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这篇文章的核心内容是关于一种图像去噪算法——非局部均值（Non-Local Means, NLM）算法的快速且开源的实现。以下是文章的主要要点：算法介绍：文章提出了一种快速且无参数的NLM算法实现，该算法用于去除图像中的噪声。NLM算法由Antoni Buades, Bartomeu Coll和Jean-Michel Morel在2005年引入，因其简单性、出色的视觉效果以及利用自然图像的非局部冗余性而受到广泛欢迎。算法改进：文章基于线之和的计算来计算块距离，这些线在块移位下是不变的。通过从一个图像数据库中计算NLM的最佳参数（以平均峰值信噪比PSNR为标准），实现了一个无需参数调整的NLM算法。 ### Parameter-Free Fast Pixelwise Non-Local Means Denoising #### 概述本文献提出了一种快速且开源的实现方式，旨在改进广受好评的非局部均值（Non-Local Means, NLM）去噪算法。该算法由Antoni Buades、Bartomeu Coll与Jean-Michel Morel于2005年首次提出，因其简单性和优秀的视觉效果而受到关注，特别是它充分利用了自然图像中普遍存在的非局部相似性（non-local redundancy）。本文的核心贡献在于提供了一种参数自由（parameter-free）的NLM算法实现，并且显著提高了运算速度。 #### 算法介绍 - **非局部均值算法原理**：NLM算法的基本思想是将每个像素与其周围像素组成的区域（通常称为“patch”或“窗口”）与其他像素的相应区域进行比较，从而找到具有相似特征的区域。这些相似区域的加权平均值用于估计当前像素的真实值，以此来去除噪声。 - **算法优化**：为了提高计算效率并简化参数设置过程，本文提出了一种基于“线之和”的块距离计算方法。这种方法的关键在于利用了线的平移不变性，即当两个块之间存在相对平移时，可以通过它们之间的水平和垂直线的累加值得到块间的相似度。这种优化不仅减少了计算量，还避免了对特定参数的手动调整。 - **参数自适应策略**：通过从一个包含多种图像类型的数据库中计算出最优的NLM参数（以平均峰值信噪比PSNR为标准），该算法实现了真正的参数自由。这意味着用户无需手动调整任何参数即可获得良好的去噪效果，极大地降低了使用门槛，同时也使得算法更加易于集成到其他应用中。 #### 实验结果分析 - **性能对比**：与Buades等人先前提出的块级（blockwise）NLM算法相比，尽管在较高噪声水平下块级NLM算法能够提供更好的峰值信噪比（PSNR），但在实际视觉效果上两者并无显著差异。这表明像素级（pixelwise）NLM算法在大多数情况下都能达到满意的去噪效果。 - **运行速度提升**：值得注意的是，本文提出的像素级NLM算法在运行速度上有显著优势，其运行速度是块级NLM算法的6至49倍。这对于实时或大规模图像处理应用而言非常重要。 #### 结论本文介绍了一种快速且参数自由的像素级NLM去噪算法实现。通过采用基于线之和的计算方法来优化块间距离的计算，以及通过对大量图像数据集进行预训练来确定最优参数，该算法能够在保证去噪质量的同时大幅提高处理速度。这种改进不仅降低了用户的使用难度，也使得NLM算法更适用于实时图像处理场景。对于那些寻求高效、易用且高质量去噪解决方案的研究人员和开发者来说，这一研究成果极具价值。

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Published in Image Processing On Line on 2014–11–19.

Submitted on 2014–07–18, accepted on 2014–08–05.

ISSN 2105–1232

 2014 IPOL & the authors CC–BY–NC–SA

This article is available online with supplementary materials,

software, datasets and online demo at

https://doi.org/10.5201/ipol.2014.120

2015/06/16 v0.5.1 IPOL article class

Parameter-Free Fast Pixelwise Non-Local Means Denoising

Jacques Froment

Univ. Bretagne - Sud, UMR 6205, LMBA, F-56000 Vannes, France (Jacques.Froment@univ-ubs.fr)

Communicated by Bartomeu Coll Demo edited by Jacques Froment

Abstract

This article proposes a fast and open-source implementation of the well-known Non-Local Means

(NLM) denoising algorithm, in its original pixelwise formulation. The fast implementation is

based on the computation of patch distances using sums of lines that are invariant under a

patch shift. The optimal parameters of NLM (in the average peak signal to noise ratio - PSNR

- sense) are computed from an image database, thereby leading to a parameter-free NLM imple-

mentation. Comparison is performed with the parameter-free blockwise NLM implementation

already proposed in IPOL journal by Buades, Coll and Morel. As expected the blockwise imple-

mentation oﬀers better PSNR, at least when the noise standard deviation is large enough, but

there is no signiﬁcant diﬀerence in quality when performing visual inspection. The highlight is

that the proposed parameter-free pixelwise NLM implementation is faster than the patchwise

one by a factor of 6 to 49.

