Results 1  10
of
150
A review of image denoising algorithms, with a new one
 Simul
, 2005
"... Abstract. 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 outstand ..."
Abstract

Cited by 297 (2 self)
 Add to MetaCart
(Show Context)
Abstract. 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 (NLmeans) 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 NLmeans 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; perceptualmathematical: 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.
The Digital TV Filter and Nonlinear Denoising
 IEEE Trans. Image Process
, 2001
"... Motivated by the classical TV (total variation) restoration model, we propose a new nonlinear filterthe digital TV filter for denoising and enhancing digital images, or more generally, data living on graphs. The digital TV filter is a data dependent lowpass filter, capable of denoising data witho ..."
Abstract

Cited by 115 (14 self)
 Add to MetaCart
(Show Context)
Motivated by the classical TV (total variation) restoration model, we propose a new nonlinear filterthe digital TV filter for denoising and enhancing digital images, or more generally, data living on graphs. The digital TV filter is a data dependent lowpass filter, capable of denoising data without blurring jumps or edges. In iterations, it solves a global total variational optimization problem, which differs from most statistical filters. Applications are given in the denoising of onedimensional (1D) signals, twodimensional (2D) data with irregular structures, gray scale and color images, and nonflat image features such as chromaticity.
A new alternating minimization algorithm for total variation image reconstruction
 SIAM J. IMAGING SCI
, 2008
"... We propose, analyze and test an alternating minimization algorithm for recovering images from blurry and noisy observations with total variation (TV) regularization. This algorithm arises from a new halfquadratic model applicable to not only the anisotropic but also isotropic forms of total variati ..."
Abstract

Cited by 109 (17 self)
 Add to MetaCart
(Show Context)
We propose, analyze and test an alternating minimization algorithm for recovering images from blurry and noisy observations with total variation (TV) regularization. This algorithm arises from a new halfquadratic model applicable to not only the anisotropic but also isotropic forms of total variation discretizations. The periteration computational complexity of the algorithm is three Fast Fourier Transforms (FFTs). We establish strong convergence properties for the algorithm including finite convergence for some variables and relatively fast exponential (or qlinear in optimization terminology) convergence for the others. Furthermore, we propose a continuation scheme to accelerate the practical convergence of the algorithm. Extensive numerical results show that our algorithm performs favorably in comparison to several stateoftheart algorithms. In particular, it runs orders of magnitude faster than the Lagged Diffusivity algorithm for totalvariationbased deblurring. Some extensions of our algorithm are also discussed.
A Variational Method In Image Recovery
 SIAM J. Numer. Anal
, 1997
"... This paper is concerned with a classical denoising and deblurring problem in image recovery. Our approach is based on a variational method. By using the LegendreFenchel transform, we show how the nonquadratic criterion to be minimized can be split into a sequence of halfquadratic problems easier t ..."
Abstract

Cited by 108 (22 self)
 Add to MetaCart
(Show Context)
This paper is concerned with a classical denoising and deblurring problem in image recovery. Our approach is based on a variational method. By using the LegendreFenchel transform, we show how the nonquadratic criterion to be minimized can be split into a sequence of halfquadratic problems easier to solve numerically. First we prove an existence and uniqueness result, and then we describe the algorithm for computing the solution and we give a proof of convergence. Finally, we present some experimental results for synthetic and real images.
An iterative regularization method for total variationbased image restoration, Multiscale Model
 Simul
"... Abstract. We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total varia ..."
Abstract

Cited by 101 (20 self)
 Add to MetaCart
(Show Context)
Abstract. We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total variation regularization. We obtain rigorous convergence results and effective stopping criteria for the general procedure. The numerical results for denoising appear to give significant improvement over standard models, and preliminary results for deblurring/denoising are very encouraging.
Euler's Elastica And Curvature Based Inpaintings
 SIAM J. Appl. Math
, 2002
"... Image inpainting is a special image restoration problem for which image prior models play a crucial role. Euler's elastica was first introduced by Mumford [21] to computer vision as a prior curve model. By functionalizing the elastica energy, Masnou and Morel [19] proposed an elastica based var ..."
Abstract

Cited by 96 (25 self)
 Add to MetaCart
Image inpainting is a special image restoration problem for which image prior models play a crucial role. Euler's elastica was first introduced by Mumford [21] to computer vision as a prior curve model. By functionalizing the elastica energy, Masnou and Morel [19] proposed an elastica based variational inpainting model. The current paper is intended to contribute to the development of its mathematical foundation, and the study of its properties and connections to the earlier works of Bertalmio, Sapiro, Caselles, and Ballester [2] and Chan and Shen [6, 7]. A computational scheme based on numerical PDEs is presented, which allows the handling of topologically complex inpainting domains.
Variational Restoration Of Nonflat Image Features: Models And Algorithms
, 2000
"... We develop both mathematical models and computational algorithms for variational denoising and restoration of nonflat image features. Nonflat image features are those that live on Riemannian manifolds, instead of on the Euclidean spaces. Familiar examples include the orientation feature (from optica ..."
Abstract

