NON-LOCAL FILTER FOR REMOVING A MIXTURE OF
GAUSSIAN AND IMPULSE NOISES
Haijuan Hu
1
, Bing Li
2
and Quansheng Liu
1
1
UMR 6205, Laboratoire de Math´ematiques de Bretagne Atlantique, Universit´e de Bretagne-Sud,
Campus de Tohannic, 56017 Vannes, France
2
Department of Math.,Zhongshan Polytechnic, 528404 Zhongshan, P.R.China
Keywords:
Gaussian Noise, Impulse Noise, Mixed Noise, Image Restoration, Denoising, Trilateral Filter, Non-local
Means Filter, Law of Large Numbers, Central Limit Theorem.
Abstract:
In this paper we first present two convergence theorems which give a theoretical justification of the Non-Local
Means Filter. Based on these theorems, we propose a new filter, called Non-Local Mixed Filter, to remove a
mixture of Gaussian and random impulse noises. This filter combines the essential ideas of the Trilateral Filter
and the Non-Local Means Filter. It improves the Trilateral Filter and extends the Non-Local Means Filter.
Our experiments show that the new filter generally outperforms two other recent proposed methods. A careful
discussion and simple formulas are given for the choice of parameters for the proposed filter.
1 INTRODUCTION
The main objective of this paper is to extend the Non-
Local Means Filter (Buades et al., 2005) for removing
Gaussian noise to the case where the image is contam-
inated by a mixture of Gaussian and random impulse
noises, based on two convergence theorems for the
Non-Local Means Filter that we will present.
Let us first introduce the Gaussian and impulse
noise models. As usual, we denote a digital im-
age by a N × N matrix u = {u(i) : i ∈ I}, where
I = {0,1,...,N−1}
2
and 0 ≤u(i) ≤255. The additive
Gaussian noise model is: v(i) = u(i) + η(i), where
u = {u(i) : i ∈ I} is the original image, v = {v(i) : i ∈
I} is the noisy one, and η is the Gaussian noise: η(i)
are independent and identically distributed Gaussian
random variables with mean 0 and standard deviation
σ > 0. We always denote by u the original image, v
the noisy one. The random impulse noise model is:
v(i) =

η(i) with probability p,
u(i) with probability (1−p),
where p is the impulse probability (the proportion
of the occurrence of impulse noise), and η(i) are in-
dependent random variables uniformly distributed on
the interval [min{u(i) : i ∈ I}, max{u(i) : i ∈ I} ].
There is a large literature for removing Gaussian
noise. A very important progress in this classical
research field was marked by the proposition of the
Non-Local Means Filter (NL-means) by Buades, Coll
and Morel. The key idea of this filter is to estimate
the original image by weighted means along simi-
lar local patches. Since then a series of important
works have been done by many authors in various
contexts using this interesting idea, see e.g. the op-
timal spatial adaptive patch-based filter in (Kervrann
and Boulanger, 2006), the K-SVD (Elad and Aharon,
2006) and BM3D (Dabov et al., 2007) algorithms.
There are also many methods to remove impulse
noise, see e.g. the variational methods in (Nikolova,
2004; Chan et al., 2004; Dong et al., 2007).
However, few filters are known to remove a mix-
ture of Gaussian and impulse noises, although such
noises can take place quite often. On this subject,
in (Garnett et al., 2005) an interesting statistic called
ROAD is introduced to detect impulse noisy pix-
els; this statistic is combined with the Bilateral Fil-
ter (Smith and Brady, 1997; Tomasi and Manduchi,
1998) leading to the so-called Trilateral Filter (TriF).
The performance of TriF is related to the efficiency
of the ROAD statistic for detecting impulse noise and
the performance of the Bilateral Filter for removing
Gaussian noise. A slightly different version of the
ROAD statistic is proposed in (Dong et al., 2007).
In this paper, we first (cf. Section 2) present two
convergence theorems, which gives a good theoreti-
cal justification for NL-means with a probabilistic in-
terpretation of the similarity phenomenon which ex-
145
Hu H., Li B. and Liu Q..
NON-LOCAL FILTER FOR REMOVING A MIXTURE OF GAUSSIAN AND IMPULSE NOISES.
DOI: 10.5220/0003864801450150
In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2012), pages 145-150
ISBN: 978-989-8565-03-7
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

