A Second-Order Model for Image Denoising

We present a variational model for image denoising and/or texture identification. Noise and textures may be modelled as oscillating components of images. The model involves a L ²-data fitting term and a Tychonov-like regularization term. We choose the BV ² norm instead of the classical BV norm. Here BV ² is the bounded hessian function space that we define and describe. The main improvement is that we do not observe staircasing effects any longer, during denoising process. Moreover, texture extraction can be performed with the same method. We give existence results and present a discretized problem. An algorithm close to the one set by Chambolle (J Math Imaging Vis 20:89–97, 2004) is used: we prove convergence and present numerical tests.     

Image denoising using the Gaussian curvature of the image surface

A number of high‐order variational models for image denoising have been proposed within the last few years. The main motivation behind these models is to fix problems such as the staircase effect and the loss of image contrast that the classical Rudin–Osher–Fatemi model [Leonid I. Rudin, Stanley Osher and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), pp. 259–268] and others also based on the gradient of the image do have. In this work, we propose a new variational model for image denoising based on the Gaussian curvature of the image surface of a given image. We analytically study the proposed model to show why it preserves image contrast, recovers sharp edges, does not transform piecewise smooth functions into piecewise constant functions and is also able to preserve corners. In addition, we also provide two fast solvers for its numerical realization. Numerical experiments are shown to illustrate the good performance of the algorithms and test results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1066–1089, 2016     

广义全变差正则化图像去噪模型研究

本文研究了全变差正则化模型在图像去噪过程中易产生阶梯效应的问题,依据图像的局部结构特利用联合高斯滤波器和边缘检测算子的方法,构建了广义全变差正则化图像去噪模型,获得了在消除噪声的同时能够保留图像边缘细节和纹理信息的结果.实验结果表明,广义全变差正则化模型在平滑噪声的同时能够保留图像的边缘轮廓等细节信息,得到的复原图像在峰值信噪比、平均结构相似度和主观视觉效果方面均有所提高.     

Image Restoration and Decomposition via Bounded Total Variation and Negative Hilbert-Sobolev Spaces

We propose a new class of models for image restoration and decomposition by functional minimization. Following ideas of Y. Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, our model decomposes a given (degraded or textured) image u ₀ into a sum u+v. Here u∈BV is a function of bounded variation (a cartoon component), while the noisy (or textured) component v is modeled by tempered distributions belonging to the negative Hilbert-Sobolev space H ^−s . The proposed models can be seen as generalizations of a model proposed by S. Osher, A. Solé, L. Vese and have been also motivated by D. Mumford and B. Gidas. We present existence, uniqueness and two characterizations of minimizers using duality and the notion of convex functions of measures with linear growth, following I. Ekeland and R. Temam, F. Demengel and R. Temam. We also give a numerical algorithm for solving the minimization problem, and we present numerical results of denoising, deblurring, and decompositions of both synthetic and real images.     

Adaptive total variation regularization based scheme for Poisson noise removal

To better preserve the edge features, this paper investigates an adaptive total variation regularization based variational model for removing Poisson noise. This edge‐preserving scheme comprises a spatially adaptive diffusivity coefficient, which adjusts the diffusion strength automatically. Compared with the classical total variation based one, numerical simulations distinctly indicate the superiority of our proposed strategy in maintaining the small details while denoising Poissonian image. Copyright © 2012 John Wiley & Sons, Ltd.     

Vector total fractional-order variation and its applications for color image denoising and decomposition

In this paper, a vector total fractional-order variation (VTV-β) is proposed. Then, VTV-β model and G^β(Ω) model are proposed for color image denoising and decomposition, respectively. Some properties of the VTV-β are investigated and an alternative algorithm is used to solve the two models. Some experimental results are given to show the effectiveness and advantages of our methods.     

