摘要
本文主要用几何分析方法,包括变分法,PDE等讨论视频图像的去抖动、运动目标跟踪,图像分割,彩色图像的色度去噪、着色及分割,复杂噪声去除等问题.为此,我们首先把所要处理的问题转化为能量泛函极小化问题,建立变分模型,并讨论能量极小化问题解的存在唯一性;然后得到其相应的Euler-Lagrange方程,再由负梯度下降法得到热方程,或者为了快速求解能量极小化问题,采用如中间变量法的技术,将能量泛函极小化问题分解为若干较为容易处理的子问题,再利用新近的快速算法分别加以求解.最后求此方程的数值解,得到相应的实验结果.
本文的主要研究成果有以下几个方面:
1.基于变分法的抖动视频图像分析
现实生活中的视频往往存在抖动现象,以往的方法一般先将视频图像进行去抖动的预处理,然后再进行分析,如跟踪运动目标等,我们在改进了KDA模型的基础上给出一个统一的变分模型框架,在泛函中添加一个仿射变换,使之能够处理抖动视频,同时做到去除抖动,目标跟踪以及背景修复.我们先在约束空间考虑所提出的能量泛函极小解的存在性,然后证明该极小解等价于无约束条件下能量泛函的极小解.在极小化问题的求解过程中,我们利用交替迭代方法,将能量泛函极小化问题看作为三个子问题,并利用全变差的对偶方法快速求解极小化问题.对于上述视频图像的研究,文中给出的数值实验结果很好的说明了模型及交替迭代算法的有效性.
2.Chan-Vese分割模型的改进
经典的Chan-Vese模型无法有效的分割带纹理图像,本文利用信息论中常用的Kullback-Leibler散度改进Chan-Vese模型,使之能够分割复杂图像.我们首先在给出基于KL散度的灰度图像分割模型,然后将其推广到彩色图像分割上.并且我们证明了能量泛函极小解的存在性,给出相应的E-L方程,利用基于全变差范数的快速整体极小方法求解该模型,在数值实验部分,与现有一些方法进行了比较.
3.基于色度亮度分解的彩色图像处理
基于色度亮度分解模型的彩色图像处理方法往往会比基于RGB模型的方法会取得更好的效果.但是该方法要考虑关于色度信息的球面约束问题,本文在研究了现有方法的基础上,采用中间变量法,给出了一个新的方法处理该约束,并且分别应用到色彩图像的去噪问题及着色问题上.此外我们首次提出了一种基于色度亮度分解的彩色图像分割模型,对亮度信息利用Wasserstein距离来处理,对色度信息利用带球面约束的向量值Chan-Vese保真项处理,并介绍了变分泛函极小解的存在性相关定理以及相应的E-L方程,然后通过引入一个附加变量,给出了泛函极小化问题的快速整体最小方法.
4.空间变化噪声的去除
在雾,人体组织等散射介质中所得到的图像,往往噪声不是均匀的,而是空间变化的,本文分析了空间变化噪声模型,并提出一个能够在去除空间变化噪声的同时保持图像纹理信息的变分去噪方法.然后给出了泛函极小的存在性证明,用负梯度下降方法求解极小化问题.在数值实验部分与ROF模型及非局部平均(Non-localMeans)方法进行了比较.
The main idea of this paper is to use geometric analysis methods, including calculus of variation and PDE methods, to study the problems, e.g., video/image sequences stabilization and moving objects tracking, image segmentation, chromaticity denoising, colorization and segmentation of color image based on chromaticity-brightness decomposition, spatially varying noise removing. Firstly, we transform the discussed problem to an energy minimization problem, establish the variational model and discuss the existence and uniqueness of the minimizer; Secondly, the Euler-Lagrange equation of the problem is derived; then we use the steepest method to derive the heat flow equation, or use intermediate method to separate the proposed energy minimization problem into some manageable sub-problems, and then solve them by fast algorithm developed recently; Finally, we get the numerical solution and experimental results.
The main research results are as follows:
1. Shaky video analysis based on calculus of variation
Generally, there exists shake phenomenon in real-world videos. Former methods pre-process the shaky video by video stabilization techniques and then analysis it, such as moving object tracking. We propose a uniform variational framework by modifying the KDA model. We add an affine transform in the energy functional so as to stabilize the shaky video, track the moving objects, and restore the background at the same time. We firstly prove the existence of minimizer of the energy in constraint space, and then show the equivalence between this minimizer and the minimizer in unconstraint space. When solving the minimization problem, we utilize an alternate iterative method by regard the energy minimization problem as three sub-problems, and solve them by the fast dual method of total variation norm. The efficiency of the above-mentioned study of video images is verified by the numerical experiments.
2. Improvement of the Chan-Vese segmentation model
The classical Chan-Vese model cannot provide good results in textured image segmentation. In this paper, we use the KuUback-Leibler divergence in information theory to improve the Chan-Vese model. Firstly, we propose a segmentation model for gray-level image, then we extend it to color image segmentation. Moreover, we prove the existence of minimizer of the proposed energy functional, derive the Euler-Lagrange equations, and solve the minimization problem by fast global minimization method. In experimental part, we compare our method with some state of arts.
3. Color image processing using chromaticity-brightness decomposition
The color image processing methods based on chromaticity-brightness decomposition model provide more satisfactory results than the RGB model. However, such methods have to deal with the unit sphere constraint of chromaticity information. Based on the study of existed methods, we propose a new method to handle the unit sphere constraint by using an intermediate variable. Then we show the applications in chromaticity denoising and image colorization. In the rest of this part, we propose a novel color image segmentation model by using chromaticity-brightness decomposition, the chromaticity term of the proposed functional follows the data term of the color Chan-Vese model with constraint on unit sphere, and the brightness term is formulated by the Wasserstein distance. Then we prove the existence of minimizer of the proposed energy functional, derive the Euler-Lagrange equations, and solve the minimization problem by fast global minimization method.
4. Removal of spatially varying noise
The image obtained from scatter media such as frog or human body usually contains inhomogeneous noise, so called spatially varying noise. In this paper, we discuss this kind of noise, and propose a variational model to remove spatially varying noise and meanwhile keep the image details (texture). Then we give a proof of the existence of the minimizer of the energy functional, and solve the minimization problem by steepest descent method. The comparisons between our results with ROF model and non-local means are shown in the experimental part.
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