Bayesian Image Denoising with Multiple Noisy Images

Abstract

In this paper, we propose a fast image denoising method based on discrete Markov random fields and the fast Fourier transform. The purpose of the image denoising is to infer the original noiseless image from a noise corrupted image. We consider the case where several noisy images are available for inferring the original image and the Bayesian approach is adopted to create the posterior probability distribution of the denoised image. In the proposed method, the estimation of the denoised image is achieved using belief propagation and an expectation–maximization algorithm. We numerically verified the performance of the proposed method using several standard images.

Full text

Vol.:(0123456789)
The Review of Socionetwork Strategies
https://doi.org/10.1007/s12626-019-00043-3
1 3
ARTICLE
Bayesian Image Denoising withMultiple Noisy Images
ShunKataoka1 · MunekiYasuda2
Received: 3 February 2019 / Accepted: 20 June 2019
© The Author(s) 2019
Abstract
In this paper, we propose a fast image denoising method based on discrete Markov
random fields and the fast Fourier transform. The purpose of the image denoising is
to infer the original noiseless image from a noise corrupted image. We consider the
case where several noisy images are available for inferring the original image and
the Bayesian approach is adopted to create the posterior probability distribution of
the denoised image. In the proposed method, the estimation of the denoised image is
achieved using belief propagation and an expectation–maximization algorithm. We
numerically verified the performance of the proposed method using several standard
images.
Keywords Image denoising· Discrete Markov random field· Belief propagation·
EM algorithm· FFT
1 Introduction
Bayesian image processing based on Markov random fields (MRFs) is an important
framework in the field of image processing [1, 2]. An MRF is a undirected graph
representation of probability distribution, and many applications of MRFs exist
in the image processing and computer vision fields [3–5]. MRFs have also been
applied to other research fields, including traffic engineering [6, 7] and earth science
[8, 9]. In Bayesian image processing, the objective image can be inferred based on
the posterior probability distribution.
Recently, we proposed a fast image denoising method for the case where multiple
noisy images are available for inferring the original noiseless image that is based on
* Shun Kataoka
xskataoka@res.otaru‑uc.ac.jp
Muneki Yasuda
muneki@yz.yamagata‑u.ac.jp
1 Faculty ofCommerce, Otaru University ofCommerce, Otaru‑shi047‑8571, Japan
2 Graduate School ofScience andEngineering, Yamagata University, Yonezawa‑shi992‑8510,
Japan

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1 3
Gaussian MRFs [10]. However, in the study in [10], for ease of mathematical treat‑
ment, we made an unnatural assumption that pixel values are continuous. In general,
a pixel takes a discrete value from 0 to 255, and an additional framework is required
to treat pixel values as discrete instead of continuous values. Therefore, in this paper,
we focus on the Bayesian image denoising problem of inferring the original noiseless
image from multiple noisy images when the pixel values are treated as discrete values.
We created a probability model for image denoising based on the discrete MRF and
Bayesian perspective. A major disadvantage of an image processing model based on
discrete MRFs is the computational complexity. In fact, the inference problem from
discrete MRFs belongs to the NP‑hard class. Therefore, an approximate inference tech‑
nique is required to infer the objective image from a discrete MRF. Belief propagation
[11] is known as one such effective technique. In this paper, we propose an effective
image denoising algorithm for multiple noisy images that applies belief propagation.
The main contributions of this paper are that an MRF model for image denoising with
multiple noisy images is defined and a fast effective denoising algorithm based on our
discrete MRF model and the fast Fourier Transform (FFT) is proposed.
The remainder of this paper is organized as follows. In Sect. 2, we define a prob‑
ability model for image denoising with multiple noisy images based on the discrete
MRF and Baysian perspective. In Sect.3, we derive an image denoising algorithm
based on the posterior probability distribution defined in Sect.2. In Sect.4, we describe
the framework for estimating the parameters in the posterior probability distribution.
We explain the implementation of our denoising method using the FFT in Sect. 5. In
Sect. 6, we describe the numerical verification of the performance of the proposed
method. Finally, in Sect.7, we present our concluding remarks.
2 Framework ofBayesian Image Denoising Method
In this section, we briefly explain the framework of the Bayesian image denoising
method for the case where multiple noisy images are available. Suppose that we have K
degraded images that are independently obtained by adding additive white Gaussian
noise (AWGN) to the original image. We assume that the images are composed of
N=h×w
pixels. Let
x
=
[
x
1
x
2
⋯x
N]T
and y(k)=
[
y(k)
1y(k)
2
⋯y(k)
N
]T
be N dimen‑
sional vectors corresponding to the original image and the k‑th noisy image, respec‑
tively. Vectors
x
and
y(k)
can be easily obtained by raster scanning the images. We
assume that
xi(i=1, 2, …N)
takes L discrete values from 0 to
L−1
.
The purpose of the image denoising is to infer the original noiseless image
x
from K
noisy images
Y
=
{
y
(
1
)
,y
(
2
)
,…,y
(
K
)}
. In the Bayesian framework, the original image
x
can be inferred using the posterior probability distribution
P(x|Y)
that is expressed as
where
∑x
denotes the multiple summations over all the possible
LN
states of
x
. The
framework of the proposed Bayesian image denoising method is illustrated in Fig.1.
From the definition of
y(k)
, the probability density function
P(Y|x)
is expressed as
(1)
P
(x
�
Y)=
P(Y�
x
)P(
x
)
∑x
P(Y
�
x)P(x)
,