Modeling Very Oscillating Signals. Application to Image Processing

This article is a companion paper of a previous work where we have developed the numerical analysis of a variational model first introduced by Rudin et al. and revisited by Meyer for removing the noise and capturing textures in an image. The basic idea in this model is to decompose an image f into two components (u + v) and then to search for (u,v) as a minimizer of an energy functional. The first component u belongs to BV and contains geometrical information, while the second one v is sought in a space G which contains signals with large oscillations, i.e. noise and textures. In Meyer carried out his study in the whole ² and his approach is rather built on harmonic analysis tools. We place ourselves in the case of a bounded set of ² which is the proper setting for image processing and our approach is based upon functional analysis arguments. We define in this context the space G, give some of its properties, and then study in this continuous setting the energy functional which allows us to recover the components u and v. We present some numerical experiments to show the relevance of the model for image decomposition and for image denoising.     

Graduated adaptive image denoising: local compromise between total variation and isotropic diffusion

We introduce variants of the variational image denoising method proposed by Blomgren et al. (In: Numerical Analysis 1999 (Dundee), pp. 43–67. Chapman & Hall, Boca Raton, FL, 2000), which interpolates between total-variation denoising and isotropic diffusion denoising. We study how parameter choices affect results and allow tuning between TV denoising and isotropic diffusion for respecting texture on one spatial scale while denoising features assumed to be noise on finer spatial scales. Furthermore, we prove existence and (where appropriate) uniqueness of minimizers. We consider both L ² and L ¹ data fidelity terms.     

On a class of scalar variational inequalities with measure data

We consider a class of variational inequalities defining the so-called hysteresis play operator. We propose a new approach to discontinuous BV-solutions based on measure theoretical arguments, which enable us to infer the existence of solutions as a simple consequence of the classical theory. In this way, we generalize a recent result where only continuous BV-solutions were studied. We also provide a representation formula which allows to deduce the continuity of the play operator from general theorems on hysteresis operators.     

Functions of bounded variation on “good” metric spaces

In this paper we give a natural definition of Banach space valued BV functions defined on complete metric spaces endowed with a doubling measure (for the sake of simplicity we will say doubling metric spaces) supporting a Poincaré inequality (see Definition 2.5 below). The definition is given starting from Lipschitz functions and taking closure with respect to a suitable convergence; more precisely, we define a total variation functional for every Lipschitz function; then we take the lower semicontinuous envelope with respect to the L¹ topology and define the BV space as the domain of finiteness of the envelope. The main problem of this definition is the proof that the total variation of any BV function is a measure; the techniques used to prove this fact are typical of Γ-convergence and relaxation. In Section 4 we define the sets of finite perimeter, obtaining a Coarea formula and an Isoperimetric inequality. In the last section of this paper we also compare our definition of BV functions with some definitions already existing in particular classes of doubling metric spaces, such as Weighted spaces, Ahlfors-regular spaces and Carnot–Carathéodory spaces.     

Variational image inpainting

Inpainting is an image interpolation problem with broad applications in image and vision analysis. Described in the current expository paper are our recent efforts in developing universal inpainting models based on the Bayesian and variational principles. Discussed in detail are several variational inpainting models built upon geometric image models, the associated Euler‐Lagrange PDEs and their geometric and dynamic interpretations, as well as effective computational approaches. Novel efforts are then made to further extend this systematic variational framework to the inpainting of oscillatory textures, interpolation of missing wavelet coefficients as in the wireless transmission of JPEG2000 images, as well as light‐adapted inpainting schemes motivated by Weber's law in visual perception. All these efforts lead to the conclusion that unlike many familiar image processors such as denoising, segmentation, and compression, the performance of a variational/Bayesian inpainting scheme much more crucially depends on whether the image prior model well resolves the spatial coupling (or geometric correlation) of image features. As a highlight, we show that the Besov image models appear to be less interesting for image inpainting in the wavelet domain, highly contrary to their significant roles in thresholding‐based denoising and compression. Thus geometry is the single most important keyword throughout this paper. © 2005 Wiley Periodicals, Inc.     

