利用分裂Bregman方法的图像恢复组合模型

ROF模型是图像恢复中的经典模型,具有保留图像边缘的优点,但同时也存在梯子现象.而利用二次范数fΩ｜▽u｜2 dxdy的模型可以避免梯子现象,但容易使图像变得模糊.针对两种方法的优缺点,提出了一种新的通过设置边缘检测开关函数的组合模型,在图像平坦区利用二次范数模型处理,而在强边缘处利用ROF模型处理,而且应用分裂的Br...     

An analysis of wavelet frame based scattered data reconstruction

In real world applications many signals contain singularities, like edges in images. Recent wavelet frame based approaches were successfully applied to reconstruct scattered data from such functions while preserving these features. In this paper we present a recent approach which determines the approximant from shift invariant subspaces by minimizing an ${?}_1$-regularized least squares problem which makes additional use of the wavelet frame transform in order to preserve sharp edges. We give a detailed analysis of this approach, i.e., how the approximation error behaves dependent on data density and noise level. Moreover, a link to wavelet frame based image restoration models is established and the convergence of these models is analyzed. In the end, we present some numerical examples, for instance how to apply this approach to handle coarse-grained models in molecular dynamics.     

A Compound Algorithm of Denoising Using Second-Order and Fourth-Order Partial Differential Equations

In this paper, we propose a compound algorithm for the image restoration. The algorithm is a convex combination of the ROF model and the LLT model with a parameter function 6. The numerical experiments demonstrate that our compound algorithm is efficient and preserves the main advantages of the two models. In particular, the errors of the compound algorithm in L2 norm between the exact images and corresponding restored images are the smallest among the three models. For images with strong noises, the restored images of the compound algorithm are the best in the corresponding restored images. The proposed algorithm combines the fixed point method, an improved AMG method and the Krylov acceleration. It is found that the combination of these methods is efficient and robust in the image restoration.     

A hybrid convex variational model for image restoration

We propose a new hybrid model for variational image restoration using an alternative diffusion switching non-quadratic function with a parameter. The parameter is chosen adaptively so as to minimize the smoothing near the edges and allow the diffusion to smooth away from the edges. This model belongs to a class of edge-preserving regularization methods proposed in the past, the ?-function formulation. This involves a minimizer to the associated energy functional. We study the existence and uniqueness of the energy functional of the model. Using real and synthetic images we show that the model is effective in image restoration.     

Partial differential equations and finite-difference methods in image processing,part 1: Image representation

Stochastic representation of discrete images by partial differential equation operators is considered. It is shown that these representations can fit random images, with nonseparable, isotropic covariance functions, better than other common covariance models. Application of these models in image restoration, data compression, edge detection, image synthesis, etc., is possible.Different representations based on classification of partial differential equations are considered. Examples on different images show the advantages of using these representations. The previously introduced notion of fast Karhunen-Loeve transform is extended to images with nonseparable or nearly isotropic covariance functions, or both.     

Data-Driven Tight Frame Construction for Impulsive Noise Removal

The method of data-driven tight frame has been shown very useful in image restoration problems.We consider in this paper extending this important technique,by incorporating L₁ data fidelity into the original data-driven model,for removing impulsive noise which is a very common and basic type of noise in image data.The model contains three variables and can be solved through an efficient iterative alternating minimization algorithm in patch implementation,where the tight frame is dynamically updated.It constructs a tight frame system from the input corrupted image adaptively,and then removes impulsive noise by the derived system.We also show that the sequence generated by our algorithm converges globally to a stationary point of the optimization model.Numerical experiments and comparisons demonstrate that our approach performs well for various kinds of images.This benefits from its data-driven nature and the learned tight frames from input images capture richer image structures adaptively.     

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.     

Generic singularities of implicit systems of first order differential equations on the plane

For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well-known singularities of generic vector fields on the plane and the singularities described by a generic first order implicit differential equations, there exists only one generic singularity described by the implicit first order equation supplied by Whitney umbrella surface generically embedded to the space of directions on the plane.     

