基于变分PDE和多重分形的图像建模理论、算法与应用
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摘要
从机器视觉和感知科学角度看，图像处理是一种由2D图像(或者图像序列)重建3D世界中几何关系、拓扑结构、模式和动力学行为的基本工具。图像处理中图像恢复和图像分割构成了低层视觉处理的基本问题，同时作为数学上的反问题也成为现代数学研究的热点。由于图像是3D世界在2D成像平面的投影，并以二维强度分布的形式作用于人的视觉，其非平稳性和非高斯特征，很难用线性算法进行处理，迫切需要建立合适的能够处理边缘到纹理各层面奇异性的图像模型。图像模型作为一般的高层先验知识，有助于设计性能优越的图像处理器，有助于主动视觉实现数据约束与先验知识约束的介入与交互过程。因此图像建模的研究在计算机视觉和图像处理中具有重大的理论意义和应用价值。
本文以图像建模研究为主线，针对图像恢复和图像分割的基本问题以及目前兴起的图像修补这一热点问题，着重研究相应变分PDE模型及其算法；然后针对数字景像图仿真生成及其地面景像图的奇异性分析，研究基于大气辐射传输理论和增强型CCD的成像建模、仿真，研究自然图像多重分形理论及其算法；最后针对图像版权保护问题，研究图像视觉感知建模及其鲁棒性数字水印技术。论文所做的主要工作和研究成果如下：
(1)系统地研究了图像建模的概念和意义，综述了图像建模理论与方法在国内外的研究现状，从机器视觉和感知理解的角度研究了图像模型、低层视觉和图像处理的关系。分析和对比了各类模型的表现形式、之间的联系及各自的优势与不足。
(2)根据有界变差函数空间理论，全面系统地研究了主TV、对偶TV模型和扩展TV模型及相关算法。针对数字灰度图像和彩色图像，深入研究了标量图像-数字TV模型和向量图像-数字TV模型，通过简捷和新颖地证明该模型的严格凸性和最优化能量函数应满足的数字欧拉-拉格朗日方程，揭示了数字TV模型的三个基本性质。并结合应用分别设计了与经典梯度倒数加权滤波器类似的Scalar-NGIW-Filter和Vector-NGIW-Filter。
在自然图像处理中，由于噪声和小尺度纹理并存，TV模型对纹理丰富图像的恢复效果并不好。本文又提出了基于分数阶奇异性提取的耦合TV恢复模型。
(3)系统地论述了贝叶斯模型、鲁棒性估计理论与变分PDE模型的统一性。从半二次规整化出发，针对变分模型中位势函数的选择问题，提出了过渡位势函数的概念，并通过研究该过渡位势函数的性质以及与位势函数的关系，导出了一系列有价值的变分模型；最后分析了一类基于目标与边缘的耦合变分模型及其相应的推广与改进形
摘要
博士论文
式。
(4)针对分片光滑Mumford一Shah水平集模型的数值解问题，通过构造具有柔性
距离函数，对迭代步骤中水平集函数重新初始化，提出了基于ENO格式和预测校正方
法的新水平集算法。
将Ambrosio和Tortorelli提出的Mumford一shah辅助变量模型推广到彩色图像，
提出了一种彩色图像祸合变分模型。新模型将彩色图像建模为黎曼流形上的嵌入曲
面，并据此将刻画不同颜色通道间梯度方向差异的物理量一向量积项引入目标的正则
化部分，提出了新的能量泛函。理论上证明了最优化该能量泛函应满足的欧拉一拉格
朗日方程。由于直接将标量图像的Mumford一Shah模型推广到向量图像，往往存在孤
立地对待每个通道的问题;而基于曲面方法的几何图像模型，却能够精细的刻画各通
道之间的相关性和相互影响。基于时间步进法和半点格式，本文又提出了新模型的一
种数值计算方法。理论分析和实验结果都表明，彩色图像藕合变分模型在图像恢复和
分割效果上都优于直接推广的模型。
(5)针对图像修补问题，通过分析人类视觉系统(HVS)和心脑认知机制修补图像
的一些启发性原则，提炼了目前非纹理图像局部性修补的指导性原则;系统地对比了
目前基于变分PDE的结构修补方法以及纹理合成方法。基于结构与纹理分治策略，提
出了包含四个语义推断形式的通用修补模型;结合扩散和传输机制，提出了一个三阶
形态不变结构修补的PDE。最后研究了图像修补两个主要应用，提出了基于数字TV
模型的数字图像放大算法和基于纹理匹配与边缘勾连的错误隐匿方法。
(6)针对一个实际的“仿真图像生成与分析系统”工程项目，提出了基于大气辐
射传输和图像增强型CCD相机的图像与图形混合的数字景像匹配图仿真生成模型，
并实现了一个功能强大的仿真系统SMART-SS。为了分析地面景像不同奇异性结构对
景像匹配性能的影响，提出了应用多重分形分解和重构的不同奇异性结构提取方法，
并系统地证明了图像多重分形测度在一类速降函数上的投影应满足的幂指数形式。
(7)针对图像水印问题，基于小波域视觉感知模型，提出用多数字基实现小波域
多重水印嵌入;基于脊波域视觉感知模型，提出了一种在脊波变换域基于图像内容的
数字水印模型，统计分析了脊波域数字水印方法中最优检测闭值、虚警和漏警概率估
计等相关问题。
From the machine vision and cognitive science, image processing is a basic tool used to reconstruct the geometrical relationship, topology structure, patterns and dynamics of the three-dimensional (3-D) world from two-dimensional (2-D) images or image sequences. Image restoration and image segmentation are two basic problems in the low-level vision processing. Belonging to the inverse problems, they are also two of the most challenging, active subjects in modern mathematics. However, since each image is a 2-D projection of a window of our 3-D world and is a matrix of positive integers which represents a pattern of radiant energy emitted by objects in space, it is difficult to handle with linear algorithms due to its non-Gaussian behavior and non-stationary property. Hence, it is desiderated to construct image models, which is allowed to deal with not only edge-like set, but also to other texture-like structure. As the general high-level priori knowledge, image models are very useful for designing image proce