An Aumann–Shapley type operator in Pseudo-D-lattices

Let L be a \(\sigma \)-complete pseudo-D-lattice and let BV be the Banach space of all real-valued, vanishing at zero, functions of bounded variation on L endowed with the variation norm. We prove the existence of a continuous Aumann–Shapley type operator \(\phi \) on the closed subspace of BV spanned by powers of nonatomic \(\sigma \)-additive positive modular measures on L. Moreover we give an integral representation of \(\phi \) on a class of functions that correspond to measure games.     

Image restoration with a high-order total variation minimization method

In this paper, we propose a fast and efficient way to restore blurred and noisy images with a high-order total variation minimization technique. The proposed method is based on an alternating technique for image deblurring and denoising. It starts by finding an approximate image using a Tikhonov regularization method. This corresponds to a deblurring process with possible artifacts and noise remaining. In the denoising step, a high-order total variation algorithm is used to remove noise in the deblurred image. We see that the edges in the restored image can be preserved quite well and the staircase effect is reduced effectively in the proposed algorithm. We also discuss the convergence of the proposed regularization method. Some numerical results show that the proposed method gives restored images of higher quality than some existing total variation restoration methods such as the fast TV method and the modified TV method with the lagged diffusivity fixed-point iteration.     

Sobolev classes of mappings on a Carnot-Carathéodory space: Various norms and variational problems

We prove the equivalence of various definitions of Sobolev classes and the BV-class of metric-space-valued mappings on some domain in a Carnot-Carathéodory space. Moreover, we prove the existence and uniqueness of the solution to some variational problem.     

Boundedness and unboundedness results for some maximal operators on functions of bounded variation

We characterize the space BV(I) of functions of bounded variation on an arbitrary interval I⊂R, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator MR from BV(I) into the Sobolev space W1,1(I). By restriction, the corresponding characterization holds for W1,1(I). We also show that if U is open in Rd, d>1, then boundedness from BV(U) into W1,1(U) fails for the local directional maximal operator , the local strong maximal operator , and the iterated local directional maximal operator . Nevertheless, if U satisfies a cone condition, then boundedly, and the same happens with , , and MR.     

基于分数阶微积分正则化的图像处理

全变分正则化方法已被广泛地应用于图像处理,利用此方法可以较好地去除噪声,并保持图像的边缘特征,但得到的优化解会产生"阶梯"效应.为了克服这一缺点,本文通过分数阶微积分正则化方法,建立了一个新的图像处理模型.为了克服此模型中非光滑项对求解带来的困难,本文研究了基于不动点方程的迫近梯度算法.最后,本文利用提出的模型与算法进行了图像去噪、图像去模糊与图像超分辨率实验,实验结果表明分数阶微积分正则化方法能较好的保留图像纹理等细节信息.     

Separability of the space of continuous functions that are continuous in Λ-variation

The Shao-Sablin index of a Λ-sequence Λ=(λi) is defined by . The main result of the paper states that the Banach space CΛBV of continuous functions of bounded Λ-variation with the standard Λ-variation norm is separable if and only if SΛ<2. Also, ΛBV=ΛBVc if and only if SΛ<2, where ΛBVc denotes the space of functions continuous in Λ-variation. A number of corollaries is drawn, and one of them being that the Garsia-Sawyer class GS is a dense subset of the Banach space HBV of functions of bounded harmonic variation.     

Absolute continuity for Banach space valued mappings

We consider the notion ofp, λ, δ-absolute continuity for Banach space valued mappings introduced in [2] for real valued functions and for λ=1. We investigate the validity of some basic properties that are shared byn, λ-absolutely continuous functions in the sense of Maly and hencl. We introduce the class δ-BV _λ,loc ^p and we give a characterization of the functions belonging to this class.     

Split Bregman iteration algorithm for total bounded variation regularization based image deblurring

Many existing algorithms taking the seminorm in BV(Ω) for regularization have achieved great success in image processing. However, this paper considers the total bounded variation regularization based approach to perform image deblurring. Based on this novel model, we introduce an extended split Bregman iteration to obtain the optimum solution quickly. We also provide the rigorous convergence analysis of the iterative algorithm here. Compared with the results of the ROF method, numerical simulations illustrate the more excellent reconstruction performance of the proposed algorithm.