A Variational Method to Remove the Combination of Poisson and
Gaussian Noises
D. N. H. Thanh and S. D. Dvoenko
Tula State University, Lenin Ave. 92, Tula, Russian Federation
Keywords: Total Variation, ROF Model, Gaussian Noise, Poisson Noise, Mixed Poisson-Gaussian Noise, Image
Processing, Euler-Lagrange Equation.
Abstract: In this paper, we propose a method to remove noise in digital images. Our method is based on the well-
known variational approach. The novelty of proposed method consists in removing of mixed Poisson-
Gaussian noise. This is the actual problem for many types of real raster images, for example, biomedical
images. Our method is developed with the goal to combine two famous models: ROF for removing Gaussi-
an noise and modified ROF for removing Poisson noise. As a result, our proposed method can be also ap-
plied to remove Gaussian or Poisson noise separately. We develop procedure to perform noise removal with
automatically evaluated parameters to get the best result of denoising.
1 INTRODUCTION
Digital image is a type of a signal that is obtained
from a real analogous signal by discretization and
quantization. Many digital devices can create digital
images, such as digital camera, X-ray scanner, and
so on. In practice, these devices can give unexpected
effects. One of them is noise. Noise reduces image
quality and efficiency of image processing.
The problem of noise removal from digital imag-
es is very actual today. In order to solve this prob-
lem, many different strong approaches were already
developed.
The variational approach (Chan, 2005, Burger,
2008, Chambolle, 2009, Xu, 2014, Rankovic, 2012,
Lysaker, 2006, Li, 2006, Zhu, 2012, Tran, 2012,
Getreuer, 2012, Caselles, 2011, Rudin, 1992, Chen,
2013) is well-known and very promising.
This concept was pioneered by Rudin (1992). He
proposed the total variation to solve many problems
in image processing. Especially, he built a model for
denoising of digital images. This model is referred to
by ROF (Rudin, 1992, Chen, 2013).
It is known, ROF model is used to remove only
Gaussian noise. However, another important type
like Poisson noise is usually presented in digital
images. For example, this noise appears in medical
X-ray images. In order to remove this noise, Le T.
(2007) developed so called modified ROF model.
Gaussian and Poisson noises are popular sepa-
rately, but their combination is also important
(Luisier, 2011). This combination of noises usually
appears in biomedical images, for example, in elec-
tronic microscope images (Jezierska, 2011, 2012).
Nevertheless, ROF and modified ROF models
ineffectively treat this combination. ROF model
gives priority to Gaussian noise, but modified ROF
model gives it to Poisson noise.
Our goal is to combine ROF model (for Gaussian
noise) and modified ROF model (for Poisson noise)
to create new model that can treat this combination
effectively. Our model will treat this combination
with considering proportion of noise between them.
In experiments, we used initial images and added
noise into them. We performed denoising of digital
images by proposed method and other methods, such
as ROF model, median filter (Wang 2012) and Wie-
ner filter (Abe 2012). In order to evaluate an image
quality after denoising, we used well-known criteria
MSE (Mean Square Error), PSNR (Peak Signal-to-
Noise Ratio) and SSIM (Structure SIMilarity)
(Wang, 2004, 2006). We give priority to PSNR,
because it is most popular and used to evaluate the
quality of restored signal in signal processing in
general, and in image processing, especially.
38
Thanh D. and Dvoenko S..
A Variational Method to Remove the Combination of Poisson and Gaussian Noises.
DOI: 10.5220/0005460900380045
In Proceedings of the 5th International Workshop on Image Mining. Theory and Applications (IMTA-5-2015), pages 38-45
ISBN: 978-989-758-094-9
Copyright
c
2015 SCITEPRESS (Science and Technology Publications, Lda.)
2 DENOISING MODEL FOR
MIXED POISSON-GAUSSIAN
NOISE
Let in
2
R space a bounded domain
2
R be
given. Let us call functions
2
(, ) Ruxy and
2
(, ) Rvxy , respectively, ideal (without noise) and
observed images (noisy), where
(, )xy
.
