   
 
 for convolution operator *,  
1, 1 33 , 1 32 1, 1 31 1, 23*ij i j i j i j i ju u u u u                
22 1, 21 1, 1 13 , 1 12 1, 1 11ij i j i j i j i ju u u u u              , 
1 21,..., ; 1,..., ;i N j N   
0iju  , if 0i  , or 0j  , or 1 1i N  , or 2 1j N  . 
4.4 Initial solution 
In the iteration procedure (6), the result depends on ini-
tial parameters 0 0 01 2, ,   . If 
0 0 0
1 2, ,    are given first, 
then its unsuitable values define not so good solution iju  
and later, not so good evaluation of a probability distri-
bution parameters. If 
0 0 0
1 2, ,    are randomized, the re-
sult is unacceptable too, because of the additional noise 
added in the image. 
Of course, initial values of 
0 0 0
1 2, ,    need to be 
closed to required values. We evaluate
0 0 0
1 2, ,    as av-
erage values of neighbour pixels of the image, for exam-
ple, by the method of Immerker. 
5 Image quality evaluation 
In order to evaluate the image quality after denoising, we 
use criteria PSNR, MSE and SSIM [24, 25]: 
1 2
2
MSE
1 11 2
1
Q ( )
N N
ij ij
i j
v u
N N  
  ,
2
PSNR
MSE
Q 10lg
Q
L 
  
 
, 
1 2
SSIM 2 2 2 2
1 2
(2 )(2 )
Q
( )( )
uv
u v
u v C C
u v C C

 
 

   
, 
where 
1 2
1 11 2
1
N N
ij
i j
u u
N N  
  , 
1 2
1 11 2
1
N N
ij
i j
v v
N N  
  . 
1 2
2 2
1 11 2
1
( )
1
N N
u ij
i j
u u
N N  
 

 , 
1 2
2 2
1 11 2
1
( )
1
N N
v ij
i j
v v
N N  
 

 , 
1 2
1 11 2
1
( )( )
1
N N
uv ij ij
i j
u u v v
N N  
  

 , 
2 2
1 1 2 2 1 2( ) , ( ) ; 1; 1C K L C K L K K    . 
For example, 61 2 10K K
  , L  is an image intensi-
ty with 255L   for 8-bits greyscale image. 
The greater the value of 
PSNRQ , the better the image 
quality. If the value of 
PSNRQ  belongs to the interval 
from 20 to 25, then the image quality is acceptable, for 
example, for wireless transmission [21]. 
The MSEQ  is a mean squared error and is used to 
evaluate the difference between two images. The lower 
the value of MSEQ , the better the result of restoration. 
The value of MSEQ  directly related to the value of PSNRQ . 
The value of SSIMQ  is used to evaluate an image 
quality by comparing the similarity of two images. This 
value is between -1 and 1. The greater the value of 
SSIMQ , the better the image quality. 
6 Experiments and discussion 
In this paper, we consider cases as in [19] and additional-
ly the superposition of noises. The image size is changed 
from 300x300 pixels to 256x256 pixels specified in 
PURE-LET method [11]. We process the artificial image 
with artificial noise and the real image with artificial 
noise. The artificial image is noise-free and we need to 
add noise with high intensity (the image to be very noisy) 
to reduce its quality. Therefore, we specify 0.6 for pro-
portion of Gaussian noisy image and 0.4 for proportion 
of Poisson noisy image. The real image (captured by a 
digital device) already includes some noise. We specify 
0.5 for proportion of Gaussian noisy image and 0.5 for 
proportion of Poisson noisy image too.  
We need to point the attention in the case of Gaussi-
an noise our method sometimes can be better than ROF, 
because the method to evaluate the variance of Gaussian 
noise can be better than one included in the original ROF 
model in many cases. In the case of superposition of 
noises, our method sometimes can be better than PURE-
LET, because parameters of our method are usually more 
optimal than in original model too. 
6.1 Artificial image with artificial noise 
We use artificial image with artificial mixed noise for the 
first test. The image includes eight bars (Fig. 1a). Other 
images (Fig. 1b-j) show noisy and denoised images and 
zoomed out part of them. 
Artificial noise is generated by linear combination, 
and by superposition of Poisson and Gaussian noises.  
For both cases, we consider Poisson noise first. Its 
probability density is 2 ( | )p v u , and variance is 2 i ju  
A Mixed Noise Removal Method Based on Total Variation Informatica 40 (2016) 159–167 163 
at every pixel ( , )i j , 11,...,i N , 21,...,j N . Poisson 
noise variance is an average value 2 11.7939 . If the 
grey value of a pixel after adding of Poisson noise is out 
of the interval from 0 to 255, it needs to be reset to 
(2)
i j i jv u . For this image, there are no pixels out of this 
interval. Next, we consider the variance of Gaussian 
noise is four times greater than the variance of Poisson 
noise 21 4 47.1757  . 
For the linear combination of noises, we denote the 
intensity function of Gaussian noisy image as (1)v . As 
above, values of (1)v  need to be between 0 and 255. If the 
grey value of a pixel after adding of Gaussian noise is out 
of the interval from 0 to 255, it needs to be reset to 
(1)
i j i jv u . In this case, there are 1075 pixels out of this 
  
