Image Inpainting and Enhancement using Fractional Order Variational Model

The intention of image inpainting is to complete or fill the corrupted or missing zones of an image by considering the knowledge from the source region. A novel fractional order variational image inpainting model in reference to Caputo definition is introduced in this article. First, the fractional differential, and its numerical methods are represented according to Caputo definition. Then, a fractional differential mask is represented in 8-directions. The complex diffusivity function is also defined to preserve the edges. Finally, the missing regions are filled by using variational model with fractional differentials of 8-directions. The simulation results and analysis display that the new model not only inpaints the missing regions, but also heightens the contrast of the image. The inpainted images have better visual quality than other fractional differential filters

Fractional Entropy Based Active Contour Segmentation of Cell Nuclei in Actin-Tagged Confocal Microscopy Images

In the framework of cell structure characterization for predictive oncology, we propose in this paper an unsupervised statistical region based active contour approach integrating an original fractional entropy measure for single channel actin tagged fluorescence confocal microscopy image segmentation. Following description of statistical based active contour segmentation and the mathematical definition of the proposed fractional entropy descriptor, we demonstrate comparative segmentation results between the proposed approach and standard Shannon’s entropy obtained for nuclei segmentation. We show that the unsupervised proposed statistical based approach integrating the fractional entropy measure leads to very satisfactory segmentation of the cell nuclei from which shape characterization can be subsequently used for the therapy progress assessment

ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing

We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H−1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation

Symmetrized fractional total variation for signal and image analysis

We introduce and study a variational model for signal and image analysis based on Riemann–Liouville fractional derivatives. Both the one-dimensional and two-dimensional cases are studied. The model exploits a quadratic fitting data term together with both right and left Riemann–Liouville fractional derivatives as regularizing terms, with the aim of achieving an orientation-independent analysis

Contributions en optimisation topologique : extension de la méthode adjointe et applications au traitement d'images

De nos jours, l'optimisation topologique a été largement étudiée en optimisation de structure, problème majeur en conception de systèmes mécaniques pour l'industrie et dans les problèmes inverses avec la détection de défauts et d'inclusions. Ce travail se concentre sur les approches de dérivées topologiques et propose une généralisation plus flexible de cette méthode rendant possible l'investigation de nouvelles applications. Dans une première partie, nous étudions des problèmes classiques en traitement d'images (restauration, inpainting), et exposons une formulation commune à ces problèmes. Nous nous concentrons sur la diffusion anisotrope et considérons un nouveau problème : la super-résolution. Notre approche semble meilleure comparée aux autres méthodes. L'utilisation des dérivées topologiques souffre d'inconvénients : elle est limitée à des problèmes simples, nous ne savons pas comment remplir des trous ... Dans une seconde partie, une nouvelle méthode visant à surmonter ces difficultés est présentée. Cette approche, nommée voûte numérique, est une extension de la méthode adjointe. Ce nouvel outil nous permet de considérer de nouveaux champs d'application et de réaliser de nouvelles investigations théoriques dans le domaine des dérivées topologiques.Nowadays, topology optimization has been extensively studied in structural optimization which is a major interest in the design of mechanical systems in the industry and in inverse problems with the detection of defects or inclusions. This work focuses on the topological derivative approach and proposes a more flexible generalization of this method making it possible to address new applications. In a first part, we study classical image processing problems (restoration, inpainting), and give a common framework to theses problems. We focus on anisotropic diffusion and consider a new problem: super-resolution. Our approach seems to be powerful in comparison with other methods. Topological derivative method has some drawbacks: it is limited to simple problems, we do not know how to fill holes, ... In a second part, to overcome these difficulties, an extension of the adjoint method is presented. Named the numerical vault, it allows us to consider new fields of applications and to explore new theoretical investigations in the area of topological derivative

A Spatially Adaptive Edge-Preserving Denoising Method Based on Fractional-Order Variational PDEs

Image denoising is a basic problem in image processing. An important task of image denoising is to preserve the significant geometric features such as edges and textures while filtering out noise. So far, this is still a problem to be further studied. In this paper, we firstly introduce an edge detection function based on the Gaussian filtering operator and then analyze the filtering characteristic of the fractional derivative operator. On the basis, we establish the spatially adaptive fractional edge-preserving denoising model in the variational framework, discuss the existence and uniqueness of our proposed model solution and derive the nonlinear fractional Euler-Lagrange equation for solving our proposed model. This forms a fractional order extension of the first and second order variational approaches. Finally, we apply the proposed method to the synthetic images and real seismic data denoising to verify the effectiveness of our method and compare the experimental results of our method with the related state-of-the-art methods. Experimental results illustrate that our proposed method can not only improve the signal to noise ratio (SNR) but also adaptively preserve the structural information of an image compared with other contrastive methods. Our proposed method can also be applied to remote sensing imaging, medical imaging and so onThe work of Dehua Wang was supported in part by the Science and Technology Planning Project of Shaanxi Province under Grant 2020JM-561, in part by the Postdoctoral Foundation of China under Grant 2019M663462, in part by the Innovative Talents Cultivate Program of Shaanxi Province under Grant 2019KJXX-032, in part by the President Fund of Xi’an Technological University under Grant XAGDXJJ17026, and in part by the Teaching Reform Project of Xi’an Technological University under Grant 18JGY08. The work of Juan J. Nieto was supported in part by the Agencia Estatal de Investigacion (AEI) of Spain under Grant MTM2016-75140-P, and in part by the European Community Fund FEDER. The work of Xiaoping Li was supported in part by the NSFC under Grant 61701086, and in part by the Fundamental Research Funds for the Central Universities under Grant ZYGX2016KYQD143S

Image Structure-Preserving Denoising Based on Difference Curvature Driven Fractional Nonlinear Diffusion

