The search for efficient image denoising methods is still
a valid challenge at the crossing of functional analysis
and statistics. In spite of the sophistication of the recently
proposed methods, most algorithms have not yet attained a
desirable level of applicability. All show an outstanding performance when the image model corresponds to the algorithm assumptions but fail in general and create artifacts or
remove image fine structures. The main focus of this paper
is, first, to define a general mathematical and experimental methodology to compare and classify classical image
denoising algorithms and, second, to describe the nonlocal means (NL-means) algorithm6 introduced in 2005 and
its more recent extensions. The mathematical analysis is
based on the analysis of the “method noise,” defined as the
difference between a digital image and its denoised version.
NL-means, which uses image self-similarities, is proven to
be asymptotically optimal under a generic statistical image
model. The denoising performance of all considered methods are compared in four ways: mathematical, asymptotic
order of magnitude of the method noise under regularity
assumptions; perceptual-mathematical, the algorithms
artifacts and their explanation as a violation of the image
model; perceptual-mathematical, analysis of algorithms
when applied to noise samples; quantitative experimental, by tables of L2 distances of the denoised version to the
Formally we define a denoising method Dh as a
v = Dhv + n(Dh, v),
where v is the noisy image and h is a filtering parameter,
which usually depends on the standard deviation of the
noise. Ideally, Dhv is smoother than v and n(Dh, v) looks like
the realization of a white noise.
The goal of image denoising methods is to recover the original image from a noisy measurement:
v(i ) = u(i ) + n(i ),
where v(i) is the observed value, u (i) is the “true” value, and
n(i) is the noise perturbation at a pixel i. The best simple
way to model the effect of noise on a digital image is to add
a Gaussian white noise. In that case, n(i) are i.i.d. Gaussian
values with zero mean and variance s 2.
2. methoD noIse
Definition 1 (Method noise). Let u be an image and Dh a
denoising operator depending on a filtering parameter h. Then,
we define the method noise as the image difference u − Dhu.
The original version of this paper is entitled “A Review
of Image Denoising Algorithms, with a New One.” It was
published in Multiscale Modeling and Simulation, 2005.