a) b) 
  
c) d) 
  
e) f) 
  
g) h) 
  
i) j) 
Figure 1: Denoising of the artificial image: a)-b) 
original image, c)-d) noisy image with linear combi-
nation of noises, e)-f) denoised image (c), g)-h) noisy 
image with superposition of noises, i)-j) denoised 
image (g). 
  
a) b) 
  
c) d) 
  
e) f) 
  
g) h) 
  
i) j) 
Figure 2: Denoising of the real image: a)-b) original 
image, c)-d) noisy image with linear combination of 
noises, e)-f) denoised image (c), g)-h) noisy image 
with superposition of noises, i)-j) denoised image (g). 
 
164 Informatica 40 (2016) 159–167 N.H.T. Dang et al.  
 
interval (1.64%).  
The final noisy image (linear combination of noises 
in Fig. 1c) is created with proportion 0.6 for Gaussian 
noisy image (1)v  and proportion 0.4 for Poisson noisy 
image (2)v : (1) (2)0.6 0.4v v v  .  
Then we define proportion for linear combination as 
   1 2/ 0.6 47.1757 / 0.4 11.7939 6 /1     . Coeffi-
cients of linear combination are defined as 
1 = 6/7 = 0.8571, 2 = 1/7 = 0.1429.  
Values of QPSNR, QMSE, and QSSIM of the noisy im-
age (linear combination of noises) are, respectively, 
19.4291, 741.5963, and 0.1073. 
In the case of the image with superposition of noises, 
we add Gaussian noise over Poisson noisy image. The 
intensity function of this Gaussian noisy image is (1)v  
too. As above, the grey values of (1)v  need to be between 
0 and 255. If the grey value of a pixel after adding of 
Gaussian noise is out of the interval from 0 to 255, it 
needs to be reset to 
(1) (2)
i j i jv v .  
There are 1220 pixels out of this interval (1.86%). 
The noisy image (superposition of noises, Fig. 1g) is also 
Gaussian noisy image (1)v v . In this case, we don’t 
know 1 and 2, therefore we use the algorithm with au-
tomatically defined parameters. 
Values of QPSNR, QMSE, and QSSIM of the noisy im-
age are, respectively, 14.9211, 2093.9827, and 0.0439. 
Tables 1 – 4 show results for linear combination of 
noises, Gaussian noise, Poisson noise, and superposition 
of noises for the artificial image. 
6.2 Real image with artificial noise 
The artificial noise is generated by linear combination 
and superposition of Poisson and Gaussian noises. 
For both cases, we consider Poisson noise first. Pois-
son noise variance is an average value 2 9.0882 . If 
the grey value of a pixel after adding of Poisson noise is 
out of the interval from 0 to 255, it needs to be reset to 
 QPSNR QSSIM QMSE 
Noisy 19.4291 0.1073 741.5963 
ROF 34.1236 0.8978 25.1606 
Modified ROF 32.4315 0.8703 37.8791 
PURE-LET  33.0309 0.9277 32.3587 
Proposed method  
1=0.8571, 
2=0.1429, 
 = 0.5003, 
 = 47.1757 
41.1209 0.9841 4.9905 
Proposed method 
with automatically 
defined parameters 
1=0.8414, 
2=0.1586, 
 = 0.5112, 
 = 41.0314 
41.0998 0.9840 5.0478 
Table 1: Quality of noise removing for the artificial 
image with linear combination of noises. 
 QPSNR QSSIM QMSE 
Noisy 15.1406 0.0457 1990.8 
ROF 31.4797 0.8364 21.2502 
Modified ROF 28.4591 0.7871 27.5694 
PURE-LET  28.9451 0.7986 25.9883 
Proposed method  
1=1, 2=0, 
 = 0.3033, 
 = 47.1757 
35.8011 0.9598 16.8122 
Proposed method 
with automatically 
defined parameters 
1=0.9715, 
2=0.0285, 
 = 0.3021, 
 = 46.0314 
35.7589 0.9596 17.2658 
Table 2: Quality of noise removing for the artificial 
image with Gaussian noise. 
 QPSNR QSSIM QMSE 
Noisy 26.6776 0.3640 139.7396 
ROF 36.4958 0.9381 14.5715 
Modified ROF 44.6347 0.9897 2.2001 
PURE-LET  37.4485 0.9404 10.5692 
Proposed method  
1=0,  
2=1, 
 = 0.8012, 
 = 0.0001 
44.6343 0.9897 2.2014 
Proposed method 
with automatically 
defined parameters 
1=0.0524, 
2=0.9476, 
 = 0.7923, 
 = 2.0544 
44.6156 0.9896 2.2466 
Table 3: Quality of noise removing for the artificial 
image with Poisson noise. 
 QPSNR QSSIM QMSE 
Noisy 14.9211 0.0439 2093.983 
ROF 31.2913 0.8346 48.3008 
Modified ROF 30.5471 0.8232 56.5601 
PURE-LET  33.9889 0.9298 25.9534 
Proposed method 
with automatically 
defined parameters 
1=0.8014, 
2=0.1986, 
 = 0.4812, 
 = 40.0314 
37.3366 0.9677 12.0066 