1 3
The Review of Socionetwork Strategies
where
V={1, 2, …,N}
and
𝜎2
is the variance of the AWGN. We express the param‑
eters of the probability model by its arguments after the semicolon as Eq. (2). We
define the prior probability distribution as
where E is a set of edges of the
h×w
lattice graph and
𝜙(x)
is a downward convex
even function taking its minimum at
x=0
. In this study, we assumed the periodic
boundary condition on the graph structure E, as demonstrated in Fig.2.
𝛼>0
is the
parameter of the prior probability distribution; if
𝛼
is set to a large value, neighbor‑
ing
xi
and
xj
tend to take close values.
Zprior(𝛼)
is a normalization constant defined as
By substituting Eqs. (2) and (3) into Eq. (1), the posterior probability distribution
P(x|Y)
is expressed as
(2)
P
Yx;𝜎2=
K

k=1
P

y(k)x;𝜎2

=
K

k=1

i∈V
1
2𝜋𝜎
2
exp

−1
2𝜎2

y(k)
i−xi

2
,
(3)
P
(x;𝛼)=1
Zprior(𝛼)exp

−𝛼

ij∈E
𝜙

xi−xj
,
(4)
Z
prior(𝛼)=

x
exp

−𝛼

ij∈E
𝜙

xi−xj
.
(5)
P
xY;𝛼,𝜎
2
=1
Zpost

𝛼,𝜎2

exp

−1
2𝜎2

i∈V
𝜓i

xi

−𝛼

ij∈E
𝜙

xi−xj
,
Fig. 1 Illustration of the proposed Bayesian image denoising method

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1 3
where
and
respectively.
3 Inference Algorithm Based onBelief Propagation
The image denoising is achieved by finding the image
̂
x
that maximizes the posterior
probability distribution in Eq. (5):
However, the problem of determining such an image is intractable, because this
maximization problem belongs to the NP‑hard class. Therefore, we need an approxi‑
mate inference method to find
̂
x
. In this section, we explain an effective approximate
inference method called belief propagation for inferring the denoised image
̂
x
.
Belief propagation is a method of computing the approximate marginal distributions
bi(
x
i)
and
bij(
x
i
,x
j)
for each
i∈V
and
ij ∈E
. In the belief propagation framework, the
approximate marginal distributions
bi(
x
i)
and
bij(
x
i
,x
j)
are given by
(6)
𝜓
i

xi

=
K

k=1
xi−y(k)
i

2
,
(7)
Z
post

𝛼,𝜎2

=

x
exp

−1
2𝜎2

i∈V
𝜓i

xi

−𝛼

ij∈E
𝜙

xi−xj
,
(8)
̂
x=argmax
x
P
(
x
|
Y;𝛼,𝜎
2).
(9)
b
i

xi

=
1
Z
i
exp

−
1
2𝜎2𝜓i

xi

k∈𝜕i
Mpost
k→i

xi
,
(10)
Z
i=

xi
exp

−
1
2𝜎2𝜓i

xi

k∈𝜕i
Mpost
k→i

xi
,
Fig. 2 Periodic boundary condi‑
tion for
h×w
lattice graph E,
when
h=4
and
w=4