ssors with good performance, and they are also helpful for implementing the human interaction in active vision through combining data constraints coming from image (bottom-up) with priori knowledge from high-level (top-down). Therefore the research of image modeling has great significance in the field of computer vision and image processing.
This dissertation focuses on the research on image modeling and corresponding algorithms for some problems of the low-level vision process. We firstly research on the variational PDE image models for three major problems of image restoration, segmentation and image inpainting, in which the last problem is the research hotspot recently.
Corresponding to the Image Intensified CCD (IICCD) camera, we then discuss the
modeling and simulation of digital scene image synthesis based on the atmospheric propagation modeling theory. Finally, we focus on the image visual perceptual modeling and robust watermarking techniques for copyright protection problem of multimedia. The primary work and remarks of the paper are:
(1) The paper takes a systematic research of the general ideas, basic concepts of image models and presents the state of arts on image modeling theory and approaches. The relationships among image models, low-level vision and image processing are established. A detailed analysis and comparative study of the models' representation, advantages and disadvantages is made and discussed.
(2) Based on the theory of bounded variation functional space, the paper investigates the prime Total Variation (TV), prime-dual TV, extended TV models and related algorithms.
We develop a new digital TV model for scalar-valued and vector-valued images, and
provide novel and simple proofs for its convex property and the digital Euler-Lagrange equation for minimizing the digital energy functional.
Although total variation restoration model is good to de-noise non-texture images, it is not sufficient for real nature images since it removes small scale texture as well as noise. The drawbacks of total variation restoration model are analyzed in this paper. The paper investigates the possibility of extracting the texture with fractional order derivatives filter. And an improved total variation restoration model combined with fractional order derivatives singularities extracting is proposed.