Source Code

The reviewed source code and documentation for the parameter-free fast pixelwise NLM algo-

rithm are available from the web page of this article

. Compilation and usage instructions are

included in the README.txt ﬁle of the archive.

Keywords: image denoising; non-local means

1 Introduction

The Non-Local Means (NLM) image denoising algorithm was introduced in 2005 by Antoni Buades,

Bartomeu Coll and Jean-Michel Morel [1] and the success was such that this method has inspired

a great number of variants and articles, see [3] for some updated references. Three factors largely

explain why the image denoising community has been immersed in the NLM approach: the original

algorithm remains simple; it provides great visual quality; it introduces a basic tool to exploit

the non-local redundancy of natural images. Despite the subsequent introduction of more eﬃcient

denoising algorithms (see [20, 16] for recent comparisons between NLM and some state of the art

https://doi.org/10.5201/ipol.2014.120

Jacques Froment, Parameter-Free Fast Pixelwise Non-Local Means Denoising, Image Processing On Line, 4 (2014), pp. 300–326.

https://doi.org/10.5201/ipol.2014.120

Parameter-Free Fast Pixelwise Non-Local Means Denoising

denoising schemes), NLM remains a reference method for image denoising. It is therefore essential

that the image denoising community has freely available a documented, veriﬁed and open-source

implementation of NLM in its original version.

Despite the number of NLM codes that can be found in the Internet, it seems that there are only

two implementations that fulﬁll these conditions: the nlmeans module included in MegaWave2 [11]

and written by Lionel Moisan and the IPOL article [4] from Buades, Coll and Morel. The ﬁrst code

implements the pixelwise NLM algorithm given in [1] (Section 5.1) for gray-level images only and

using the Euclidean norm instead of the uniform one to deﬁne neighbor patches. The parameters’

default values correspond to those in [1], but they turn out to be non optimal for most images.

Finally, this code does not implement any algorithmic trick or parallelization of computing, making

it a slow running program. While the IPOL article [4] describes both the pixelwise NLM (NLM-P)

([1] Section 5.1) and the blockwise NLM (NLM-B) ([1] Section 5.5.2), the published demo and code

implement the blockwise version only

The present article aims to propose a fast and parameter-free implementation of the pixelwise

NLM as described in [1] (pages 510-512). In order to ﬁx the notation, let us ﬁrst recall the equations

that lead to this algorithm.

The input, noisy image v, is assumed to come from the classical additive noise model

v(x) = u(x) + (x), x ∈ Ω, (1)

where u is the original image, Ω is the set of pixels and  is the noise perturbation: ((x))

x∈Ω

are independent and identically distributed (i.i.d.) Gaussian variables with mean 0 and standard

deviation σ > 0. In the case of gray-level images, u and v take real values (integer values in the

discrete model) whereas for color images, u and v are vector-valued.

The output, denoised image, is the estimator ˜u computed as

˜u(x) =

y∈Ω

w(x, y)v(y), (2)

where the weight w(x, y) estimates the similarity between the pixel x and y in the original image

and the sequence of weights satisﬁes

w(x, y) ≥ 0 and

y∈Ω

w(x, y) = 1, ∀x ∈ Ω, y ∈ Ω

. (3)

The set of pixels Ω

is a neighborhood of x ∈ Ω, it is deﬁned as

Ω

= {y ∈ Ω : kx − yk

∞

≤ D}, (4)

for D > 0 a ﬁxed size parameter. The window Ω

is called the search window at x and, for simplicity

and faster computations, in [1] a choice of a square shape of ﬁxed size is made while, according to

authors, the search window should cover the entire image plane, hence the non-local nature of the

algorithm. However, it has been reported that using for NLM a neighborhood instead of the whole

image plane allows to increase the denoising performance [12, 13, 21, 22], see also the discussion in [9]

and the speciﬁc study in [19] where it is experimentally established that the optimal window size D

is very small, when using a variant of the pixelwise NLM. As this present article will establish the

best parameters, it will give an answer for the original pixelwise NLM.

This is the following particular form of weights w(x, y) that makes the diﬀerence between NLM

and classical denoising methods known as neighborhood ﬁlters:

NLM-Pa

(x, y) =

n(x)

−

kV (x)−V (y)k

2,a

, (5)

Blockwise NLM is called patchwise NLM in [4].