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The Review of Socionetwork Strategies
and
respectively, where
𝜕i={k∈V|ik ∈E}
is a set of all the neighboring pixels of pixel
i.
Mpost
k→i(
x
i)
in Eqs. (9) and (10) is a message from pixel k to pixel i and is obtained
by the convergence point of the message update rule
where
Zj
→
i
is a normalization constant to ensure algorithmic stability. The estima‑
tion of the denoised image
̂
x
=
{̂
x
i|
i∈V
}
is achieved by finding
for
i∈V
in the belief propagation framework.
4 Parameter Estimation Using Expectation–Maximization Algorithm
In the preceding section, we explained the method for inferring the denoised image
̂
x
based on belief propagation. In our framework, the denoised image is inferred from the
posterior probability distribution in Eq. (5), which has two parameters,
𝛼
and
𝜎2
. It is
obvious that the inferred denoised image
̂
x
depends on these parameters. In this sec‑
tion, we explain the method for determining these parameters from degraded images Y
based on the expectation–maximization (EM) algorithm [12].
The EM algorithm is a statistical inference method to infer the maximum likelihood
estimates
by an iterated method. In the EM algorithm framework, the parameters
𝛼
and
𝜎2
are
estimated by iterative maximization of the Q function defined as
(11)
b
ij
(
xi,xj
)
=
1
Z
ij
exp
(
−𝛼𝜙
(
xi−xj
))
mpost
�j→i
(
xi
)
mpost
�i→j
(
xj
),
(12)
Z
ij =
∑
x
i
∑
x
j
exp
(
−𝛼𝜙
(
xi−xj
))
m
post
�j→i
(
xi
)
m
post
�i→j
(
xj
),
(13)
M
post
j→i
(
xi
)
=
1
Zj→i
∑
x
j
exp
(
−𝛼𝜙
(
xi−xj
))
mpost
�i→j
(
xj
),
(14)
m
post
�j→i

xi

=exp

−
1
2𝜎2𝜓i

xi

k∈𝜕i�{j}
Mpost
k→i

xi
,
(15)
̂x
i=argmax
x
i
bi
(
xi
),
(16)
̂𝛼
,̂𝜎 2=argmax
𝛼,𝜎
2
∑
x
P
(
x,Y;𝛼,𝜎2
),

The Review of Socionetwork Strategies
1 3
where
𝛼t
and
𝜎2
t
are estimates of the parameters at the t‑th iteration and
Using belief propagation, we can approximate the expectations in Eq. (17) as
and
respectively, where
b(t)
i(
x
i)
and
b(t)
ij (
xi,xj
)
are the approximate marginal distributions
of the posterior probability distribution
P(
x
|
Y;𝛼
t
,𝜎2
t)
computed using Eqs. (9) and
(11). The parameter update at iteration t is given by
and the maximum likelihood estimates in Eq. (16) are given as the convergence point
of the above iterative estimation. By differentiating the Q function with respect to
𝛼
and
𝜎2
and considering the conditions for the extremal value, the updated parameter
𝛼t+1
in Eq. (21) is expressed as the solution of the equation
and the updated parameter
𝜎2
t+1
is calculated as
where

𝜙

xi−xj

;𝛼
prior
=

x𝜙

xi−xj

P(x;𝛼
)
; this expectation can also be com‑
puted approximately using belief propagation similarly to Eq. (20). Using the bisec‑
tion method, we can easily find
𝛼t+1
that satisfies Eq. (22).
(17)
Q(
𝛼,𝜎2;𝛼t,𝜎2
t
)
=
∑
x
P
(
x|Y;𝛼t,𝜎2
t
)
log P
(
x,Y;𝛼,𝜎2
)
=− 1
2𝜎2∑
i∈V
⟨𝜓i(xi);𝛼t,𝜎2
t⟩post −NK
2log 𝜎2
−𝛼∑
ij∈E
⟨𝜙(xi−xj);𝛼t,𝜎2
t⟩post −log Zprior(𝛼
)
+Const.,
(18)
⟨
f(x);𝛼t,𝜎2
t
⟩
post =
∑
x
f(x)P
(
x
|
Y;𝛼t,𝜎2
t
).
(19)
⟨
𝜓i
(
xi
)
;𝛼t,𝜎2
t
⟩
post =
∑
xi
𝜓i
(
xi
)
b
(t)
i
(
xi
),
(20)
⟨
𝜙
(
xi−xj
)
;𝛼t,𝜎2
t
⟩
post =
∑
x
i
∑
x
j
𝜙
(
xi−xj
)
b
(t)
ij
(
xi,xj
),
(21)
𝛼
t+1,𝜎
2
t+1=argmax
𝛼,𝜎
2
Q
(
𝛼,𝜎
2
;𝛼t,𝜎
2
t
),
(22)
∑
ij∈E
⟨
𝜙
(
xi−xj
)
;𝛼t,𝜎2
t
⟩
post =
∑
ij∈E
⟨
𝜙
(
xi−xj
)
;𝛼
⟩
prior
,
(23)
𝜎
2
t+1=
1
NK
∑
i∈V
⟨
𝜓i
(
xi
)
;𝛼t,𝜎2
t
⟩
post
,

