On image denoising methods Antoni Buades *† Bartomeu Coll * Jean Michel Morel † Abstract The search for efficient image denoising methods still is 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, second, to propose an algorithm (Non Local Means) 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 also 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. 1 Introduction 1.1 Digital images and noise The need for efficient image restoration methods has grown with the massive production of digital images and movies of all kinds, often taken in poor conditions. No matter how good cameras are, an image improvement is always desirable to extend their range of action. A digital image is generally encoded as a matrix of grey level or color values. In the case of a movie, this matrix has three dimensions, the third one corresponding to time. Each pair (i, u(i)) where u(i) is the value at i is called pixel, for “picture element”. In the case of grey level images, i is a point on a 2D grid and u(i) is a real value. In the case of classical color images, u(i) is a triplet of values for the red, green and blue components. All of what we shall say applies identically to movies, 3D images and color or multispectral images. For a sake of simplicity in notation and display of experiments, we shall here be contented with rectangular 2D grey-level images. The two main limitations in image accuracy are categorized as blur and noise. Blur is intrinsic to image acquisition systems, as digital images have a finite number of samples and must respect the Shannon-Nyquist sampling conditions [34]. The second main image perturbation is noise. Each one of the pixel values u(i) is the result of a light intensity measurement, usually made by a CCD matrix coupled with a light focusing system. Each captor of the CCD is roughly a square * Universitat de les Illes Balears, Ctra. Valldemossa Km. 7.5, 07122 Palma de Mallorca, Spain † Centre de Math´ ematiques et Leurs Applications. ENS Cachan 61, Av du Pr´ esident Wilson 94235 Cachan, France 1