If the function
u is smooth, then its total varia-
tion is defined by
[] | |
T
Vu udxdy
,
where
(, )
x
y
uuu is a gradient (nabla operator),
/
x
uux
, /
y
uuy ,
22
||
x
y
uuu
. In this
paper, we only consider function
u that always has
limited total variation
[]
T
Vu
.
2.1 Denoising Model
According to results (Chang, 2005, Burger, 2008,
Rudin, 1992, Chen, 2013, Scherzer, 2009), image
smoothness is characterised by the total variation.
The total variation of noisy image is always greater
than the total variation of smoothed image.
When Rudin solved the problem
[] min
T
Vu
,
he used this characteristic and assumed, that Gaussi-
an noise variance is fixed by the additional con-
straint
2
()u v dxdy const
He proposed the ROF model to remove Gaussian
noise from an image
*2
arg min | | ( )
2
u
u u dxdy u v dxdy
,
where
0
is a Lagrange multiplier.
Le T. (2007) proposed another model to remove
Poisson noise based on ROF model:
*
arg min | | ( ln( ))
u
uudxdyuvudxdy
,
where
is a regularization coefficient. We call it a
modified ROF model for Poisson noise.
In order to develop the denoising model for
mixed noise, we also solve the problem based on the
smooth characteristic of the total variation
[] min
T
Vu
.
And we also define a constrained condition. We
assume that with given image
u , the mixed noise in
image is fixed too (Poisson noise is unchangeable,
and Gaussian noise only depends on noise variance):
ln( ( | ))
p
v u dxdy const
,
(1)
where
(|)pv u
is a conditional probability.
Let us consider Gaussian noise. Its probability
density function (
p.d.f.) is
2
1
2
()
(|) exp /( 2)
2
vu
pvu
.
For Poisson noise the p.d.f. is
2
exp( )
(|)
!
v
uu
pvu
v
.
We have to note that intensity levels of image col-
ours are integer (for example, the intensity interval
for an 8-bit grayscale image is from 0 to 255), so we
regard
u
as an integer value, but this will ultimately
be unnecessary (Le 2007).
In order to treat combination of Gaussian and
Poisson noises, we assume the following linear
combination
11 2 2
ln((|)) ln( (|)) ln( (|))pv u p v u p v u
,
where
1
0
,
2
0
,
12
1
.
According to (1), we obtain the denoising prob-
lem with constrained condition as following:
*
2
1
2
2
arg min | |
() (ln()) ,
2
u
uudxdy
vu uv u dxdy
where
is a constant value.
We can transform this constrained optimization
problem to the unconstrained optimization problem
by using Lagrange functional
2
1
2
(,) | | ( )
2
L u u dxdy v u dxdy
2
(ln())uv udxdy
to find
**
,
(,) argmin(,)
u
uLu
,
(2)
AVariationalMethodtoRemovetheCombinationofPoissonandGaussianNoises
39
where 0
is a Lagrange multiplier.
This is our proposed model to remove mixed
Poisson-Gaussian noise from digital image. We have
to notice that, if
1
0
and
2
, we obtain
modified ROF model for removing Poisson noise. If
2
0
and
2
1
/(2 )
, then we obtain ROF
model for removing Gaussian noise. In the case of
12
0, 0
we get the model for removing mixed
Poisson-Gaussian noise.
2.2 Model Discretization
In order to solve the problem (2), we can use the
Lagrange multipliers method (Zeidler, 1985,
Rubinov, 2003, Gill, 1974).
However, in this paper, we will solve it by using
the Euler-Lagrange equation (Zeidler, 1985).
Let function
(, )
f
xy
be defined in limited do-
main
2
R
and be the second-order continuous
differentiable one by
x
and y for
(, )xy
.