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The Review of Socionetwork Strategies
5 Proposed Algorithm: Fast Implementation Based ontheFast
Fourier Transform
The image denoising algorithm based on our probabilistic model in Eq. (5) described
in Sects. 2–4 is summarized in Algorithm 1, together with the differences in the
computation times of the naive implementation and the proposed method, which is
explained in this section. The worst computation time of the naive implementation
of this algorithm is
O(
T
EM
T
BP
NL
2)
, where
TEM
and
TBP
are the maximum number of
updates for the parameter update in Eq. (21) and the message update in Eq. (13),
respectively. In Algorithm 1, we terminate the parameter and message updates in
iteration
TEM
and
TBP
, respectively. Because we assume the periodic boundary con‑
dition for the graph structure E, the number of edges is
|E|=2N
. Therefore, the
number of messages passing each edge is O(N). The computation time of the naive
message update from pixel j to pixel i is
O(
L
2)
, because Eq. (13) must be computed
for each
xi=0, 1, …,L−1
to update a message
Mpost
j
→
i
(xi
)
.
Algorithm 1:Image DenoisingAlgorithm
Require:
Y=y(1),y(2) ,...,y(K)
1: Initializeα0andσ2
0
2: fort=0,1,2,...,TEM −1do
3: initializeall messagesMj→i(xi)
4: fort=0,1,2,...,TBP −1do
5: forall j→ido
6: update Mj→i(xi)using Eq.(13) (naive:O L2→propose: O(LlogL))
7: if allmessagesMj→i(xi)are convergedthen break
8: forall i∈Vdo
9: computeψi(xi);αt,σ
2
tpost usingEq. (19)
10:
forall ij ∈Edo
11:
computeφ(xi−xj);αt,σ
2
tpost usingEq. (20) (naive:OL2→prop
ose:
O(LlogL))
12:
computeαt+1 usingEq. (22) andbisectionmethod(naive: O L2→prop
ose:
O(LlogL))
13:
computeσ2
t+1 usingEq. (23)
14:
if αandσ2areconverged then break
15:
initializeall messagesMj→i(xi)
16:
fort=0,1,2,...,T
BP −1do
17:
forall j→ido
18:
update Mj→i(xi)using Eq.(13) (naive:O L2→propose: O(LlogL))
19:
if allmessagesMj→i(xi)are convergedthen break
20:
forall i∈Vdo
21:
computebi(xi)using Eq.(9)
22:
computexiusingEq. (15)
It should be noted that the computation time of the message update can be
reduced to
O(Llog L)
using the FFT [13]. It has been confirmed that this FFT‑based
method in fact accelerates the message computation for the probabilistic image
denoising model in Eq. (5) for the case where
K=1
[14]. However, there exist