Singularity formation in thin jets with surface tension

We derive and study asymptotic models for the dynamics of a thin jet of fluid that is separated from an outer immiscible fluid by fluid interfaces with surface tension. Both fluids are assumed to be incompressible, inviscid, irrotational, and density-matched. One such thin jet model is a coupled system of PDEs with nonlocal terms—Hilbert transforms—that result from expansion of a Biot-Savart integral. In order to make the asymptotic model well-posed, the Hilbert transforms act upon time derivatives of the jet thickness, making the system implicit. Within this thin jet model, we demonstrate numerically the formation of finite-time pinching singularities, where the width of the jet collapses to zero at a point. These singularities are driven by the surface tension and are very similar to those observed previously by Hou, Lowengrub, and Shelley in large-scale simulations of the Kelvin-Helmholtz instability with surface tension and in other related studies. Dropping the nonlocal terms, we also study a much simpler local model. For this local model we can preclude analytically the formation of certain types of singularities, though not those of pinching type. Surprisingly, we find that this local model forms pinching singularities of a very similar type to those of the nonlocal thin jet model. © 1998 John Wiley & Sons, Inc.     

High-order total variation-based Poissonian image deconvolution with spatially adapted regularization parameter

Images captured by image acquisition systems using photon-counting devices such as astronomical imaging, positron emission tomography and confocal microscopy imaging, are often contaminated by Poisson noise. Total variation (TV) regularization, which is a classic regularization technique in image restoration, is well-known for recovering sharp edges of an image. Since the regularization parameter is important for a good recovery, Chen and Cheng (2012) proposed an effective TV-based Poissonian image deblurring model with a spatially adapted regularization parameter. However, it has drawbacks since the TV regularization produces staircase artifacts. In this paper, in order to remedy the shortcoming of TV of their model, we introduce an extra high-order total variation (HTV) regularization term. Furthermore, to balance the trade-off between edges and the smooth regions in the images, we also incorporate a weighting parameter to discriminate the TV and the HTV penalty. The proposed model is solved by an iterative algorithm under the framework of the well-known alternating direction method of multipliers. Our numerical results demonstrate the effectiveness and efficiency of the proposed method, in terms of signal-to-noise ratio (SNR) and relative error (RelRrr).     

Fraction-order total variation blind image restoration based on L1-norm

A fraction-order total variation blind image restoration algorithm based on L1-norm was proposed for restoring the images blurred by unknown point spread function (PSF) during imaging. According to the form of total variation, this paper introduced an arithmetic operator of fraction-order total variation and generated a mathematical model of cost. Semi-quadratic regularization was used to solve the model iteratively so that the solution of this algorithm became easier. This paper also analyzed the convergence of this algorithm and then testified its feasibility in theory. The experimental results showed the proposed algorithm can increase the PSNR of the restored image by 1 dB in relation to the first order total variation blind restoration method and Bayesian blind restoration method. The details in real blurred image were also pretty well restored. The effectiveness of the proposed algorithm revealed that it was practical in the blind image restoration.     

Wavelet frame based blind image inpainting

Image inpainting has been widely used in practice to repair damaged/missing pixels of given images. Most of the existing inpainting techniques require knowing beforehand where those damaged pixels are, either given as a priori or detected by some pre-processing. However, in certain applications, such information neither is available nor can be reliably pre-detected, e.g. removing random-valued impulse noise from images or removing certain scratches from archived photographs. This paper introduces a blind inpainting model to solve this type of problems, i.e., a model of simultaneously identifying and recovering damaged pixels of the given image. A tight frame based regularization approach is developed in this paper for such blind inpainting problems, and the resulted minimization problem is solved by the split Bregman algorithm first proposed by Goldstein and Osher (2009) [1]. The proposed blind inpainting method is applied to various challenging image restoration tasks, including recovering images that are blurry and damaged by scratches and removing image noise mixed with both Gaussian and random-valued impulse noise. The experiments show that our method is compared favorably against many available two-staged methods in these applications.     

Adaptive shock-diffusion model for restoration of degraded document images

This paper presents an adaptive enhancement model with selective smoothing for restoration of degraded document images with blur, noise and bleed-through, which involves adaptive shock filtering and selective diffusion processes. A novel hybrid scheme is developed to solve the proposed model numerically, which combines explicit finite difference and exponential smoothing. Numerical experiments show that the proposed model is very effective for restoration of degraded document images with blur, noise and bleed-through, and has averagely the best performance on the DIBCO (Document Image Binarization Competition) series datasets, compared to five PDE (partial differential equation)-based models for restoration of degraded document images.     