We consider the special convex functional
(, , , , )
x
y
F
xy f f f , where
/
x
f
fx
, /
y
f
fy .
The solution of the optimization problem
(, , , , ) min
xy
Fxyf f f dxdy
satisfies the following Euler-Lagrange equation
(, , , , ) (, , , , )
x
fxyf xy
F xyf f f F xyf f f
x
(, , , , ) 0
y
fxy
Fxyfff
y
,
where
/
f
F
Ff ,
/
x
f
x
F
Ff
,
/
y
f
y
F
Ff
.
We use the result above to solve the problem (2).
The solution of the problem (2) is given by the fol-
lowing Euler-Lagrange equation:
1
2
2
()(1)
v
vu
u
22 22
0,
y
x
xy xy
u
u
xy
uu uu
(3)
where
1/
. We can reduce (3) to
1
2
2
()(1)
v
vu
u
22
223/2
2
0
()
xx y x y xy x yy
xy
uu uuu uu
uu
,
(4)
where
2
2
xx
u
u
x
,
2
2
yy
u
u
y
,
x
yyx
uu
uu
xy yx
.
In order to discretize the equation (4), we add an
artificial time parameter and consider the function
(, ,)uuxyt
. Then the equation (4) relates to the
following diffusion equation
1
2
2
()(1)
t
v
uvu
u
22
223/2
2
()
x
xy xyxy xyy
xy
uu uuu uu
uu
,
(5)
where
/
t
uut
.
We can write the discretized form of the equation
(5) as following:
1
1
2
()
kk k
ij ij ij ij
uu vu
2
(1 )
ij
k
ij
k
ij
v
u
,
(6)
where
2
223/2
()( ())
(( ()) ( ()))
kk
xx ij y ij
k
ij
kk
xij yij
uu
uu
2
223/2
2()() ()(()) ()
(( ( )) ( ( )) )
kk k k k
x
ij yij xyij xij yyij
kk
xij yij
uu u u u
uu
,
1, 1,
()
2
kk
ij ij
k
xij
uu
u
x
,
,1 ,1
()
2
kk
ij ij
k
yij
uu
u
y
,
1, 1,
2
2
()
()
kkk
ij ij ij
k
xx ij
uuu
u
x
,
,1 ,1
2
2
()
()
kkk
ij ij ij
k
yy ij
uuu
u
y
,
1, 1 1, 1 1, 1 1, 1
()
4
kkkk
ij ij ij ij
k
xy ij
uuuu
u
xy
,
11 22
01 1, ,01,1,
;;;;
kkk k kkk k
j j N j N j i i iN iN
uuu u uuu u
IMTA-52015-5thInternationalWorkshoponImageMining.TheoryandApplications
40
12
1,..., ; 1,..., ;iNjN
0,1,..., ; 1; 0 1kKxy
.
Here
K
is enough great number. In this paper, we
use
500K . For initial condition, we have
0
12
; 1,..., ; 1,...,
ij ij
uvi Nj N
.
2.3 Finding Optimal Parameters
We can use the procedure (6) to perform image
denoising. In this procedure, values of parameters
12
,,,
need to be given. In some cases, we
have to define these parameters to perform image
denoising automatically. Then parameters
12
,,
in process (6) need to be changed into
12
,,
kkk
for
each step
k
. So we obtain new procedure that al-
lows us to calculate values of these parameters au-
tomatically in iteration steps.
2.3.1 Optimal Parameters
1
and
2
Let
(,)u
be a solution of the problem (2). Then we
get the condition
(,)
0
Lu
u
.
This condition gives us the optimal parameters
12
,
:
1
2
(1 )
1
() (1)
v
dxdy
u
v
v u dxdy dxdy
u
,
21
1
.
Its discretized form is
12
12
11
1
2
11
(1 )
(1)
NN
ij
k
ij
ij
k
k
NN
ij ij ij
k
ij
ij
v
u
vu v
u
,
21
1
kk
,
where
0,1,...,kK
.