The Review of Socionetwork Strategies
1 3
additional
O(L2)
computation terms in Algorithm1: Steps 11 and 12. In this section,
we show that these
O(L2)
computation terms can also be computed in
O(Llog L)
using the FFT. Therefore, we can reduce the worst computation time of Algorithm1
to
O(
T
EM
T
BP
NL log L
)
.
The key idea for accelerating the message updates in Eq. (13) is to consider the
update rule a convolution calculation. If we define the function
f(x;𝛼)
as
we can reformulate the message update rules as
The calculation of
mpost
j
→
i(
xi
)
in Eq. (25) is a convolution calculation. Therefore, we
can calculate
mpost
j
→
i(
xi
)
for
xi=0, 1, …,L−1
in
O(Llog L)
computation time using
the FFT, and the computation time of M
post
j
→
i(
xi
)
in Eq. (26) is linear with respect to
L. Therefore, we can update a message
Mpost
j
→
i(
xi
)
in
O(Llog L)
computation time.
Now, we show that the expectation in Eq. (20) can be calculated in
O(Llog L)
computation time by using the FFT. By substituting Eqs. (11) into (20), the expecta‑
tion calculation can be expressed as
If we define functions
g(xi)
and
h(xi)
as
respectively, we can reformulate the expectation calculation as
(24)
f(x;𝛼)=exp (−𝛼𝜙(x)),
(25)
mpost
j→i
(
xi
)
=
∑
x
j
f(xi−xj;𝛼)m
post
�i→j
(
xj
),
(26)
M
post
j→i
(
xi
)
=mpost
j→i
(
xi
)/L−1
∑
l=0
mpost
j→i(l)
.
(27)
⟨
𝜙
(
xi−xj
)
;𝛼,𝜎2
⟩
post
=
G
H,
(28)
G
=
∑
x
i
∑
x
j
𝜙
(
xi−xj
)
f(xi−xj;𝛼)m
post
�i→j
(
xj
)
m
post
�j→i
(
xi
),
(29)
H
=Zij =
∑
x
i
∑
x
j
f(xi−xj;𝛼)m
post
�i→j
(
xj
)
m
post
�j→i
(
xi
).
(30)
g
(xi)=
∑
x
j
𝜙
(
xi−xj
)
f(xi−xj;𝛼)m
post
�i→j
(
xj
),
(31)
h
(xi)=
∑
x
j
f(xi−xj;𝛼)m
post
�i→j
(
xj
),
(32)
G
=
∑
xi
g(xi)m
post
�j→i
(
xi
),

1 3
The Review of Socionetwork Strategies
and
respectively. Therefore, the computation time of the expectation in Eq. (27) is
O(Llog L)
, because the total computation time to calculate convolutions
g(xi)
and
h(xi)
in Eqs. (30) and (31) for all
xi=0, 1, …,L−1
is
O(Llog L)
, and Eqs. (32) and
(33) can be computed in
O(L)
computation time.
In Eq. (22), we need to calculate the messages and expectation of prior probability
distribution
P(
x;𝛼
t)
to find
𝛼t+1
that satisfies this equation using the bisection method.
It should be noted that, because we assume the periodic boundary condition for the
graph structure E, we can calculate the messages and expectation of prior probabil‑
ity distribution faster than those of posterior probability distribution by considering the
translational symmetry assumption. If we assume both a periodic boundary condition
and translational symmetry, the messages and expectation of prior probability distribu‑
tion become not dependent on the position of the edges
ij ∈E
. Therefore, the message
update rule and expectation calculation for prior probability distribution are expressed
as
and
respectively, where
Mprior(
x
i)
is a message of the prior probability distribution. The
calculation of the message and the expectation of the prior probability distribution
in Eqs. (34) and (36) can be computed in
O(Llog L)
computation time by the same
calculation method as Eqs. (25, 26) and Eqs. (27)–(33), respectively.
(33)
H
=
∑
xi
h(xi)m
post
�j→i
(
xi
),
(34)
M
prior
(
xi
)
=
1
Zprior
∑
x
j
𝜙
(
xi−xj
)
f(xi−xj;𝛼)
(
Mprior
(
xj
))
3
,
(35)
Z
prior =
∑
x
i
∑
x
j
exp
(
−𝛼𝜙
(
xi−xj
))(
Mprior
(
xj
))3,
(36)
∑
ij∈E
⟨
𝜙
(
xi−xj
)
;𝛼
⟩
prior =2NG
prior
Hprior
,
(37)
G
prior =
∑
x
i
∑
x
j
𝜙
(
xi−xj
)
f(xi−xj;𝛼)
(
Mprior
(
xi
))3(
Mprior
(
xj
))3,
(38)
H
prior =
∑
x
i
∑
x
j
f(xi−xj;𝛼)
(
Mprior
(
xi
))3(
Mprior
(
xj
))3,