Image Denoising by a Local Clustering Framework

Images often contain noise due to imperfections in various image acquisition techniques. Noise should be removed from images so that the details of image objects (e.g., blood vessels, inner foldings, or tumors in the human brain) can be clearly seen, and the subsequent image analyses are reliable. With broad usage of images in many disciplines—for example, medical science—image denoising has become an important research area. In the literature, there are many different types of image denoising techniques, most of which aim to preserve image features, such as edges and edge structures, by estimating them explicitly or implicitly. Techniques based on explicit edge detection usually require certain assumptions on the smoothness of the image intensity surface and the edge curves which are often invalid especially when the image resolution is low. Methods that are based on implicit edge detection often use multiresolution smoothing, weighted local smoothing, and so forth. For such methods, the task of determining the correct image resolution or choosing a reasonable weight function is challenging. If the edge structure of an image is complicated or the image has many details, then these methods would blur such details. This article presents a novel image denoising framework based on local clustering of image intensities and adaptive smoothing. The new denoising method can preserve complicated edge structures well even if the image resolution is low. Theoretical properties and numerical studies show that it works well in various applications.     

A unifying approach to isotropic and anisotropic total variation denoising models

Total variation minimisation is a well-established method for digital image restoration. Its implicit preservation of edges permits the derivation of anisotropic models for a qualitative improvement at corners. This paper is a synopsis of anisotropic models with state-of-the-art insights into the numerics of isotropic models. We generalise two representative models from both branches of research. This formulation leads to a general convergent algorithm and a general highly efficient algorithm which apply for both cases. A transfer of the discretisation from the anisotropic model to the isotropic setting results in an improvement of rotational invariance.     

Image restoration and edge detection by topological asymptotic expansion

A new method for edge detection and image restoration is proposed. This method is based on topological gradient approach. Experimental results obtained on noisy images illustrate the capabilities of this promising method in image processing. To cite this article: L.J. Belaid et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

An algebra of meromorphic corner symbols

Operators on manifolds with corners that have base configurations with geometric singularities can be analysed in the frame of a conormal symbolic structure which is in spirit similar to the one for conical singularities of Kondrat'ev's work. Solvability of elliptic equations and asymptotics of solutions are determined by meromorphic conormal symbols. We study the case when the base has edge singularities which is a natural assumption in a number of applications. There are new phenomena, caused by a specific kind of higher degeneracy of the underlying symbols. We introduce an algebra of meromorphic edge operators that depend on complex parameters and investigate meromorphic inverses in the parameter-dependent elliptic case. Among the examples are resolvents of elliptic differential operators on manifolds with edges.     

灰度图像复原的一种空间适应性向前向后扩散模型

该文考虑退化灰度图像复原问题. 首先, 作者利用时滞正则化方法定义退化图像去噪过程和去模糊过程之间的权重函数, 将激波过滤器边缘增强模型与水平集运动去噪模型相结合, 建立一种新的图像磨光增强偏微分方程. 然后, 证明该偏微分方程初值问题黏性弱解的存在唯一性. 最后, 给出该模型的部分数值算例.     

Improved Harmonic Incompatibility Removal for Susceptibility Mapping via Reduction of Basis Mismatch

In quantitative susceptibility mapping (QSM), the background field removal is an essential data acquisition step because it has a significant effect on the restoration quality by generating a harmonic incompatibility in the measured local field data. Even though the sparsity based first generation harmonic incompatibility removal (1GHIRE) model has achieved the performance gain over the traditional approaches, the 1GHIRE model has to be further improved as there is a basis mismatch underlying in numerically solving Poisson’s equation for the background removal. In this paper, we propose the second generation harmonic incompatibility removal (2GHIRE) model to reduce a basis mismatch, inspired by the balanced approach in the tight frame based image restoration. Experimental results shows the superiority of the proposed 2GHIRE model both in the restoration qualities and the computational efficiency.     

Separation of Edges in X-Ray Images by Microlocal Analysis

In many x-ray images, we find crossing structures, for example the rib bones of a thorax image. Hence, many edges in such images intersect with other edges. We propose a new algorithm, which simultaneously detects and separates such crossing edges. We prove the functionality of the algorithm under the geometric assumption that the edges possess different orientations at the points of intersection. To this end, we model edges as singularities of distributions with a submanifold structure and define the edge orientation to be the orientation of the corresponding singularity in the sense of microlocal analysis. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)