ists very often in natural images. We then (cf. Sec-
tion 3) propose a new filter called Non-Local Mixed
Filter (NLMixF) to remove mixed noises, using the
presented convergence theorems in an adaptive way.
This filter improves the Trilateral Filter and extends
NL-means. Our experimental results (cf. Section
4) also show that for removing mixed noise, our fil-
ter NLMixF outperforms the two algorithms recently
proposed in (Yang and Wu, 2009) and (Xiao et al.,
2011) which are based respectively on the ideas of
BM3D (Dabov et al., 2007) and K-SVD (Elad and
Aharon, 2006).
2 CONVERGENCE THEOREMS
FOR NON-LOCAL MEANS
The Non-Local Means Filter (NL-means) (Buades
et al., 2005) is mainly based on the similarity of
local patches. For i ∈ I and d an odd integer, let
N
i
(d) = {j ∈ I : |j −i| ≤ (d − 1)/2} be the win-
dow with center i and size d ×d, where |j −i| =
max(|j
1
−i
1
|,|j
2
−i
2
|) for i = (i
1
,i
2
) and j = ( j
1
, j
2
).
Set N
0
i
(d) = N
i
(d)\{i}. We sometimes simply write
N
i
and N
0
i
for N
i
(d) and N
0
i
(d), respectively. De-
note v(N
i
) = {v(k) : k ∈ N
i
} as the vector composed
of the gray values of v in the window N
i
arranged lex-
icographically.
The denoised image by NL-means is given by
¯v(i) =
∑
j∈N
i
(D)
w(i, j)v( j)
∑
j∈N
i
(D)
w(i, j)
with
w(i, j) = e
−||v(N
i
)−v(N
j
)||
2
a
/(2σ
2
r
)
( j 6= i), (1)
where σ
r
> 0 is a control parameter,
||v(N
i
)−v(N
j
)||
2
a
=
∑
k∈N
i
(d)
a(i,k)|v(k) −v(T (k))|
2
∑
k∈N
i
(d)
a(i,k)
,
(2)
a(i,k) > 0 being some fixed weights usually chosen
to be a decreasing function of the Euclidean norm
ki−kk or |i−k|, and T = T
ij
is the translation map-
ping of N
i
onto N
j
: T (k) = k−i + j,k ∈ N
i
. Origi-
nally, N
i
(D) in (1) is chosen as the whole image I, but
in practice, it is better to choose N
i
(D) with an appro-
priate number D. We call N
i
(D) searches windows,
and N
i
= N
i
(d) local patches.
We now present some convergence theorems for
NL-means via probability theory. For simplicity, we
use the same notation v(N
i
) to denote both the ob-
served image patches and the corresponding random
variables (in fact the observed image is just a real-
ization of the corresponding variable). Therefore the
distribution of the observed image v(N
i
) is just that of
the corresponding random variable.
Definition 2.1. Two patches v(N
i
) and v(N
j
) are
called similar if they have the same probability dis-
tribution.
We sometimes simply say that the two windows
N
i
and N
j
are similar in the same sense. Defini-
tion 2.1 is a probabilistic interpretation of the simi-
larity phenomenon that occurs very often in natural
images. According to this definition, two observed
patches v(N
i
) and v(N
j
) are similar if they are issued
from the same probability distribution. In practice, we
consider that two patches v(N
i
) and v(N
j
) are simi-
lar if their Euclidean distance is small enough, say
kv(N
i
) −v(N
j
)k < T for some threshold T.
The following theorem is a kind of Marcinkiewicz
law of large numbers. It gives an estimation of the
almost sure convergence rate of the estimator to the
real image in NL-Means.
Theorem 2.1. Let i ∈ I and let I
i
be the set of j ∈ I
such that the patches N
i
and N
j
are similar (in the
sense of Definition 2.1). Set
v
0
(i) =
∑
j∈I
i
w
0
(i, j)v( j)
∑
j∈I
i
w
0
(i, j)
,
where
w
0
(i, j) = e
−kv(N
0
i
)−v(N
0
j
)k
2
a
/(2σ
2
r
)
. (3)
Then for any ε ∈ (0,
1
2
], as |I
i
| → ∞,
v
0
(i) −u(i) = o(|I
i
|
−(
1
2
−ε)
) almost surely, (4)
where |I
i
| denotes the cardinality of I
i
.
Theorem 2.1 improves the similarity principle in
(Li et al., 2011) which is just (4) with ε = 1/2. It
shows that v
0
(i) is a good estimator of the original
image u(i) if the number of similar patches |I
i
| is suf-
ficiently large. Here we use the weight w
0
(i, j) in-
stead of w(i, j), as w
0
(i, j) has the nice property that
it is independent of v( j) if j 6∈ N
i
. This property is
used in the proof, and makes the estimator v
0
(i) to be
“almost” non-biased: in fact, if the family {v( j)}
j
is
independent of the family {w
0
(i, j)}
j
(e.g. this is the
case when the similar windows are disjoint), then it
is evident that Ev
0
(i) = u
i
. We can consider that this
non-biased property holds approximately as for each
j there are few pixels k such that w
0
(i,k) are depen-
dent of v( j). A different explanation about the biased
estimation of NL-means can be found in (Xu et al.,
2008).
Notice that when v(N
j
) is not similar to v(N
i
),
then the weight w
0
(i, j) is small and negligible.
Therefore in practice we can take all windows. But
VISAPP 2012 - International Conference on Computer Vision Theory and Applications
146