2.3.2 Optimal Parameter
In order to find an optimal parameter
, we multiply
(3) by
()uv
and integrate by parts over
. Final-
ly, we obtain the formula to find the optimal pa-
rameter
:
2
2
1
2
2
22
22
()
(() )
()
xx yy
xy
xy
uv
u v dxdy
u
uv uv
u u dxdy
uu
.
Its discretized form is
12
12
2
2
1
2
2
11
11
()
(( ) )
k
NN
k
ij ij
kk
ij ij
k
ij
ij
k
NN
k
ij
ij
uv
uv
u
,
where
22
( ( )) ( ( ))
kk k
ij x ij y ij
uu
22
() () () ()
(()) (())
kk
x
ij x ij y ij y ij
kk
xij yij
uv uv
uu
,
1, 1,
()
2
kk
ij ij
k
xij
uu
u
x
,
,1 ,1
()
2
kk
ij ij
k
yij
uu
u
y
,
1, 1,
()
2
kk
ij ij
k
xij
vv
vv
x
,
,1 ,1
()
2
kk
ij ij
k
yij
vv
v
y
,
11 22
01 1, ,01,1,
;;;;
kkk k kkk k
j j N j N j i i iN iN
uuu u uuu u
11 22
01 1, 01,1,
;;; ;
jjNjNjiiiN iN
vvv vvvv v
12
1,..., ; 1,..., ;iNjN
0,1,..., ; 1kKxy
.
2.3.3 Optimal Parameter
In order to evaluate this parameter
, we use the
result of Immerker (1996):
12
11
12
/2
|*|
6( 2)( 2)
NN
ij
ij
u
NN
, where
121
24 2
121
is the mask of an image.
Operator * is a convolution operator, where
1,133 ,132 1,131 1, 23
*
ij i j i j i j i j
uu u u u
22 1, 21 1, 1 13 , 1 12 1, 1 11ij ij ij ij ij
uu u u u
,
12
1,..., ; 1,..., ;iNjN
AVariationalMethodtoRemovetheCombinationofPoissonandGaussianNoises
41
0
ij
u
, if
0i , or 0j , or
1
1iN
,
or
2
1jN
.
We have to notice, that the parameter
is just
evaluated at first time of the iteration process.
2.4 Image Quality Evaluation
In order to evaluate image quality after denoising,
we use criteria MSE (Mean Square Error), PSNR
(Peak Signal-to-Noise Ratio) and SSIM (Structure
SIMilarity) (Wang 2004, 2006):
12
2
11
12
1
()
NN
MSE ij ij
ij
Quv
NN
,
12
2
12
2
11
10lg
()
PSNR
NN
ij ij
ij
NNL
Q
uv
,
12
22 2 2
12
(2 )(2 )
()( )
uv
SSIM
uv
uv C C
Q
uvC C
,
where
12
11
12
1
NN
ij
ij
uu
NN
,
12
11
12
1
NN
ij
ij
vv
NN
.
12
22
11
12
1
()
1
NN
uij
ij
uu
NN
,
12
22
11
12
1
()
1
NN
vij
ij
vv
NN
,
12
11
12
1
()()
1
NN
uv ij ij
ij
uuvv
NN
,
22
11 2 2 1 2
(), ( ); 1; 1CKLCKLK K .
For example,
6
12
10KK
,
L is image intensity,
where, for example,
8
21255L
for an 8-bit
greyscale images.
The greater value
P
SNR
Q
, the better image quali-
ty. If
P
SNR
Q
is between 20 and 25, then an image
quality is acceptable, for example, for the wireless
transmission (Thomos 2006).
SSIM
Q
is used to evaluate image quality by com-
paring similarity of both images. Its value is between
-1 and 1. The greater value
SSIM
Q
, the better image
quality.