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6 Numerical Experiments
In this section, we describe the numerical verification of the proposed method. We
used the standard images in Fig.3, which are widely used in the image processing
research field. The pixel values of these images take
L=256
different values. All
the experiments were implemented using C++ and were run single‑threaded on an
Ubuntu 18.04.1 LTS (64 bit) machine with an Intel Core i7‑6850K CPU running at
3.60 GHz and 128 GB RAM. In this experiment, we defined the function
𝜙(x)
as
and set the parameters of the proposed method as follows. The initial parameter
𝛼0
was set at 0.005 and
𝜎2
0
was set at the sample variance calculated from Y. The maxi‑
mum numbers of iterations
TEM
and
TBP
were set at 100 and 1000, respectively. We
considered that the messages converged if the absolute value of the average change
in the messages was smaller than
10−4
; the same applied to the parameters
𝛼
and
𝜎2
.
We set the search interval of the bisection method for computing
𝛼t+1
as
[0, 2𝛼t]
.
First, we compared the computation time of the proposed method with that of the
previous methods (belief propagation FFT (BP‑FFT) and Naive). The Naive method
is the naive implementation version of Algorithm 1; the worst computation time is
O(
T
EM
T
BP
NL
2)
. BP‑FFT is the method used in the study presented in [14], where only
the message updates in Eqs. (13) and (14) were speeded‑up by using the FFT. Tables1
and 2 show the average computation times over 10 trials for each method where K
noisy images Y were generated by adding an AWGN of
𝜎=15
to the original noise‑
less images. According to the results, the proposed method was faster than the other
methods. It should be noted that the difference between the three methods is in whether
Algorithm1 is implemented using the FFT. Therefore, the image denoising results of
these method are all the same.
(39)
𝜙(x)=x2,
Fig. 3 Gray scale standard images used in experiments (
L=256
)
Table 1 Average computation
time over 10 trials with
𝜎=15
,
K=1
, and image size
128 ×128
Fig.3a Fig.3b Fig.3c Fig.3d
Proposed (min) 15.24 15.24 11.56 15.74
BP‑FFT (min) 45.14 45.10 49.79 47.55
Naive (min) 275.17 249.64 230.18 263.57

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Figure4 shows the average computation time versus image size for
K=1
and
K=5
over 10 trials, where the noise level of AWGN was
𝜎=15
. The computation time of
our denoising methods grows approximately linearly with the increase in the image
size (it dose not grow strictly linearly, because we break off the message and parameter
updates according to the convergence condition).
Figure5 shows the denoising performance of the proposed method versus various
values of K for two levels of AWGN (
𝜎=15
and
𝜎=30
). We evaluated the perfor‑
mance of the method according to the average peak signal‑to‑noise ratio (PSNR) over
10 trials. The PSNR is defined as
(40)
PSNR
=10 log10
255
2
MSE ,
Table 2 Average computation
time over 10 trials with
𝜎=15
,
K=5
, and image size
128 ×128
Fig.3a Fig.3b Fig.3c Fig.3d
Proposed (min) 4.86 6.46 4.22 6.05
BP‑FFT (min) 14.02 18.92 16.24 16.18
Naive (min) 100.92 119.43 97.54 103.63
Fig. 4 Average computation time and peak signal‑to‑noise ratio over 10 trials versus various image sizes
(
128 ×128
,
256 ×256
, and
512 ×512
) for
K=1
and
K=5
. The noise level applied in these experiments
was
𝜎=15
. a Fig.3a. b Fig.3b. c Fig.3c. d Fig.3d

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1 3
where MSE is the mean squared error between the original noiseless image and
the inferred denoised image
̂
x
. Figure 5 conforms that the image denoising results
improve as the value of K is increased. Example of image denoising results for
Fig.3a are shown in Fig.6 for
K=1
and
K=5
, respectively.
Fig. 5 Average peak signal‑to‑noise ratio over 10 trials against K for
𝜎=15
and
𝜎=30
. a Fig. 3a. b
Fig.3b. c Fig.3c. d Fig.3d
Fig. 6 Examples of the image denoising results for Fig.3a. a Example of a noisy image when
𝜎=30
(peak signal‑to‑noise ratio (PSNR):18.61). b Denoised image obtained by the proposed method for
K=1
(PSNR:25.49). c Denoised image obtained by the proposed method for
K=5
(PSNR:30.33)