selecting only similar windows can slightly improve
the restoration result, and can also speed up the algo-
rithm. The difference between ¯v(i) and v
0
(i) is also
small, so that Theorem 2.1 shows that ¯v(i) is also a
good estimator of u(i). But very often v
0
(i) gives bet-
ter restoration result.
The following result is a generalized central limit
theorem; it states that v
0
|I
i
|
(i) tends to u(i) just like
1/
p
|I
i
| in the sense of probability distribution.
Theorem 2.2. Under the condition of Theorem 2.1,
assume additionally that {v(N
j
) : j ∈I
i
}with the lexi-
cographical order is a stationary sequence of random
vectors. Then as |I
i
| → ∞,
p
|I
i
|(v
0
(i) −u(i))
d
→ L ,
where
d
→ means the convergence in distribution, L is
a mixture of centered Gaussian laws in the sense that
it has a density of the form
f(t) =
Z
R
|N
0
i
|
1
√
2πc
x
e
−
t
2
2c
2
x
ν(dx),
ν being the law of v(N
0
i
) and c
x
> 0.
By Theorems 2.1 and 2.2, the larger the value of
|I
i
|, the better the approximation of v
0
(i) to u(i). This
will be confirmed in another paper where we shall in-
troduce the notion of degree of similarity for images,
showing that the larger the degree of similarity, the
better the quality of restoration. Due to the limitation
of space, the proofs of theorems will be given else-
where.
3 NON-LOCAL MIXED FILTER
In this section, we will define our new filter. Be-
fore this we first recall the Trilateral Filter (Garnett
et al., 2005). This filter is based on the statistic ROAD
(Rank of Ordered Absolute Differences) defined by
ROAD(i) = r
1
(i) + ···+ r
m
(i), (5)
r
k
(i) being the k-th smallest term in {|u(i) −u( j)| :
j ∈ N
i
(d)\{i}}, m a constant taken as m = 4 in (Gar-
nett et al., 2005). The ROAD statistic serves to detect
noisy points: in fact, if i is an impulse noisy point,
then ROAD(i) is large; otherwise it is small. The Tri-
lateral Filter (TriF) is by definition
TriF(v)(i) =
∑
j∈N
i
(D)
w(i, j)v( j)
∑
j∈N
i
(D)
w(i, j)
, (6)
where
w(i, j) = w
S
(i, j)w
R
(i, j)
J
I
(i, j)
w
I
( j)
1−J
I
(i, j)
contains the spatial factor w
S
(i, j) = e
−|i−j|
2
/(2σ
2
S
)
,
the radiometric factor w
R
(i, j) = e
−(v(i)−v( j))
2
/(2σ
2
R
)
(which measure the similarity between the pixels i and
j), the impulse factor w
I
(i) and the joint impulse fac-
tor J
I
(i, j) defined by
w
I
(i) = e
−
ROAD(i)
2
2σ
2
I
, (7)
J
I
(i, j) = e
−
((ROAD(i)+ROAD( j))/2)
2
2σ
2
J
, (8)
σ
S
,σ
R
,σ
I
and σ
J
being control parameters. (In fact,
Garnett et al. (2005) initially defined the joint impulse
factor as J(i, j) = 1−J
I
(i, j). We found that it is more
convenient to use J
I
(i, j) instead of J(i, j).) Notice
that if either i or j is an impulse noisy point, then the
value of J
I
(i, j) is close to 0; otherwise it is close to
1. Similarly, w
I
(i) is close to 0 if i is an impulse noisy
point, and to 1 otherwise.
Our new filter will be based on the following
weighted norm that we call mixed norm:
||v(N
0
i
) −v(N
0
j
)||
2
M
(9)
=
∑
k∈N
0
i
w
S,M
(i,k)J
I
(k,T (k))|v(k) −v(T (k))|
2
∑
k∈N
0
i
w
S,M
(i,k)J
I
(k,T (k))
,
where w
S,M
(i,k) = e
−|i−k|
2
/(2σ
2
S,M
)
, and J
I
(k,T (k)) is
defined in (8). Recall that if k or T (k) is an impulse
noisy point, then J
I
(k,T (k)) is close to 0, so that
the concerned point contributes little to the weighted
norm (9). Therefore the mixed norm (9) filters im-
pulse noisy points. Clearly, it also measures the simi-
larity between the patches v(N
i
) and v(N
j
) and takes
into account the spatial factor. Our new filter that we
call Non-Local Mixed Filter (NLMixF) is by defini-
tion
NLMixF(v)(i) =
∑
j∈N
i
(D)
w(i, j)v( j)
∑
j∈N
i
(D)
w(i, j)
,
where
w(i, j) = w
S
(i, j)w
I
( j)w
M
(i, j)
contains the spatial factor w
S
(i, j) = e
−|i−j|
2
/(2σ
2
S
)
, the
similarity factor w
M
(i, j) = e
−||v(N
i
)−v(N
j
)||
2
M
/(2σ
2
M
)
,
and the impulse factor w
I
( j) defined in (7), with σ
S
and σ
M
being parameters. Notice that NLMixF re-
duces to NL-means when σ
I
= σ
J
= σ
S
= ∞. This
filter NLmixF is an improved version of the filter in-
troduced in (Li et al., 2011). Compared to the filter
of (Li et al., 2011), it improves the quality of restora-
tion and contains entirely NL-means filter thanks to
the added spatial factor w
S,M
in the mixed norm (9).
Notice that for each impulse noisy point j in N
i
(D),
the weight w(i, j) is close to 0. Hence our new filter
can be regarded as an application of Theorems 2.1 and
NON-LOCAL FILTER FOR REMOVING A MIXTURE OF GAUSSIAN AND IMPULSE NOISES
147