301

Jacques Froment

where n(x) is a generic normalization factor and h > 0 is a ﬁltering parameter. We write here the

denominator of the fraction in the exponential as h

, thus following the notation in [3, 1, 4], whereas

one reads 2h

in most subsequent articles about NLM.

The estimation of the oracle |u(x) − u(y)| is performed using the Gaussian Euclidean norm

kV (x) − V (y)k

2,a

, where a is the standard deviation of the Gaussian and where V (x) is the patch of

size d × d centered at x in image v:

V (x) = {v(y) : y ∈ Ω, kx − yk

∞

≤ d}. (6)

Another signiﬁcant diﬀerence with some subsequent articles is in the fact that weights are computed

without subtracting 2σ

to the square of this norm. Such an alternative is written

NLM-Pa-variant

(x, y) =

n(x)

−

max{kV (x)−V (y)k

2,a

−2σ

,0}

(7)

and this variant is motivated by the fact that

E{kV (x) − V (y)k

2,a

} = kU(x) − U(y)k

2,a

+ 2σ

, (8)

where the expectation is taken over the noise distribution. The choice to consider (5) rather than (7)

is related to the concern to implement the original NLM algorithm. Note that in [4] and unlike [1, 3],

the authors do subtract 2σ

to the square of the norm.

In the discrete setting, the Gaussian Euclidean norm writes

kV (x) − V (y)k

2,a

{z∈Z

:kzk

∞

≤d

}

(z) kv(x + z) − v(y + z)k

(9)

with

(z) =

−

kzk

{t∈Z

:ktk

∞

≤d

}

−

ktk

(10)

and d

= (d − 1)/2 being the patch side half-length, d being an odd number. In the right-hand of

Equation (9) the symbol kv(x)k

must be understood as the Euclidean norm of v(x), which is simply

the absolute value of v(x) in the case of a gray-level image. In the case of a color image, v(x) is a

vector of N

components, N

being the number of channels, and kv(x)k

is the sum of the squares of

each component.

We denote by NLM-Pa this pixelwise NLM denoising scheme, given by equations (2) and (5). The

letter “a” recalls the use of the Gaussian Euclidean norm (9) of parameter a. Besides the original

articles on NLM [3, 1], the Gaussian kernel is often dropped and the norm between patches becomes

the standard Euclidean one, being in the discrete case the mean square error between V (x) and V (y):

kV (x) − V (y)k

{z∈Z

:kzk

∞

≤d

}

kv(x + z) − v(y + z)k

. (11)

In such a case, NLM-weights write

NLM-P

(x, y) =

n(x)

−

kV (x)−V (y)k

. (12)

We denote by NLM-P the pixelwise NLM denoising scheme with the Euclidean norm instead of the

Gaussian Euclidean one. When mentioned in the literature, the reason for using NLM-P rather than

302

Parameter-Free Fast Pixelwise Non-Local Means Denoising

NLM-Pa is that the increase in quality provided by the Gaussian kernel is low, while it requires the

additional parameter a to be tuned. However, it seems that there is no experimental study using

image databases to quantify the eﬀect of the Gaussian kernel. It is a goal of this article to examine

to what extend the previous assertion is valid.

To unify the presentation of NLM-P and NLM-Pa, we may denote by K the kernel used to weight

the norm between patches (K = K

for NLM-Pa, K = 1/d

for NLM-P) and the corresponding

norm will be written k.k

2,K

. Before considering optimization, let us present the basic algorithm.

2 Basic Algorithm for the Pixelwise NLM

The pixelwise NLM method may be implemented in a straightforward manner using the above

equations. To denoise a pixel x ∈ Ω, one has to compute the patch V (y) for all y ∈ Ω

and

therefore to scan each pixel z such that ky − zk

∞

≤ d. A window of size (D + d)

centered in x is

then to be accessed and this requires adding to the borders of the input image v a strip of width

D−1

d−1

= D

. As usual in image processing, the padding is performed by mirror reﬂection and

the resulting symmetrized image will be denoted Vsym in the following pseudo-codes (to assist the

reader, Table 1 lists the main notations specifying those used in the mathematical context and those

used for pseudo and source codes). The kernel K is precomputed as an array of size d

using (10)

for NLM-Pa (or as a constant K = 1/d

for NLM-P).

The main loop consists in going through each pixel x ∈ Ω and as explained in the next section,

parallelization may be performed at this stage by assigning a speciﬁc x to a speciﬁc thread. A ﬁrst

pass of all neighboring pixels y ∈ Ω

is performed in order to compute the array of NLM-weights,

using (5) and (9) or (12) and (11). A second pass of all neighboring pixels y ∈ Ω

is performed to get

the denoised pixel ˜u(x) using the estimation in (2). Note that, in case of gray-level images, these two

passes can be merged together. A more detailed sketch of this algorithm is proposed as pseudo-code

in Algorithm 1.

