针对全变分图像去噪的半光滑牛顿法研究

发布时间：2018-04-11 01:16

本文选题：数字图像处理 + 全变分　；参考：《昆明理工大学》2017年硕士论文

【摘要】：在数字图像去噪处理中,基于变分法的思想是根据泛函知识进行排除图像噪声的惯用思想。由此思想发展的方法中最典型的模型是全变分(Total Variation:TV)模型,它的最大优点在于去除图像噪声的同时保持图像的周围边界。但由于TV模型的正则项是不可微项的,导致整个模型是不可微优化(非光滑优化)的,传统的基于微分定义的优化方法不再适用。推广经典的微分定义,建立广义微分定义和对应的优化理论和算法就成为非光滑优化的研究重点。研究者们在钻研过程中提出了半光滑牛顿法,之后迅速成为非光滑优化方法中热门的研究方向之一。本文继续发展半光滑牛顿法消除噪声,恢复图像,提高图像的质量和视觉效果,以TV模型为数学模型,以一维信号问题、二维逆源问题、二维图像问题为研究对象,研究数值优化算法。本文开展的主要工作有:(1)综述了数字图像处理理论中的一些去噪基础知识,概述了近年来国内外关于图像去噪的研究现状及图像去噪在工程等应用中的实际意义。(2)叙述了最优化的基本理论和全变分模型,根据优化理论将其转变为最优化问题。阐述目前广泛用于求解TV问题的原对偶算法和交替迭代法。(3)介绍了带约束条件的最小二乘问题,将有约束条件的优化问题通过投影转化为没有约束条件的优化问题的整个过程,并提出了基于投影的梯度下降法。(4)研究了非线性非光滑方程组和半光滑牛顿法的相关定义,提出一种新算法:结合定点迭代法的半光滑牛顿法,并叙述它的局部超线性收敛性。(5)将论文中描述过的算法在Matlab中写出对应代码做数值模拟实验,通过对实验现象和结果作对比分析得出实验结论:在同级噪声和相同参数的情况下,所提新算法优于其他三种算法。
[Abstract]:In digital image denoising, the idea based on variational method is to eliminate image noise according to functional knowledge.The most typical model developed from this idea is the Total variation: (TTV) model, which has the greatest advantage of removing image noise while preserving the peripheral boundaries of the image.However, because the regular term of TV model is non-differentiable, the whole model is non-differentiable (non-smooth optimization), so the traditional optimization method based on differential definition is no longer applicable.Generalizing the classical differential definition and establishing the generalized differential definition and corresponding optimization theory and algorithm becomes the research focus of non-smooth optimization.Researchers put forward semi-smooth Newton method in the process of research, and then became one of the hot research directions in non-smooth optimization methods.In this paper, we continue to develop a semi-smooth Newton method to eliminate noise, restore images, improve image quality and visual effect, take TV model as mathematical model, take one-dimensional signal problem, two-dimensional inverse source problem and two-dimensional image problem as research object.Numerical optimization algorithm is studied.The main work carried out in this paper is to summarize some basic knowledge of denoising in the theory of digital image processing.The research status of image denoising at home and abroad in recent years and the practical significance of image denoising in engineering and other applications are summarized. The basic theory of optimization and total variational model are described. According to the optimization theory, the optimization problem is transformed into an optimization problem.In this paper, the primal dual algorithm and alternating iteration method, which are widely used to solve TV problem at present, are introduced. The constrained least squares problem is introduced. The whole process of transforming the constrained optimization problem into an optimization problem without constraints by projection is introduced.A gradient descent method based on projection is proposed. (4) the definitions of nonlinear nonsmooth equations and semi-smooth Newton method are studied, and a new algorithm is proposed: a semi-smooth Newton method based on fixed-point iterative method.Its local superlinear convergence. 5) the algorithm described in this paper is written out in Matlab for numerical simulation.The experimental results show that the proposed algorithm is superior to the other three algorithms in the case of the same level noise and the same parameters.
【学位授予单位】：昆明理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TP391.41

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