2.2 to the remained image (which can be considered
to contain only Gaussian noise) obtained after filter-
ing the impulse noisy points by the mixed norm (9).
4 SIMULATIONS AND CHOICES
OF PARAMETERS
In this section, we present some experimental results
to compare the new filter NLMixF with NL-means,
TriF, and two recently algorithms proposed in (Yang
and Wu, 2009) and (Xiao et al., 2011). As usual we
use PSNR (Peak Signal-to-Noise Ratio) defined by
PSNR ( ¯v) = 10log
10
255
2
|I|
∑
i∈I
( ¯v(i) −u(i))
2
to measure the quality of a restored image, where u
is the original image, ¯v the restored one. In our ex-
periments we use the 512×512 images Lena, Bridge,
Boats and the 256×256 image Peppers. They are all
available on line.
1
In our implementations, image boundaries are
handled by assuming symmetric boundary conditions.
In the original image Peppers, there are black bound-
aries of width of one pixel, we therefore compute the
PSNR value for the image of size 254×254 obtained
after removing the four boundaries.
There are several parameters to be tuned in
NLMixF. Recall that NLMixF reduces to NL-means
when σ
I
= σ
J
= σ
S
= ∞. So for removing Gaus-
sian noise, a reasonable choice is to take σ
I
,σ
J
and
σ
S
large enough (though this choice is not necessar-
ily optimal). To apply our filter easily in practice, we
look for a simple and uniform formula in terms of p
and σ. We first look for a linear relation; when this
does not seem possible we test some slightly more
complicated functions. To obtain the formulas, we
consider Gaussian noise with σ = 10, 20,30, impulse
noise with p = 0.2,0.3,0.4,0.5 and their mixture. We
have done our best to choose the formulas, but we can
not guarantee that our formulas are always optimal
due to the complexity of the subject. Our choices of
parameters for NLMixF are shown in the following,
where Gaussian noise, impulse noise and mixed noise
are abbreviated respectively as Gau, Imp and Mix:
σ
I
= 60 + 2σ −50p, σ
J
= 45 + 0.5σ −50p,
σ
M
= 4+ 0.4σ + 30p−
p
2σp,
1
for Lena, Peppers and Boats, cf.
http://decsai.ugr.es/∼javier/denoise/test images/index.htm;
for Bridge, cf. www.math.cuhk.edu.hk/∼rchan/paper/dcx/.
σ
S
=