(3) This paper establishes the connections and unification among Bayesian inference, the variational PDE models and robust statistics theory. Starting from the half-quadratic regularization, the paper proposes the concept of transitional potential function in order to give a solution for the choice of the potential function in the variational restoration model. Investigating the properties of the transitional potential function, a series of valuable variational models are easily derived by half-quadratic theorem. Finally, the paper advances a class of generalizing and improved coupled "object-edge" models with variational formulation.
(4) In order to solve the problem of the piece-wise smoothing Mumf
本文以图像建模研究为主线，针对图像恢复和图像分割的基本问题以及目前兴起的图像修补这一热点问题，着重研究相应变分PDE模型及其算法；然后针对数字景像图仿真生成及其地面景像图的奇异性分析，研究基于大气辐射传输理论和增强型CCD的成像建模、仿真，研究自然图像多重分形理论及其算法；最后针对图像版权保护问题，研究图像视觉感知建模及其鲁棒性数字水印技术。论文所做的主要工作和研究成果如下：
(1)系统地研究了图像建模的概念和意义，综述了图像建模理论与方法在国内外的研究现状，从机器视觉和感知理解的角度研究了图像模型、低层视觉和图像处理的关系。分析和对比了各类模型的表现形式、之间的联系及各自的优势与不足。
(2)根据有界变差函数空间理论，全面系统地研究了主TV、对偶TV模型和扩展TV模型及相关算法。针对数字灰度图像和彩色图像，深入研究了标量图像-数字TV模型和向量图像-数字TV模型，通过简捷和新颖地证明该模型的严格凸性和最优化能量函数应满足的数字欧拉-拉格朗日方程，揭示了数字TV模型的三个基本性质。并结合应用分别设计了与经典梯度倒数加权滤波器类似的Scalar-NGIW-Filter和Vector-NGIW-Filter。
在自然图像处理中，由于噪声和小尺度纹理并存，TV模型对纹理丰富图像的恢复效果并不好。本文又提出了基于分数阶奇异性提取的耦合TV恢复模型。
(3)系统地论述了贝叶斯模型、鲁棒性估计理论与变分PDE模型的统一性。从半二次规整化出发，针对变分模型中位势函数的选择问题，提出了过渡位势函数的概念，并通过研究该过渡位势函数的性质以及与位势函数的关系，导出了一系列有价值的变分模型；最后分析了一类基于目标与边缘的耦合变分模型及其相应的推广与改进形
摘要
博士论文
式。
(4)针对分片光滑Mumford一Shah水平集模型的数值解问题，通过构造具有柔性
距离函数，对迭代步骤中水平集函数重新初始化，提出了基于ENO格式和预测校正方
法的新水平集算法。
将Ambrosio和Tortorelli提出的Mumford一shah辅助变量模型推广到彩色图像，
提出了一种彩色图像祸合变分模型。新模型将彩色图像建模为黎曼流形上的嵌入曲
面，并据此将刻画不同颜色通道间梯度方向差异的物理量一向量积项引入目标的正则
化部分，提出了新的能量泛函。理论上证明了最优化该能量泛函应满足的欧拉一拉格
朗日方程。由于直接将标量图像的Mumford一Shah模型推广到向量图像，往往存在孤
立地对待每个通道的问题;而基于曲面方法的几何图像模型，却能够精细的刻画各通
道之间的相关性和相互影响。基于时间步进法和半点格式，本文又提出了新模型的一
种数值计算方法。理论分析和实验结果都表明，彩色图像藕合变分模型在图像恢复和
分割效果上都优于直接推广的模型。
(5)针对图像修补问题，通过分析人类视觉系统(HVS)和心脑认知机制修补图像
的一些启发性原则，提炼了目前非纹理图像局部性修补的指导性原则;系统地对比了
目前基于变分PDE的结构修补方法以及纹理合成方法。基于结构与纹理分治策略，提
出了包含四个语义推断形式的通用修补模型;结合扩散和传输机制，提出了一个三阶
形态不变结构修补的PDE。最后研究了图像修补两个主要应用，提出了基于数字TV
模型的数字图像放大算法和基于纹理匹配与边缘勾连的错误隐匿方法。
(6)针对一个实际的“仿真图像生成与分析系统”工程项目，提出了基于大气辐
射传输和图像增强型CCD相机的图像与图形混合的数字景像匹配图仿真生成模型，
并实现了一个功能强大的仿真系统SMART-SS。为了分析地面景像不同奇异性结构对
景像匹配性能的影响，提出了应用多重分形分解和重构的不同奇异性结构提取方法，
并系统地证明了图像多重分形测度在一类速降函数上的投影应满足的幂指数形式。
(7)针对图像水印问题，基于小波域视觉感知模型，提出用多数字基实现小波域
多重水印嵌入;基于脊波域视觉感知模型，提出了一种在脊波变换域基于图像内容的
数字水印模型，统计分析了脊波域数字水印方法中最优检测闭值、虚警和漏警概率估
计等相关问题。
From the machine vision and cognitive science, image processing is a basic tool used to reconstruct the geometrical relationship, topology structure, patterns and dynamics of the three-dimensional (3-D) world from two-dimensional (2-D) images or image sequences. Image restoration and image segmentation are two basic problems in the low-level vision processing. Belonging to the inverse problems, they are also two of the most challenging, active subjects in modern mathematics. However, since each image is a 2-D projection of a window of our 3-D world and is a matrix of positive integers which represents a pattern of radiant energy emitted by objects in space, it is difficult to handle with linear algorithms due to its non-Gaussian behavior and non-stationary property. Hence, it is desiderated to construct image models, which is allowed to deal with not only edge-like set, but also to other texture-like structure. As the general high-level priori knowledge, image models are very useful for designing image proce