Table 4: Quality of noise removing for the artificial 
image with superposition of noises. 
A Mixed Noise Removal Method Based on Total Variation Informatica 40 (2016) 159–167 165 
(2)
i j i jv u . Here there are no pixels out of this interval.  
For Gaussian noise, we consider the variance of 
Gaussian noise is four times greater than the variance of 
Poisson noise 21 4 36.3529  . The real image is a 
human skull [14] (Fig. 2a). Others (Fig. 2b-j) show noisy 
and denoised images and zoomed out part of them.  
For the case of linear combination of noises, we de-
note the intensity function of Gaussian noisy image as 
(1)v . As above, the grey values of intensity function (1)v  
also need to be between 0 and 255. If the grey value of a 
pixel after adding of Gaussian noise is out of the interval 
from 0 to 255, it needs to be reset to 
(1)
i j i jv u . In this 
case, there are 5355 pixels out of this interval (8.17%). 
The final image (linear combination of noises, Fig. 2c) is 
created with proportion 0.5 for Gaussian noisy image (1)v  
and proportion 0.5 for Poisson noisy image (2)v : 
(1) (2)0.5 0.5v v v  . The proportion for linear combina-
tion is: 1 2 (0.5 36.3529) / (0.5 9.0882 4 1/ ) /     . 
Hence, coefficients of linear combination are defined 
as 1 = 4/5 =0.8, 2 = 1/5 = 0.2. Values of QPSNR, QMSE, 
and QSSIM of final noisy image are, respectively, 23.6878, 
278.1619, and 0.5390. 
For superposition of noises, we add Gaussian noise 
over Poisson noisy image. We denote the intensity func-
tion of Gaussian noisy image as (1)v . As above, grey val-
ues of (1)v  need to be between 0 and 255. If the grey val-
ue after adding of Gaussian noise is out of the interval 
from 0 to 255, it needs to be reset to 
(1) (2)
i j i jv v . In this 
case, there are 5621 pixels out of this interval (8.58%). 
The final noisy image (superposition of noises, Fig. 2g) 
is also the Gaussian noisy image (1)v v .  
In this case, we don’t know 1 and 2, therefore we 
use the algorithm to find them. Values of QPSNR, QMSE, 
and QSSIM of the final noisy image (superposition) are, 
respectively, 17.8071, 1077.3831, and 0.3242. 
Tables 5 – 8 show results for linear combination of 
noises, Gaussian noise, Poisson noise, and superposition 
of noises for the real image. 
 QPSNR QSSIM QMSE 
Noisy 23.6878 0.5390 278.1619 
ROF 27.3974 0.8295 118.3975 
Modified ROF 25.5644 0.7513 197.5403 
PURE-LET  25.7781 0.8105 191.0341 
Proposed method  
1=0.8, 2=0.2, 
 = 0.0524, 
 = 36.3529 
27.6641 0.8331 110.9451 
Proposed method 
with automatically 
defined parameters 
1=0.7804, 
2=0.2196, 
 = 0.0512, 
 = 34.2311 
27.6039 0.8325 112.8984 
Table 5: Quality of noise removing for the real im-
age with linear combination of noises. 
 QPSNR QSSIM QMSE 
Noisy 18.0693 0.3337 1014.3 
ROF 24.0246 0.7299 257.4095 
Modified ROF 23.2511 0.7019 311.8742 
PURE-LET  23.8712 0.7989 265.6153 
Proposed method  
1=1, 2=0, 
 = 0.0956, 
 = 36.3529 
24.2011 0.8029 242.5101 
Proposed method 
with automatically 
defined parameters 
1=0.9538, 
2=0.0462, 
 = 0.0902, 
 = 35.0633 
24.1882 0.8028 247.8894 
Table 6: Quality of noise removing for the real im-
age with Gaussian noise. 
 QPSNR QSSIM QMSE 
Noisy 28.4991 0.7625 91.8683 
ROF 31.0567 0.9457 50.9818 
Modified ROF 31.1992 0.9022 48.9375 
PURE-LET  30.8955 0.8678 53.1066 
Proposed method  
1=0, 2=1, 
 = 0.0541, 
 = 0.0001 
31.1334 0.8986 49.7922 
Proposed method 
with automatically 
defined parameters 
1=0.0491, 
2=0.9509, 
 = 0.0567, 
 = 4.2012 
31.1316 0.8986 50.1094 
Table 7: Quality of noise removing for the real image 
with Poisson noise. 
 QPSNR QSSIM QMSE 
Noisy 17.8077 0.3242 1077.383 
ROF 23.1936 0.7062 311.6856 
Modified ROF 23.0413 0.7033 319.3831 
PURE-LET  23.6278 0.7072 282.0349 
Proposed method 
with automatically 
defined parameters 
1=0.7704, 
2=0.2296, 
 = 0.1102, 
 = 36.3412 
23.7292 0.7094 275.5229 
Table 8: Quality of noise removing for the real image 
with superposition of noises. 
166 Informatica 40 (2016) 159–167 N.H.T. Dang et al.  
 