0.6+ p σ = 0 (Imp)
15 σ > 0 (Gau or Mix)
σ
S,M
=



15 σ = 0 p > 0 (Imp)
1.5 σ > 0 p = 0 (Gau)
2 σ > 0 p > 0 (Mix)
d =







9 σ = 0 p > 0 (Imp or Mix)
5 σ = 10 p = 0
7 σ = 20 p = 0
30



(Gau)
D =



















7 σ = 0 p > 0 (Imp)
9 σ = 10 p = 0
13 σ = 20 p = 0
15 σ = 30 p = 0



(Gau)
7 σ = 10 p > 0
11 σ = 20 p > 0
15 σ = 30 p > 0



(Mix)
In the calculation of ROAD, we choose 3 ×3 neigh-
borhoods and m = 4. For impulse noise or mixed
noise with p = 0.4, 0.5, to further improve the restora-
tion results, we use 5 ×5 neighborhoods and m = 12
to calculate ROAD (5). Consistently, the choice of
σ
I
,σ
J
depend on m, thus they should be multiplied
by a factor empirically chosen as 4.2. Evidently, our
choice of parameters is not restricted to σ = 10, 20,30
and p = 0.2,0.3, 0.4,0.5. This choice can also be ap-
plied to any value of σ in the interval [10,30] and p in
the interval [0.2, 0.5], or even larger intervals. Note
that when σ
S
= 15 or σ
S,M
= 15, we get w
S
(i, j) ≈ 1
or w
S,M
(i, j) ≈ 1. This means that for impulse noise
we can omit the factor w
S,M
(i, j), and for Gaussian
noise and mixed noise we can omit the factor w
S
(i, j).
A full discussion of the roles of the different choices
of parameters goes beyond of the scope of this paper.
The problem of choice of parameters for NL-means
has been considered in the literature, see for example
(Xu et al., 2008) and (Duval et al., 2011).
For TriF, we choose parametersand apply the filter
according to the suggestion of (Garnett et al., 2005).
We use σ
I
= 40,σ
J
= 50,σ
S
= 0.5,σ
R
= 2σ
QGN
,
where σ
QGN
is an estimator for the standard deviation
of “quasi-Gaussian” noise defined in (Garnett et al.,
2005). For impulse noise, when p > 0.25, it was
proposed in (Garnett et al., 2005) to apply the filter
with two to five iterations. We apply two iterations
for p = 0.3,0.4, and four iterations for p = 0.5. For
mixed noise, we apply TriF twice with different val-
ues of σ
S
as suggested in (Garnett et al., 2005): with
all impulse noise levels p, for σ = 10, we use first
σ
a
= 0.3, then σ
S
= 1; for σ = 20, first σ
S
= 0.3, then
σ
Sb
= 15; for σ = 30, first σ
S
= 15, then σ
S
= 15.
Note that when σ
S
= 15, we can omit the spatial fac-
tor.
VISAPP 2012 - International Conference on Computer Vision Theory and Applications
148