ssors with good performance, and they are also helpful for implementing the human interaction in active vision through combining data constraints coming from image (bottom-up) with priori knowledge from high-level (top-down). Therefore the research of image modeling has great significance in the field of computer vision and image processing.
This dissertation focuses on the research on image modeling and corresponding algorithms for some problems of the low-level vision process. We firstly research on the variational PDE image models for three major problems of image restoration, segmentation and image inpainting, in which the last problem is the research hotspot recently.
Corresponding to the Image Intensified CCD (IICCD) camera, we then discuss the
modeling and simulation of digital scene image synthesis based on the atmospheric propagation modeling theory. Finally, we focus on the image visual perceptual modeling and robust watermarking techniques for copyright protection problem of multimedia. The primary work and remarks of the paper are:
(1) The paper takes a systematic research of the general ideas, basic concepts of image models and presents the state of arts on image modeling theory and approaches. The relationships among image models, low-level vision and image processing are established. A detailed analysis and comparative study of the models' representation, advantages and disadvantages is made and discussed.
(2) Based on the theory of bounded variation functional space, the paper investigates the prime Total Variation (TV), prime-dual TV, extended TV models and related algorithms.
We develop a new digital TV model for scalar-valued and vector-valued images, and
provide novel and simple proofs for its convex property and the digital Euler-Lagrange equation for minimizing the digital energy functional.
Although total variation restoration model is good to de-noise non-texture images, it is not sufficient for real nature images since it removes small scale texture as well as noise. The drawbacks of total variation restoration model are analyzed in this paper. The paper investigates the possibility of extracting the texture with fractional order derivatives filter. And an improved total variation restoration model combined with fractional order derivatives singularities extracting is proposed.
(3) This paper establishes the connections and unification among Bayesian inference, the variational PDE models and robust statistics theory. Starting from the half-quadratic regularization, the paper proposes the concept of transitional potential function in order to give a solution for the choice of the potential function in the variational restoration model. Investigating the properties of the transitional potential function, a series of valuable variational models are easily derived by half-quadratic theorem. Finally, the paper advances a class of generalizing and improved coupled "object-edge" models with variational formulation.
(4) In order to solve the problem of the piece-wise smoothing Mumf
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