非局部均值算法：图像去噪新策略

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非局部均值算法（Non-Local Means, NLM）是一种在图像去噪领域备受关注的方法，特别是在21世纪初，它作为一种新型的图像处理技术引起了广泛的研究兴趣。这篇综述性的论文发表于2005年，由A. Bua\`{e}s、B. Coll和J. M. Morel合著，发表在《工业与应用数学学会杂志》上。该文章的背景指出，尽管近年来提出了一系列复杂的图像去噪算法，但它们普遍面临一个挑战：如何在满足实际应用需求的同时，既能有效去除噪声，又能保留图像的细节结构。论文的核心目标有两个：首先，作者提出了一种通用的数学和实验方法，旨在对比和分类传统的图像去噪算法，通过量化“方法噪声”，即原始图像与去噪后图像之间的差异，来评估算法的性能。这种方法论的建立有助于客观评价算法在各种情况下的适应性和有效性。其次，论文重点介绍了非局部均值算法。NL-means算法的创新之处在于其非局部性，即它不局限于邻域内的像素处理，而是考虑了整个图像中相似特征区域的信息。这种全局策略使得NL-means能够在去噪过程中更好地捕捉图像中的纹理和结构，避免了传统算法可能产生的边缘模糊或过度平滑的问题。实际上，作者证明了NL-means算法在理想条件下具有渐近最优性，这意味着它理论上能够达到最佳的去噪效果，同时最小化对图像结构的破坏。通过深入的理论分析和实证研究，本文不仅提升了对图像去噪算法的理解，而且推动了NL-means算法在后续的科研和实际应用中的广泛应用，尤其是在卫星遥感、医学影像分析、数字摄影等领域，NL-means因其出色的结构保持能力和鲁棒性而成为首选的去噪工具。此外，论文还启发了其他基于非局部策略的图像处理技术的发展，如深度学习中的卷积神经网络（CNN）中的自注意力机制，这些都进一步拓展了图像处理的技术边界。

ON IMAGE DENOISING ALGORITHMS 497

In other terms, the standard deviation of the noise n is multiplied by a factor at most

equal to (

√

2πh

)

1/2

, this maximal value being attained in the diagonals.

Proof. Let L be the line x + t

⊥

(x)

|Du(x)|

passing by x, parameterized by t ∈ R,

and denote by P

, i ∈ I, the pixels which meet L, n(i) the noise value, constant on

pixel P

, and ε

the length of the intersection of L ∩ P

. Denote by g(i) the average

of G

(x + t

⊥

(x)

|Du(x)|

)onL ∩ P

. Then one has

n(x) 



n(i)g(i).

The n(i) are i.i.d. with standard variation σ, and therefore

Var(AF

(n)) =



g(i)

≤ σ

max(ε

)



g(i)

This yields

Var (AF

(n)) ≤

√

2εσ



(t)

dt =

√

2πh

There are many versions of AF

, all yielding an asymptotic estimate equivalent

to the one in Theorem 2.3: the famous median ﬁlter [14], an inf-sup ﬁlter on segments

centered at x [5], and the clever numerical implementation of the mean curvature

equation in [21]. So all of those ﬁlters have in common the good preservation of edges,

but they perform poorly on ﬂat regions and are worse there than a Gaussian blur.

This fact derives from the comparison of the noise reduction estimates of Theorems

2.1 and 2.4 and is experimentally patent in Figure 3.

2.3. Total variation. The total variation minimization was introduced by Ru-

din and Osher [29] and Rudin, Osher, and Fatemi [30]. The original image u is

supposed to have a simple geometric description, namely a set of connected sets, the

objects, along with their smooth contours, or edges. The image is smooth inside

the objects but with jumps across the boundaries. The functional space modeling

these properties is BV (Ω), the space of integrable functions with ﬁnite total variation

(u)=



|Du|, where Du is assumed to be a Radon measure. Given a noisy image

v(x), the above-mentioned authors proposed to recover the original image u(x)asthe

solution of the constrained minimization problem

arg min

(u),(2.1)

subject to the noise constraints



(u(x) − v(x))dx = 0 and



|u(x) − v(x)|

dx = σ

The solution u must be as regular as possible in the sense of the total variation,

while the diﬀerence v(x) −u(x) is treated as an error, with a prescribed energy. The

constraints prescribe the right mean and variance to u − v but do not ensure that

it is similar to a noise (see a thorough discussion in [22]). The preceding problem is

naturally linked to the unconstrained problem

arg min

(u)+λ



|v(x) − u(x)|

dx(2.2)

498 A. BUADES, B. COLL, AND J. M. MOREL

for a given Lagrange multiplier λ. The above functional is strictly convex and lower

semicontinuous with respect to the weak-star topology of BV . Therefore the minimum

exists, is unique, and is computable (see, e.g., [6]). The parameter λ controls the

tradeoﬀ between the regularity and ﬁdelity terms. As λ gets smaller the weight of

the regularity term increases. Therefore λ is related to the degree of ﬁltering of the

solution of the minimization problem. Let us denote by TVF

(v) the solution of

problem (2.2) for a given value of λ. The Euler–Lagrange equation associated with

the minimization problem is given by

(u(x) − v(x)) −

2λ

curv (u )(x)=0

(see [29]). Thus, we have the following theorem.

Theorem 2.5. The image method noise of the total variation minimization (2.2)

u(x) − TVF

(u)(x)=−

2λ

curv (TVF

(u))(x).

As in the anisotropic case, straight edges are maintained because of their small

curvature. However, details and texture can be oversmoothed if λ is too small, as is

shown in Figure 3.

2.4. Iterated total variation reﬁnement. In the original total variation model

the removed noise, v(x) − u(x), is treated as an error and is no longer studied. In

practice, some structures and texture are present in this error. Several recent works

have tried to avoid this eﬀect [35, 25].

2.4.1. The Tadmor–Nezzar–Vese approach. In [35], the authors have pro-

posed to use the Rudin–Osher–Fatemi model iteratively. They decompose the noisy

image, v = u

+ n

, by the total variation model. So taking u

to contain only ge-

ometric information, they decompose by the very same model n

= u

+ n

, where

is assumed to be again a geometric part and n

contains less geometric informa-

tion than n

. Iterating this process, one obtains u = u

+ u

+ ···+ u

as a

reﬁned geometric part and n

as the noise residue. This strategy is in some sense

close to the matching pursuit methods [20]. Of course, the weight parameter in the

Rudin–Osher–Fatemi model has to grow at each iteration, and the authors propose a

geometric series λ, 2λ,...,2

λ. In that way, the extraction of the geometric part n

becomes twice more taxing at each step. Then the new algorithm is as follows:

1. Starting with an initial scale λ = λ

v = u

+ n

, [u

] = arg min

v =u +n



|Du| + λ



|v(x) − u(x)|

dx.

2. Proceed with successive applications of the dyadic reﬁnement n

= u

j+1

] = arg min

=u+n



|Du| + λ

j+1



(x) − u(x)|

dx.

3. After k steps, we get the following hierarchical decomposition of v:

v = u

+ n

= u

+ u

+ n

= .....

= u

+ u

+ ···+ u

+ n

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