From this pseudo-code, one can easily deduce the complexity of the pixelwise NLM in its ba-

sic implementation. Let us count the number of operations following the standard uniform cost

model [27]. Let N = |Ω| = N

be the number of pixels of the input image v, N

the number of

color channels, N

= (N

+ 2(D

+ d

))(N

+ 2(D

+ d

)) the number of pixels of the symmetrized

image and c a generic constant. The symmetrization of the input image requires cN

operations

and the precomputation of the kernel K, cd

operations. Inside the main loop x ∈ Ω, the ﬁrst pass

of all neighboring pixels y ∈ Ω

used to compute NLM-weights requires cN

operations, while

the second pass that computes the estimator ˜u(x) needs cN

operations only. Therefore and using

the big O notation to hide constant factors and smaller terms, the complexity of the plain pixelwise

NLM is given by that of the ﬁrst pass of the main loop, that is O(NN

3 Fast Algorithms for the Pixelwise NLM

Diﬀerent strategies can be deployed to increase the speed of pixelwise NLM. First, it should be

noted that the algorithm is particularly well suited to parallel computing, thanks to the independent

processing of each pixel x ∈ Ω to be denoised. The easiest way to implement parallel computations

is therefore to assign, in the main loop level, a speciﬁc pixel x to a speciﬁc thread. This allows to

roughly divide the execution time by the number of available threads (the gain is actually a bit lower

due to input/output processes and because of the initialization pass, although this one may also be

parallelized). In the code associated to this article, parallelization is implemented in the main loop

level only, both for the plain and the proposed fast implementations.

303

Jacques Froment

Mathematical notation Code notation Comments

d d Patch side length is d=2*ds+1

ds Patch side half-length

d2 Number of pixels of the patch

D D Search window side length is D=2*Ds+1

Ds Search window side half-length

D2 Number of pixels of the search window D

= |Ω

kV (x) − V (y)k

or Dist2 Euclidean or Gaussian Euclidean norm between two

kV (x) − V (y)k

2,a

patches, see (11) and (9)

K K 2D-Kernel of weighted Euclidean norm, see (10)

, K

K 1D-Kernel of weighted Euclidean norm (fast NLM-Pa

using sum of invariant lines), see (20)

L2 Weighted Euclidean distance between two patches at

one line (NLM-Pa using sum of invariant lines), see (22)

N N Number of pixels of the input image N = |Ω| = N

×N

N1 Number of columns of the input image (x

axis)

N2 Number of rows of the input image (x

axis)

Nc Number of channels (planes) in the input image (1 or 3)

NNc Image’s size N × Nc

NS Number of pixels of the symmetrized image:

= (N

+ D

+ d

) × (N

+ D

+ d

)

σ sigma Noise standard deviation

u U Input noiseless image

˜u Vd Output denoised image

- Vd[c] Color channel #c of the denoised image (pseudo-code)

v V Input noisy image

- Vsym Symmetrized noisy image

- Vsym[c] Color channel #c of Vsym (pseudo-code)

w W NLM-weights image

Table 1: Main mathematical notations (used in this PDF article) and main code notations (for

pseudo-codes in this PDF article and/or source codes) and correspondence between them.

Using parallelization should be regarded as a technical trick only, insofar as it does not reduce

the algorithmic complexity. More interesting strategies seek to change the way the calculations are

carried out, so that the complexity may be eﬀectively reduced. As seen before, the biggest term

that sets the complexity of the plain pixelwise NLM is given by the ﬁrst pass of the main loop

that computes the distance between patches V (x) and V (y). Therefore, strategies are committed to

reduce the cost associated with this distance computation.

The most known approaches involve estimating patches V (y) for which distance to V (x) is prob-

ably signiﬁcant or, more generally, which are dissimilar to V (x) in some sense. In such a case the

pixel y is simply removed in the neighborhood Ω

or, equivalently, the weight w(x, y) is set to 0 and

the exact distance (9) or (11) is not computed. To be useful from the point of view of complexity,

the preselection of similar patches has to be carried out with less than cN

operations. Such

approaches have been proposed in [17] (preselection using mean and gradient similarities), [5] (using

ﬁrst and second moments), [18] (using conditional probabilities and critical pixels), [23] (using a

multi-resolution decomposition). It is important to emphasize that these approaches do not imple-

ment the exact pixelwise NLM: the denoised image is diﬀerent to the one obtained using Algorithm 1.

For this reason and even if the denoised image could be of better quality, such patches preselection

304

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