M
SE
Q
is a criteria to evaluate the difference be-
tween two images.
M
SE
Q
is mean-squared error. The
lower value
M
SE
Q
, the better restoration result. The
value of
M
SE
Q
also relates to the value of
P
SNR
Q
.
2.5 Image Sample Initialization
In experiment with artificial image, we use an image
with the size of 300x300 pixels. This image has six
vertical bars (Fig. 1a). Bar grey level intensities are
110, 130 and 160, respectively, where numbers of
pixels are same (30.000). We zoom, crop and show
the part of the original image under processing (Fig.
1b – 1f).
First, we create the noisy image by adding
Gaussian noise (Fig. 1c) and second, create noisy
image by adding Poisson noise (Fig. 1d).
We want to generate a noisy image, the quality
of which is very low, but we cannot control the Pois-
son noise intensity. So, we just only control the
variance of Gaussian noise. In order to calculate
proportion between intensities of Gaussian and Pois-
son noises, we calculate the variance of Poisson
noise. The value of variance of Gaussian noise is
calculated via Poisson noise variance. Let the vari-
ance of Gaussian noise be four times greater than the
variance of Poisson noise.
First, let us consider Poisson noise. Its distribu-
tion is
2
(|)pvu
, value of the variance of Poisson
noise is
2 ij
u
, respectively, with
ij
u
at every
pixel
(, )ij
of image, where
12
1,..., ; 1,...,iNjN
.
We denote this Poisson noisy image as
(2)
v
. Obvi-
ously, intensity value of
(2)
v ought to be between 0
and 255. If the intensity value of some pixels are out
of this interval, they need to be reset to intensity
value of respective pixel of the original image
u ,
that means
(2)
ij ij
vu
.
In this case, number of them is zero. The vari-
ance of Poisson noise can be calculated as average
value
2
( 110 130 160) 11.5130
, beca-
use this image has three intensity levels and their
numbers of pixels are identical.
Now, we consider Gaussian noise. Its variance
need to be 46.052 (because we explained above,
variance of Gaussian noise is four times over vari-
ance of Poisson noise). We denote this Gaussian
noisy image as
(1)
v . As above case, intensity value
of
(1)
v
also need to be between 0 and 255. In this
case, there are 1063 pixels out of this interval, re-
spectively 1.2% of all image pixels.
We create resulting noisy image (Fig. 1e) by
IMTA-52015-5thInternationalWorkshoponImageMining.TheoryandApplications
42
combining first noisy and second noisy images with
proportion 0.6 for Gaussian noisy image
(1)
v and 0.4
for Poisson noisy image
(2)
v
.
This means
(1) (2)
0.6 0.4vv v. Hence:
12
/
46.052 0.6 / 11.513 0.4
27.63 / 4.6 6 / 1
.
As a result:
1
6 / 7 0.8571
,
2
1/7 0.1429
.
Values of
Q
MSE
, Q
PSNR
and Q
SSIM
of noisy image
are respectively 718.8782, 19.5643, and 0.1036. The
value of
Q
PSNR
of noisy image is lower than 20. That
means quality of noisy image is very low and it
cannot be used, for example, for wireless transmis-
sion.
2.6 Experiments
In order to test our model, we consider the following
cases for above sample image: parameters are
1
=0.2,
2
=0.8 (worse restoration);
1
=
2
=0.5 (better
restoration);
1
=0.8571,
2
=0.1429 (best restoration)
and with automatically evaluated
1
=0.8102,
2
=0.1898. Result of denoising with
1
=0.8571,
2
=0.1429 is given on Fig. 1f.
a) b)
c) d)
e) f)
Figure 1: Result of noise initialization and denoising:
a) original image, b) cropped image, c) with Gaussian
noise, d) with Poisson noise, e) with mixed noise, f) after
denoising.
We also compare the result of our model with
other noise removal methods: ROF, Wiener filter,
median filter. Results are given in Table 1.