Cited by 87 (15 self)
 Add to MetaCart
(Show Context)
We develop both mathematical models and computational algorithms for variational denoising and restoration of nonflat image features. Nonflat image features are those that live on Riemannian manifolds, instead of on the Euclidean spaces. Familiar examples include the orientation feature (from optical flows or gradient flows) that lives on the unit circle S&sup1;, the alignment feature (from fingerprint waves or certain texture images) that lives on the real projective line RP&sup1; and the chromaticity feature (from color images) that lives on the unit sphere S&sup2;. In this paper, we apply the variational method to denoise and restore general nonflat image features. Mathematical models for both continuous image domains and discrete domains (or graphs) are constructed. Riemannian objects such as metric, distance and LeviCivita connection play important roles in the models. Computational algorithms are also developed for the resulting nonlinear equations. The mathematical framework can be applied to restoring general nonflat data outside the scope of image processing and computer vision.
Digital inpainting based on the MumfordShahEuler image model
 European J. Appl. Math
, 2002
"... Abstract. Image inpainting is an image restoration problem, in which image models play a critical role, as demonstrated by Chan, Kang and Shen’s recent inpainting schemes based on the bounded variation [10] and the elastica [9] image models. In the present paper, we propose two novel inpainting mode ..."
Abstract

Cited by 68 (24 self)
 Add to MetaCart
(Show Context)
Abstract. Image inpainting is an image restoration problem, in which image models play a critical role, as demonstrated by Chan, Kang and Shen’s recent inpainting schemes based on the bounded variation [10] and the elastica [9] image models. In the present paper, we propose two novel inpainting models based on the MumfordShah image model [37], and its high order correction — the MumfordShahEuler image model. We also present their efficient numerical realization based on the ¡ and De Giorgi [18]. Key words. Inpainting, Bayesian, image model, Euler’s elastica, ¡
Practical and Theoretical Aspects of Adjoint Parameter Estimation and Identifiability in . . .
, 1997
"... The present paper has two aims. One is to survey briefly the state of the art of parameter estimation in meteorology and oceanography in view of applications of 4D variational data assimilation techniques to inverse parameter estimation problems, which bear promise of serious positive impact on imp ..."
Abstract

Cited by 57 (4 self)
 Add to MetaCart
The present paper has two aims. One is to survey briefly the state of the art of parameter estimation in meteorology and oceanography in view of applications of 4D variational data assimilation techniques to inverse parameter estimation problems, which bear promise of serious positive impact on improving model prediction. The other aim is to present crucial aspects of identifiability and stability essential for validating results of optimal parameter estimation and which have not been addressed so far in either the meteorological or the oceanographic literature. As noted by Yeh (1986, Water Resour. Res. 22, 95108) in the context of ground water flow parameter estimation the inverse or parameter estimation problem is often illposed and beset by instability and nonuniqueness, particularly if one seeks parameters distributed in spacetime domain. This approach will allow one to assess and rigorously validate results of parameter estimation, i.e. do they indeed represent a real identification of physical model parameters or just compensate model errors? A brief survey of other approaches for solving the problem of optimal parameter estimation in meteorology and oceanography is finally presented. 1997 Elsevier Science B.V.
Diffusion of General Data on NonFlat Manifolds via Harmonic Maps Theory: The Direction Diffusion Case
 Journal Computer Vision
, 2000
"... Abstract. In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to denoise directional data and to obtain multiscale representati ..."
Abstract

Cited by 56 (6 self)
 Add to MetaCart
Abstract. In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to denoise directional data and to obtain multiscale representations of it. The basic idea is to apply and extend results from the theory of harmonic maps, and in particular, harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data. We show the corresponding variational and partial differential equations formulations for isotropic diffusion, obtained from an L2 norm, and edge preserving diffusion, obtained from an L p norm in general and an L1 norm in particular. In contrast with previous approaches, the framework is valid for directions in any dimensions, supports nonsmooth data, and gives both isotropic and anisotropic formulations. In addition, the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general nonflat manifolds, that is, functions between two general manifolds. We present a number of theoretical results, open questions, and examples for gradient vectors, optical flow, and color images.