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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 propose a nonlocal means (NL-means) algorithm addressing the preservation of structure in a digital image. The mathematical analysis is based on the analysis of the "method noise," defined as the difference between a digital image and its denoised version. The NL-means algorithm 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; quantitative experimental: by tables of L 2 distances of the denoised version to the original image. The most powerful evaluation method seems, however, to be the visualization of the method noise on natural images. The more this method noise looks like a real white noise, the better the method.

Neighborhood filters are nonlocal image and movie filters which reduce the noise by averaging similar pixels. The first object of the paper is to present a unified theory of these filters and reliable criteria to compare them to other filter classes. A CCD noise model will be presented justifying the involvement of neighborhood filters. A classification of neighborhood filters will be proposed, including classical image and movie denoising methods and discussing further a recently introduced neighborhood filter, NL-means. In order to compare denoising methods three principles will be discussed. The first principle, "method noise", specifies that only noise must be removed from an image. A second principle will be introduced, "noise to noise", according to which a denoising method must transform a white noise into a white noise. Contrarily to "method noise", this principle, which characterizes artifact-free methods, eliminates any subjectivity and can be checked by mathematical arguments and Fourier analysis. "Noise to noise" will be proven to rule out most denoising methods, with the exception of neighborhood filters. This is why a third and new comparison principle, the "statistical optimality", is needed and will be introduced to compare the performance of all neighborhood filters.The three principles will be applied to compare ten different image and movie denoising methods. It will be first shown that only wavelet thresholding methods and NL-means give an acceptable method noise. Second, that neighborhood filters are the only ones to satisfy the "noise to noise" principle. Third, that among them NL-means is closest to statistical optimality. A particular attention will be paid to the application of the statistical optimality criterion for movie denoising methods. It will be pointed out that current movie denoising methods are motion compensated neighborhood filters. This amounts to say that they are neighborhood filters and that the ideal neighborhood of a pixel is its trajectory. Unfortunately the aperture problem makes it impossible to estimate ground true trajectories. It will be demonstrated that computing trajectories and restricting the neighborhood to them is harmful for denoising purposes and that space-time NL-means preserves more movie details.

We present in this paper a new denoising method called non-local means. The method is based on a simple principle: replacing the color of a pixel with an average of the colors of similar pixels. But the most similar pixels to a given pixel have no reason to be close at all. It is therefore licit to scan a vast portion of the image in search of all the pixels that really resemble the pixel one wants to denoise. The paper presents two implementations of the method and displays some results. Source CodeThe source code (ANSI C), its documentation, and the online demo are accessible at the IPOL web page of this article 1 . Some of the files use algorithms possibly linked to patent [3]. These files are made available for the exclusive aim of serving as scientific tool to verify the soundness and completeness of the algorithm description. Compilation, execution and redistribution of these files may violate exclusive patents rights in certain countries. The situation being different for every country and changing over time, it is your responsibility to determine which patent rights restrictions apply to you before you compile, use, modify, or redistribute these files. The rest of files are distributed under GPL license. A C/C++ implementation is provided. Please see the readme file or the online documentation for details.

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State of the art movie restoration methods either estimate motion and filter out the trajectories, or compensate the motion by an optical flow estimate and then filter out the compensated movie. Now, the motion estimation problem is ill-posed. This fact is known as the aperture problem: trajectories are ambiguous since they could coincide with any promenade in the space-time isophote surface. In this paper, we try to show that, for denoising, the aperture problem can be taken advantage of. Indeed, by the aperture problem, many pixels in the neighboring frames are similar to the current pixel one wishes to denoise. Thus, denoising by an averaging process can use many more pixels than just the ones on a single trajectory. This observation leads to use for movies a recently introduced image denoising method, the NL-means algorithm. This static 3D algorithm outperforms motion compensated algorithms, as it does not lose movie details. It involves the whole movie isophote and not just a trajectory.

Abstract-Many classical image denoising methods are based on a local averaging of the color, which increases the signal/noise ratio. One of the most used algorithms is the neighborhood filter by Yaroslavsky or sigma filter by Lee, also called in a variant "SUSAN" by Smith and Brady or "Bilateral filter" by Tomasi and Manduchi. These filters replace the actual value of the color at a point by an average of all values of points which are simultaneously close in space and in color. Unfortunately, these filters show a "staircase effect", that is, the creation in the image of flat regions separated by artifact boundaries. In this paper, we first explain the staircase effect by finding the subjacent PDE of the filter. We show that this ill-posed PDE is a variant of another famous image processing model, the Perona-Malik equation, which suffers the same artifacts. As we prove, a simple variant of the neighborhood filter solves the problem. We find the subjacent stable PDE of this variant. Finally, we apply the same correction to the recently introduced NL-means algorithm which had the same staircase effect, for the same reason.Edics: 2-NFLT Nonlinear Filtering and Enhancement. 2-REST Restoration.

Denoising images can be achieved by a spatial averaging of nearby pixels. However, although this method removes noise it creates blur. Hence, neighborhood filters are usually preferred. These filters perform an average of neighboring pixels, but only under the condition that their grey level is close enough to the one of the pixel in restoration. This very popular method unfortunately creates shocks and staircasing effects. In this paper, we perform an asymptotic analysis of neighborhood filters as the size of the neighborhood shrinks to zero. We prove that these filters are asymptotically equivalent to the Perona-Malik equation, one of the first nonlinear PDE's proposed for image restoration. As a solution, we propose an extremely simple variant of the neighborhood filter using a linear regression instead of an average. By analyzing its subjacent PDE, we prove that this variant does not create shocks: it is actually related to the mean curvature motion. We extend the study to more general local polynomial estimates of the image in a grey level neighborhood and introduce two new fourth order evolution equations.