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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

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

where

is the search window centered at pixel and de-

note the radii of the window and patch respectively.

is the

Eucli

dean norm.

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