Table 1 shows, the proposed method with auto-
matically evaluated parameters effectively removes
noise and gives us the high quality images to use
them, for example, for wireless transmission.
Fig. 2 shows vertical cut of grey level intensities
of original image, noisy image and denoised image.
Figure 2: Intensity of original, denoised and noisy images.
Table 1: Quality comparison of noise removal methods for
the artificial image.
Q
PSNR
Q
SSIM
Q
MSE
Noisy 19.5643 0.1036 718.8782
ROF 35.1284 0.9130 19.9635
Median 31.4844 0.7797 46.1996
Wiener 30.1502 0.6018 62.8146
Proposed method
with
1
=0.2,
2
=0.8,
= 0.1140,
= 46.0520
29.1325 0.5933 79.4014
Proposed method
with
1
=
2
=0.5,
= 0.1429,
= 46.0520
37.0462 0.9453 12.8370
Proposed method
with
1
=0.8571,
2
=0.1429,
= 0.4738,
= 46.0520
42.8237 0.9902 3.3940
Proposed method
with evaluated
parameters
1
=0.8102,
2
=0.1898,
= 0.3846,
= 45.4523
42.7795 0.9900 3.4287
AVariationalMethodtoRemovetheCombinationofPoissonandGaussianNoises
43
We also use another example to test our model in
the case of processing a real image. In this case, we
use an image of human skull (Nick 2009) with the
size 300x300 pixel (Fig. 3a). We add Gaussian noise
(Fig. 3b), Poisson noise (Fig. 3c) and combine two
these images to make final noisy image (Fig. 3d).
Variance of Gaussian noise is 4 times over vari-
ance of Poisson noise and proportion of combination
is 0.5 and 0.5, and
1
=0.75,
2
=0.25. Result of de-
noising is shown in Fig. 3e, Fig. 3f. In this case, the
variance of Poisson noise is 10.0603, the variance of
Gaussian noise is 40.2412. The number of pixels
intensities of which is out of interval 0 and 255 for
Poisson noise is 5 (respectively 0.0056%), and for
Gaussian noise is 5780 (respectively 6.42%).
The Table 2 shows the result of denoising for re-
al image in both case: given parameters and auto-
matically evaluated parameters.
a) b)
c) d)
e) f)
Figure 3: Denoising of real image: a) original image, b)
cropped image, c) with Gaussian noise, d) with Poisson
noise, e) with mixed noise, f) after denoising.
We have to notice, that in the case of the real im-
age, the value of
Q
PSNR
of denoising for given ideal
parameters is better, than the value of
Q
PSNR
of de-
noising for automatically evaluated parameters, but
the value of
Q
SSIM
is inversed.
Table 2: Quality comparison of noise removal methods for
real image of human skull.
Q
PSNR
Q
SSIM
Q
MSE
Noisy 21.4168 0.4246 427.9526
Proposed method
with
1
=0.75,
2
=0.25,
= 0.1,
= 40.2412.
27.2808 0.8157 121.6189
Proposed method
with evaluated
parameters
1
=0.8095,
2
=0.1905,
= 0.0970,
= 38.2310.
27.2567 0.8383 122.2941
3 CONCLUSIONS
In this paper, we proposed the approach to remove
combination of Poisson and Gaussian noises (mixed
noise). This method is based on variational ap-
proach.
The result of denoising depends on parameters,
especially on coefficients of linear combination
1
and
2
. We can specify values of parameters or
these values can be automatically evaluated. In order
to apply this model to real image, we need to use the
proposed method with automatically evaluated pa-
rameters.
The proposed method can be applied to remove
separate Gaussian or Poisson noise (respectively
ROF model and modified ROF model for Poisson
noise), or mixed Poisson-Gaussian noise as well.
We also can use this variational approach to re-
move other kinds of noise, such as noise of magnetic
resonance images (MRI), ultrasonogram, etc.
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