The traditional integer-order partial differential equations and gradient regularization based image denoising techniques often suffer from staircase effect, speckle artifacts, and the loss of image contrast and texture details. To address these issues, in this paper, a difference curvature driven fractional anisotropic diffusion for image noise removal is presented, which uses two new techniques, fractional calculus and difference curvature, to describe the intensity variations in images. The fractional-order derivatives information of an image can deal well with the textures of the image and achieve a good tradeoff between eliminating speckle artifacts and restraining staircase effect. The difference curvature constructed by the second order derivatives along the direction of gradient of an image and perpendicular to the gradient can effectively distinguish between ramps and edges. Fourier transform technique is also proposed to compute the fractional-order derivative. Experimental results demonstrate that the proposed denoising model can avoid speckle artifacts and staircase effect and preserve important features such as curvy edges, straight edges, ramps, corners, and textures. They are obviously superior to those of traditional integral based methods. The experimental results also reveal that our proposed model yields a good visual effect and better values of MSSIM and PSNR

코시잡음 제거를 위한 변분법적 접근

학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2020. 2. 강명주.In image processing, image noise removal is one of the most important problems. In this thesis, we study Cauchy noise removal by variational approaches. Cauchy noise occurs often in engineering applications. However, because of the non-convexity of the variational model of Cauchy noise, it is difficult to solve and were not studied much. To denoise Cauchy noise, we use the non-convex alternating direction method of multipliers and present two variational models. The first thing is fractional total variation(FTV) model. FTV is derived by fractional derivative which is an extended version of integer order derivative to real order derivative. The second thing is the weighted nuclear norm model. Weighted nuclear norm has an excellent performance in low-level vision. We have combined our novel ideas with weighted nuclear norm minimization to achieve better results than existing models in Cauchy noise removal. Finally, we show the superiority of the proposed model from numerical experiments.이미지 처리에서 이미지 잡음 제거는 가장 중요한 문제 중 하나다. 이 논문에서 우리는 다양한 접근 방식에 의한 코시 잡음 제거를 연구한다. 코시 잡음은 엔지니어링 애플리케이션에서 자주 발생하나 코시 잡음의 변분법적 모델의 비 볼록성으로 인해 해결하기가 어렵고 많이 연구되지 않았다. 코시 노이즈를 제거하기 위해 우리는 곱셈기의 볼록하지 않은 교류 방향 방법(nonconvex ADMM)을 사용하였으며 두 가지 변분법적 모델을 제시한다. 첫 번째는 분수 총 변이(FTV)를 이용한 모델이다. 분수 총 변이는 일반적인 정수 도함수를 실수 도함수로 확장 한 분수 도함수에 의해 정의된다. 두 번째는 가중 핵 노름을 이용한 모델이다. 가중 핵 노름은 저수준 영상처리에서 탁월한 성능을 발휘한다. 우리는 가중 핵 노름이 코시 잡음 제거에서도 뛰어난 성능을 발휘할 것으로 예상하였고, 우리의 새로운 아이디어를 가중 핵 노름 최소화와 결합하여 현존하는 코시 잡음 제거 최신 모델들보다 더 나은 결과를 얻을 수 있었다. 마지막 장에서 실제 코시 잡음 제거 테스트를 통해 우리 모델이 얼마나 뛰어난지 확인하며 논문을 마친다.1 Introduction 1 2 The Cauchy distribution and the Cauchy noise 5 2.1 The Cauchy distribution 5 2.1.1 The alpha-stable distribution 5 2.1.2 The Cauchy distribution 8 2.2 The Cauchy noise 13 2.2.1 Analysis of the Cauchy noise 13 2.2.2 Variational model of Cauchy noise 14 2.3 Previous work 16 3 Fractional order derivatives and total fractional order variational model 19 3.1 Some fractional derivatives and integrals 19 3.1.1 Grunwald-Letnikov Fractional Derivatives 20 3.1.2 Riemann-Liouville Fractional Derivatives 28 3.2 Proposed model: Cauchy noise removal model by fractional total variation 33 3.2.1 Fractional total variation and Cauchy noise removal model 34 3.2.2 nonconvex ADMM algorithm 37 3.2.3 The algorithm for solving fractional total variational model of Cauchy noise 39 3.3 Numerical results of fractional total variational model 51 3.3.1 Parameter and termination condition 51 3.3.2 Experimental results 54 4 Nuclear norm minimization and Cauchy noise denoising model 67 4.1 Weighted Nuclear Norm 67 4.1.1 Weighted Nuclear Norm and Its Applications 68 4.1.2 Iteratively Reweighted l1 Minimization 74 4.2 Proposed Model: Weighted Nuclear Norm For Cauchy Noise Denoising 77 4.2.1 Model and algorithm description 77 4.2.2 Convergence of algorithm7 79 4.2.3 Block matching method 81 4.3 Numerical Results OfWeighted Nuclear Norm Denoising Model For Cauchy Noise 83 4.3.1 Parameter setting and truncated weighted nuclear norm 84 4.3.2 Termination condition 85 4.3.3 Experimental results 86 5 Conclusion 95 Abstract (in Korean) 105Docto

A duality based approach to the minimizing total variation flow in the space H−sH^{-s}

We consider a gradient flow of the total variation in a negative Sobolev space $H^{-s}$ $(0\leq s \leq 1)$ under the periodic boundary condition. If $s=0$, the flow is nothing but the classical total variation flow. If $s=1$, this is the fourth order total variation flow. We consider a convex variational problem which gives an implicit-time discrete scheme for the flow. By a duality based method, we give a simple numerical scheme to calculate this minimizing problem numerically and discuss convergence of a forward-backward splitting scheme. Several numerical experiments are given