IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 4, APRIL 2013 303
Analysis of Non-Local Euclidean
Medians and Its Improvement
Zhonggui Sun and Songcan Chen
Abstract—Non-Local Euclidean Medians (NLEM) has recently
been proposed and show s more effective than Non-Local Means
(NLM) in removing heavy noise. In this letter, we find the inconsis-
tency be t we en the two dissimilarity measu res in NLEM can affect
its robustness, thus develop an improved version (INLEM) to com-
pensate such an inconsistency. Further, we provide a concise con-
vergence proof for the iterative algorithm used in both NLEM and
INLEM. Finally, our experiments on synthetic and natural images
show that INLEM achieves encouraging results.
Index Terms—Improved non-local Euclidean m edians
(INLEM), image denoising, non-local Euclidean medians (NLEM),
non-local means (NLM).
I. INTRODUCTION
N
ON-LOCAL denoising method s have d rawn a lot of
attention in the imag
e processing community and chiefly
originate from the non-local means (NLM) proposed by Buades
et al. [1]. Unlike local counterparts which typically operate
within a local ne
ighborhood, NLM computes weighted means
in a non-local way by employing the between-patch dissim-
ilarity (m easure). Despite simple in idea, NLM outperforms
some popular fi
lters [4] and then motivates many successors
proposed. Such a patch-based and non-local viewpoint has
become a core of most state-of-the-art fi lters including BM3D
[2], K-SVD
[3] et al.,asreviewedin[4].
Along the line, recently, Chaudhury et al. proposed the
Non-local Euclidean medians (NLEM) to improve robust
performan
ce of NLM to heavy ( large noise level) noise b y re-
placing the Euclidean mean with Euclidean median [5]. Unlike
those weighted mean followers of NLM, NLEM is its weighted
median
variant and inherits robustness of the median filters to
outlier or heavy noise [6]. However, we find that in its im ple-
mentation, NLEM adopts two different kinds of b etween -patch
measu
res: one is the Euclidean norm in the definition of the
Euclidean median and the other is its squares in the definitio n of
weight com putation. In fact, measures are often task-dependent
[7]
and yield different robustness. Thus in NLEM, such a joint
use of inconsistent measures likely discounts its robustness in
Manuscript received October 26, 2012; revised January 07, 2013; accepted
January 30, 2013. Date of publication February 05, 2013; date of current version
February 14, 2013. This work was supported in part by the National Natural
Science Foundation of China under Grants 61170151 and 61035003 and also
by the Qing Lan Project. The associate editor coordinating the review of this
manuscript and approving it for publication w as Prof. Yiannis Andreopoulos.
Z. Sun is with College of Computer Science and Technology (CCST),
Nanjing University of Aeronautics & Astronautics (NUAA), 210016 Na njing,
China, and also with the Department of M athematics Science, Liaocheng
University, 25 200 0 L iaocheng, China (e-mail: altlp@nuaa.edu.cn).
S. Chen is with College of Computer Science and Technology (CCST), Nan-
jing University of Aeronautics & Astronautics (NUAA), 210016 Nanjing, China
(e-mail: s.chen@nuaa.edu.cn).
Dig
ital Object Identifier 10.1109/LSP.2013.2245322
denoising (as confirmed in ou r experiments). The observation
motivates us to propose an improved NLEM (INLEM) to elim-
inate such an inconsistency. Overall, our main contributions
can be summarized as follows:
1) We point ou t and analyze t he inconsisten cy of the tw o kinds
of between-patch measures used in NLEM.
2) We develop a new non-lo cal median filter (INLEM) with
consistent measures and obtain enco uragin g denoising
effect.
3) For the iterative algorith m of both NLEM and I NLEM, we
give a q uite concise convergence pro of.
The rest of this letter is structured as follows: In the next
section, we briefly review the related works including NLM and
NLEM.TheninSectionIII,wedevelopINLEMandgiveour
convergence proof for the itera tiv e algorithm. In Sections IV
and V, we provide experime nts to demonstrate the advantages
of our filter and a brief conclusion, respectively.
II. R
ELATED WORKS
A. Non-Local M eans (N LM )
Like literatures [1]–[5], we also focus on the Gaussian noise
with mean zero and variance
. Supp ose is an observed
(noisy) ima ge,
is its corresponding noise-free image to be
recovered. Let
denote t he gray value of pixel in the
image patch centered at pixel
in . and are sim -
ilarly defined. In NLM [1],
is estimated as a weighted
mean of other pixels in a non-local search window in
and
the weight between pixels
and is defined by the similar ity
between patches
and . Algorithm 1 describes the imple-
mentation flowchart of NLM.
Algorithm 1 Non-Local Means (NLM)
Input:Noisy image and paramers .
Output: Denoised im age
.
Step 1: Extract patch
with radius around every pixel
.
Step 2: For every pixel
,do
a) Set
for
every
.
b) Find
.
c) Assign
the value of the center pixel
in
.
where
is the search window centered at pixel and de-
note the radii of the window and patch respectively.
is the
Eucli
dean norm.
1070-9908/$31.00 © 2013 IEEE
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