6.3 About of initial solution 
In order to create the initial image, we use the convolu-
tion operator. The table 9 shows the dependency of re-
stored result for the initial image, where:  
(a) Initial parameters 0 01 20, 1, 1     ;  
(b) Initial parameters 0 01 2 0.5, 1     ;    
(c) Initial solution 0u  is given as a randomized matrix; 
(d) Initial solution 0u v   is given as an average val-
ue of neighbour pixels by the convolution operator 
with the mask  1/ 9   of the size 3x3. 
Table 9 shows the best result of denoising is (d) by 
criteria PSNR and MSE.  
The result (c) by SSIM looks different in contract to 
ones in Tables 1-8. It illustrates incorrectness of a ran-
domized initial solution (accidental and not stable, if a 
probability distribution is unknown).  
Next, we have to notice that the non-optimal result 
(a) has been used in experiments for Table 5. It appears 
to be enough for the good result with automatically de-
fined model parameters. 
 (a) (b) (c) (d) 
1 0.7804 0.8094 0.8733 0.8032 
2 0.2196 0.1906 0.1267 0.1968 
 0.0512 0.0573 0.0653 0.0565 
 34.2311 
QPSNR 27.6039 27.2214 26.5611 27.6523 
QMSE 112.8984 120.4355 132.0264 107.5431 
QSSIM 0.8325 0.8317 0.8395 0.8392 
Table 9: Dependency of denoising on initial solution. 
At last, the variant (b) initially looks better than (a) 
for kind of better assumption of 
0 0
1 2 0.5    to process 
the real image. Nevertheless, our assumption about 1  
is very far from the good one, and evidently the limit of 
the number of steps 500K   is insufficient in this case.  
As a result, the variant (d) is the best idea for initial 
solution. 
7 Conclusion 
In this paper, we proposed a novel method that can effec-
tively remove the mixed Poisson-Gaussian noise. Fur-
thermore, our proposed method can be also used to re-
move Gaussian or Poisson noise separately. This method 
is based on the variational approach. 
The denoising result strongly depends on values of 
coefficients of linear combination 1  and 2 . These val-
ues can be set manually or can be defined automatically. 
When processing real images, we can use the proposed 
method with automatically defined parameters. 
Although our method concentrates on removing the 
linear combination of noise, but it also efficiently re-
moves the superposition of noises. In this case, we con-
sider the superposition of noises is equivalent to some 
linear combination of them with coefficients found in 
iteration process. 
In this paper we show that our simple model “feels” 
well the wide range of proportion of two types of noises. 
As a result, it appears to be the good basis for removing 
superposition of such noises.   
It is evident, the iteration process (6) used here is in-
sufficiently effective in comparing with other possible 
computational schemes. In this paper, we try to compare 
our approach to image denoising with PURE-LET meth-
od only in possible reduction of our model complexity, 
not in others.  
We would like to express our great thanks to devel-
opers of PURE-LET method for kindly granted us the 
original executable module of it.  
8 Acknowledgements 
This work is partially supported by Russian Foundation 
for Basic Research, under grants 13-07-00529, 14-07-
00964, 15-07-02228, 16-07-01039, and by Ministry of 
Education and Training, Vietnam, under grant number 
B2015–01–90. 
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