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The Review of Socionetwork Strategies
7 Concluding Remarks
In this paper, we defined a discrete MRF model for the Bayesian image denoising
problem with multiple noisy images. We proposed a fast denoising algorithm for
inferring a denoised image in
O(
T
EM
T
BP
NL log L
)
‑time by using belief propagation
and an EM algorithm based on our MRF model and FFT. We numerically verified
the proposed denoising method using standard images. The results show that the
proposed algorithm inferred the denoised image faster than previous implementation
methods that use belief propagation.
We believe that the proposed method is the most fastest implementation of an
image denoising algorithm based on a discrete MRF model that uses belief propaga‑
tion and an EM algorithm. However, the method cannot yet be used for real‑time
processing. Therefore, we need to seek a further effective fast approximate method
that preserves the restoration quality for the discrete MRF model. In our experiment,
we adopted the quadratic function as the form of function
𝜙(x)
. However, the pro‑
posed method is not restricted to the quadratic function: we can apply it to other
types of the function
𝜙(x)
, such as the Huber prior [15] and generalized sparse prior
[16]. Moreover, because it can be used in any discrete MRF with
𝜙(xi−xj)
potential
for
ij ∈E
interaction, it is expected that the proposed method is applicable to not
only image denoising but also other inference problems such as sparse modeling
[17, 18]. We intend to develop the method in these directions.
Acknowledgements This work was partially supported by JST CREST Grant no. JPMJCR1402 and JSPS
KAKENHI Grant nos. 18K18120, 18H03303, 18K11459, and 15H03699.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna‑
tional License (http://creat iveco mmons .org/licen ses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
References
1. Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian resto‑
ration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.
2. Tanaka, K. (2002). Statistical‑mechanical approach to image processing. Journal of Physics A:
Mathematical and General, 35(37), R81–R150.
3. Li, S. Z. (2009). Markov random field modeling in image analysis. Berlin: Springer.
4. Blake, A., Kohli, P., & Rother, C. (2011). Markov random fields for vision and image processing.
Cambridge: MIT Press.
5. Lezoray, O., & Grady, L. (Eds.). (2012). Image processing and analysis with graphs. Boca Raton:
CRC Press.
6. Kataoka, S., Yasuda, M., Furtlehner, C., & Tanaka, K. (2014). Traffic data reconstruction based on
Markov random field modeling. Inverse Problem, 30(2), 025003.
7. Hara, Y., Suzuki, J., & Kuwahara, M. (2018). Network‑wide traffic state estimation using a mixture
Gaussian graphical model and graphical lasso. Transportation Research Part C, 86, 622–638.
8. Kuwatani, T., Nagata, K., Okada, M., & Toriumi, M. (2014). Markov random field modeling for
mapping geofluid distributions from seismic velocity structures. Earth, Planets and Space, 66(5),
1–9.

The Review of Socionetwork Strategies
1 3
9. Kuwatani, T., Nagata, K., Okada, M., & Toriumi, M. (2014). Markov‑random‑field modeling for
linear seismic tomography. Physical Review E, 90, 042137.
10. Yasuda, M., Watanabe, J., Kataoka, S., & Tanaka, K. (2018). Linear‑time algorithm in Bayesian
image denoising based on Gaussian Markov random field. In: IEICE Transactions on Information
and Systems, vol. E101.D, no. 6, pp. 1629‑1639.
11. Pearl, J. (1988). Probabilistic reasoning in intelligent system:networks of plausible inference. Burl‑
ington: Morgan Kaufmann.
12. Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximun likelihood from incomplete data
via the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 39(1),
1–38.
13. Felzenszwalb, P. F., & Huttenlocher, D. P. (2006). Efficient belief propagation for early vision. Inter-
national Journal of Computer Vision, 70(1), 41–54.
14. Inoue, K. (2009). Kakuritisu denpan hou ni yoru gazou shori algorithm no kairyou ni kansuru ken‑
kyu (Study on an improvement of the image processing using belief propagation), Master Thesis in
Tohoku University (unpublished) (in Japanese).
15. Schulta, R. R., & Stevenson, R. L. (1994). A Bayesian approach to image expansion for improved
definition. IEEE Transactions on Image Processing, 3(3), 233–242.
16. Tanaka, K., Yasuda, M., & Titterington, D. M. (2012). Bayesian image modeling by means of gen‑
eralized sparse prior and loopy belief propagation. Journal of the Physical Society of Japan, 81(11),
114802.
17. Krzakala, F., Mézard, M., Sausset, F., Sun, Y., & Zdeborová, L. (2012). Probabilistic reconstruction
in compressed sensing: Algorithms, phase diagrams, and threshold achieving matrices. Journal of
Statistical Mechanics: Theory and Experiment, 2012, P08009.
18. Elad, M. (2010). Sparse and redundant representations: From theory to applications in signal and
image processing. Berlin: Springer.
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