改进的非局部欧几里得中值：提升图像去噪性能

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本文主要探讨了"非局部欧氏中值分析及其改进"这一主题，发表在2013年4月的《IEEE信号处理快报》第20卷第4期上。作者Zhonggui Sun和Songcan Chen针对近年来备受关注的非局部图像去噪方法——非局部均值（NLM）提出了一种新的观点。他们发现，尽管NLM在处理重度噪声方面表现出色，但其内在的两个相似度度量之间的不一致性可能影响其稳健性。传统NLM利用全局信息计算像素点的加权平均，这使得它在处理图像噪声时具有优势。然而，论文指出这种全局策略在某些情况下可能导致处理结果的不一致性。为了弥补这一缺陷，作者提出了改进的非局部欧氏中值算法（INLEM）。INLEM旨在通过修正原有的不一致度量，提高算法的鲁棒性和去噪效果。文中还对NLM和INLEM中使用的迭代算法进行了深入分析，并提供了简洁的收敛性证明。这意味着读者可以更好地理解这两种方法的算法行为和性能保障。实验部分，作者在合成图像和自然图像上进行了大量对比测试，结果显示INLEM在去噪效果上取得了鼓舞人心的成果。关键词包括"改进的非局部欧氏中值"（Improved Non-Local Euclidean Medians, INLEM）、"图像去噪"、"非局部欧氏中值"（Non-Local Euclidean Medians, NLEM）以及"非局部均值"（Non-Local Means, NLM）。这些关键词表明了本文的核心研究内容和贡献，即在保持NLM优点的基础上，通过改进策略提升图像处理的精度和稳定性。这篇论文为非局部图像处理技术的发展提供了一个关键的进展，特别是在处理复杂噪声场景下，INLEM展示了其在提高图像质量方面的潜力，对于该领域的研究人员和实践者来说，具有重要的参考价值。

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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