Table 1: Choice of parameters for NL-means.
σ 10 20 30
d (size of local patchesN
i
(d)) 7 9 13
D (size of search windowsN
i
(D)) 9 9 11
Table 2: PSNR values for removing random impulse noise.
p 0.2 0.3 0.4 0.5
Lena TriF 34.44 32.55 31.27 29.14
NLMixF 35.35 33.09 31.52 29.85
Bridge TriF 26.70 25.20 24.43 23.31
NLMixF 27.65 25.46 24.45 23.31
Peppers TriF 30.85 28.64 27.74 26.14
NLMixF 31.94 29.40 28.34 26.47
Boats TriF 30.19 28.57 27.64 26.13
NLMixF 31.24 29.05 27.50 26.10
For NL-means, we use σ
r
= 4 + 0.4σ in accor-
dance with the choice of σ
M
in NLMixF (with p = 0).
This choice is different from the proposed one in
the original NL-means algorithm, and generally gives
better restoration results. The values for d and D are
shown in Table 1.
We now present some experimental results. Ta-
bles 2 and 3 show the performances of NLMixF for
removing impulse noise and Gaussian noise by com-
paring it with TriF and NL-means (for which we use
w(i,i) = max{w(i, j) : j 6= i, j ∈N
i
(D)} and a(i,k) =
1
(d−1)/2
∑
(d−1)/2
l=|i−k|
1
(2l+1)
2
in (2)). Table 4 compares
NLMixF with TriF for removing mixed noise. We
add Gaussian noise and then impulse noise for simu-
lation of mixed noise. Since NL-means is not suitable
for removing impulse noise, we do not include it in
Tables 2 and 4. We can see that NLMixF improves
TriF in almost all the cases, especially when p is small
(p = 0.2), or σ is large (σ = 20,30). Some examples
are shown in Figs. 1. In Table 5, we compare the
PSNR values with the two algorithms in (Yang and
Wu, 2009) and (Xiao et al., 2011), where we show
the reported PSNR values for these two algorithms.
In Fig. 2, we show the denoised images by NLMixF
and IPAMF+BM in (Yang and Wu, 2009), using the
same noisy image.
5 CONCLUSIONS
We have first presented two convergence theorems for
the Non-Local Means Filter (Buades et al., 2005).
Based on these convergence theorems and the idea
of Trilateral Filter (TriF) (Garnett et al., 2005) we
have then given a filter called Non-Local Mixed Filter
(NLMixF) to remove Gaussian noise, impulse noise
and their mixture. To make easy the application of
our filter, we have also given empirical formulas for
Table 3: PSNR values to remove Gaussian noise.
Lena Bridge
σ 10 20 30 10 20 30
TriF 33.21 29.48 26.51 30.62 27.42 25.05
NLMixF 34.92 31.73 29.81 32.65 29.33 27.56
NL-means 35.03 31.78 29.89 32.72 29.84 27.93
Table 4: PNSR values for removing mixed noise by
NLMixF and TriF.
p 0.2 0.3 0.4 0.2 0.3 0.4
Lena σ = 10 Bridge σ = 10
TriF 31.60 30.88 29.66 25.14 24.59 23.93
NLMix 32.78 31.47 29.94 26.06 24.71 23.80
Lena σ = 20 Bridge σ = 20
TriF 28.75 28.11 27.26 23.73 23.32 22.84
NLMixF 30.54 29.65 28.44 24.44 23.61 22.86
Lena σ = 30 Bridge σ = 30
TriF 26.48 25.77 25.02 22.40 22.06 21.52
NLMixF 28.87 28.07 27.34 23.38 22.79 22.24
Peppers σ = 10 Boats σ = 10
TriF 29.05 27.95 26.63 28.32 27.60 26.79
NLMixF 30.49 28.56 27.47 29.73 28.20 26.76
Peppers σ = 20 Boats σ = 20
TriF 26.70 25.87 25.29 26.35 25.74 25.07
NLMixF 28.54 27.37 26.51 27.66 26.61 25.60
Peppers σ = 30 Boats σ = 30
TriF 24.71 23.91 23.27 24.54 24.01 23.37
NLMixF 27.04 25.91 25.14 26.34 25.56 24.69
Table 5: Compare PNSR values for mixed noise.
Lena σ = 10 p = 0.1 p = 0.2 p = 0.3
(Xiao et al., 2011) 32.75 31.66 30.42
(Yang and Wu, 2009) 33.61 32.12 30.69
NLMixF 34.10 32.78 31.47
the choice of parameters which can at least be used
for Gaussian noise with σ ∈ [10,30], impulse noise
with p ∈ [0.2, 0.5], and their mixture. Our experi-
ments show that NLMixF outperforms TriF, as well as
the more recent methods proposed in (Yang and Wu,
2009),and (Xiao et al., 2011) based respectively on
the ideas of BM3D (Dabov et al., 2007) and K-SVD
(Elad and Aharon, 2006).
ACKNOWLEDGEMENTS
The authors are very grateful to the reviewers for
helpful comments and remarks. The work has been
partially supported by the National Natural Science
Foundation of China (Grant No. 10871035, No.
11101039 and No. 11171044), and by Zhong-
NON-LOCAL FILTER FOR REMOVING A MIXTURE OF GAUSSIAN AND IMPULSE NOISES
149

Noisy (σ = 0, p = 0.2) Noisy ( σ = 30, p = 0.4)
TriF: PSNR=34.44 TriF: PSNR=25.02
NLMixF: PSNR=35.35 NLMixF: PSNR=27.34
Figure 1: Images corrupted by impulse noise (left) and
mixed noise (right).
Noisy (σ = 10, p = 0.3) Original
IPAMF+BM: PSNR=30.69 NLMixF: PSNR=31.32
Figure 2: Restored images by NLmixF and IPAMF+BM
(Yang and Wu, 2009).
shan Science and Technology Department (Grant No.
2008A245).
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