Design of generalized fractional order gradient descent method

This paper focuses on the convergence problem of the emerging fractional order gradient descent method, and proposes three solutions to overcome the problem. In fact, the general fractional gradient method cannot converge to the real extreme point of the target function, which critically hampers the application of this method. Because of the long memory characteristics of fractional derivative, fixed memory principle is a prior choice. Apart from the truncation of memory length, two new methods are developed to reach the convergence. The one is the truncation of the infinite series, and the other is the modification of the constant fractional order. Finally, six illustrative examples are performed to illustrate the effectiveness and practicability of proposed methods.Comment: 8 pages, 16 figure

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

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

Support Vector Machine optimization with fractional gradient descent for data classification

Data classification has several problems one of which is a large amount of data that will reduce computing time. SVM is a reliable linear classifier for linear or non-linear data, for large-scale data, there are computational time constraints. The Fractional gradient descent method is an unconstrained optimization algorithm to train classifiers with support vector machines that have convex problems. Compared to the classic integer-order model, a model built with fractional calculus has a significant advantage to accelerate computing time. In this research, it is to conduct investigate the current state of this new optimization method fractional derivatives that can be implemented in the classifier algorithm. The results of the SVM Classifier with fractional gradient descent optimization, it reaches a convergence point of approximately 50 iterations smaller than SVM-SGD. The process of updating or fixing the model is smaller in fractional because the multiplier value is less than 1 or in the form of fractions. The SVM-Fractional SGD algorithm is proven to be an effective method for rainfall forecast decisions

Liver CT enhancement using Fractional Differentiation and Integration

In this paper, a digital image filter is proposed to enhance the Liver CT image for improving the classification of tumors area in an infected Liver. The enhancement process is based on improving the main features within the image by utilizing the Fractional Differential and Integral in the wavelet sub-bands of an image. After enhancement, different features were extracted such as GLCM, GRLM, and LBP, among others. Then, the areas/cells are classified into tumor or non-tumor, using different models of classifiers to compare our proposed model with the original image and various established filters. Each image is divided into 15x15 non-overlapping blocks, to extract the desired features. The SVM, Random Forest, J48 and Simple Cart were trained on a supplied dataset, different from the test dataset. Finally, the block cells are identified whether they are classified as tumor or not. Our approach is validated on a group of patients’ CT liver tumor datasets. The experiment results demonstrated the efficiency of enhancement in the proposed technique

On the application of partial differential equations and fractional partial differential equations to images and their methods of solution

This body of work examines the plausibility of applying partial di erential equations and time-fractional partial di erential equations to images. The standard di usion equation is coupled with a nonlinear cubic source term of the Fitzhugh-Nagumo type to obtain a model with di usive properties and a binarizing e ect due to the source term. We examine the e ects of applying this model to a class of images known as document images; images that largely comprise text. The e ects of this model result in a binarization process that is competitive with the state-of-the-art techniques. Further to this application, we provide a stability analysis of the method as well as high-performance implementation on general purpose graphical processing units. The model is extended to include time derivatives to a fractional order which a ords us another degree of control over this process and the nature of the fractionality is discussed indicating the change in dynamics brought about by this generalization. We apply a semi-discrete method derived by hybridizing the Laplace transform and two discretization methods: nite-di erences and Chebyshev collocation. These hybrid techniques are coupled with a quasi-linearization process to allow for the application of the Laplace transform, a linear operator, to a nonlinear equation of fractional order in the temporal domain. A thorough analysis of these methods is provided giving rise to conditions for solvability. The merits and demerits of the methods are discussed indicating the appropriateness of each method

Image Denoising via Nonlinear Hybrid Diffusion

A nonlinear anisotropic hybrid diffusion equation is discussed for image denoising, which is a combination of mean curvature smoothing and Gaussian heat diffusion. First, we propose a new edge detection indicator, that is, the diffusivity function. Based on this diffusivity function, the new diffusion is nonlinear anisotropic and forward-backward. Unlike the Perona-Malik (PM) diffusion, the new forward-backward diffusion is adjustable and under control. Then, the existence, uniqueness, and long-time behavior of the new regularization equation of the model are established. Finally, using the explicit difference scheme (PM scheme) and implicit difference scheme (AOS scheme), we do numerical experiments for different images, respectively. Experimental results illustrate the effectiveness of the new model with respect to other known models