\contentsline {section}{\numberline {1}引言}{3}{}%
\contentsline {section}{\numberline {2}相关工作与理论基础}{3}{}%
\contentsline {subsection}{\numberline {2.1}图像去噪问题的数学建模}{3}{}%
\contentsline {subsection}{\numberline {2.2}传统滤波方法的数学原理}{4}{}%
\contentsline {subsubsection}{\numberline {2.2.1}高斯滤波}{4}{}%
\contentsline {subsubsection}{\numberline {2.2.2}中值滤波}{5}{}%
\contentsline {subsubsection}{\numberline {2.2.3}双边滤波}{5}{}%
\contentsline {subsubsection}{\numberline {2.2.4}自适应高斯滤波}{6}{}%
\contentsline {subsection}{\numberline {2.3}基于偏微分方程的去噪方法}{7}{}%
\contentsline {subsubsection}{\numberline {2.3.1}Perona-Malik各向异性扩散}{7}{}%
\contentsline {subsubsection}{\numberline {2.3.2}Total Variation方法的深入分析}{8}{}%
\contentsline {subsubsection}{\numberline {2.3.3}Improved Total Variation方法的理论创新}{9}{}%
\contentsline {subsection}{\numberline {2.4}Euler-Lagrange方程推导}{9}{}%
\contentsline {subsection}{\numberline {2.5}各方法的比较分析}{10}{}%
\contentsline {section}{\numberline {3}数值方法与算法实现}{10}{}%
\contentsline {subsection}{\numberline {3.1}有限差分格式的详细推导}{10}{}%
\contentsline {subsubsection}{\numberline {3.1.1}梯度的数值近似}{11}{}%
\contentsline {subsubsection}{\numberline {3.1.2}散度算子的离散化}{11}{}%
\contentsline {subsubsection}{\numberline {3.1.3}TV方法的完整数值格式}{11}{}%
\contentsline {subsubsection}{\numberline {3.1.4}数值稳定性分析}{12}{}%
\contentsline {subsubsection}{\numberline {3.1.5}ITV方法的数值实现}{12}{}%
\contentsline {subsubsection}{\numberline {3.1.6}边界条件的数值处理}{12}{}%
\contentsline {subsection}{\numberline {3.2}收敛性分析}{13}{}%
\contentsline {subsubsection}{\numberline {3.2.1}TV方法的收敛理论}{13}{}%
\contentsline {subsubsection}{\numberline {3.2.2}数值收敛判据}{13}{}%
\contentsline {subsection}{\numberline {3.3}算法优化策略}{13}{}%
\contentsline {subsubsection}{\numberline {3.3.1}多尺度策略}{13}{}%
\contentsline {subsubsection}{\numberline {3.3.2}自适应时间步长}{14}{}%
\contentsline {subsection}{\numberline {3.4}参数设置的理论指导}{14}{}%
\contentsline {subsubsection}{\numberline {3.4.1}正则化参数$\lambda $的选择}{14}{}%
\contentsline {subsubsection}{\numberline {3.4.2}时间步长的优化选择}{14}{}%
\contentsline {subsection}{\numberline {3.5}数值精度分析}{15}{}%
\contentsline {subsubsection}{\numberline {3.5.1}空间离散误差}{15}{}%
\contentsline {subsubsection}{\numberline {3.5.2}时间离散误差}{15}{}%
\contentsline {subsubsection}{\numberline {3.5.3}总体误差估计}{15}{}%
\contentsline {subsection}{\numberline {3.6}计算复杂度分析}{15}{}%
\contentsline {subsubsection}{\numberline {3.6.1}每步迭代复杂度}{15}{}%
\contentsline {subsubsection}{\numberline {3.6.2}总体复杂度}{16}{}%
\contentsline {subsection}{\numberline {3.7}数值稳定性的深入分析}{16}{}%
\contentsline {subsubsection}{\numberline {3.7.1}von Neumann稳定性分析}{16}{}%
\contentsline {subsubsection}{\numberline {3.7.2}非线性稳定性}{16}{}%
\contentsline {section}{\numberline {4}实验设计与实现}{16}{}%
\contentsline {subsection}{\numberline {4.1}测试图像与噪声模型的详细分析}{16}{}%
\contentsline {subsubsection}{\numberline {4.1.1}测试图像特性}{16}{}%
\contentsline {subsubsection}{\numberline {4.1.2}高斯噪声模型的数学理论}{17}{}%
\contentsline {subsubsection}{\numberline {4.1.3}噪声水平的定量描述}{18}{}%
\contentsline {subsubsection}{\numberline {4.1.4}其他噪声模型的对比}{18}{}%
\contentsline {subsection}{\numberline {4.2}评估指标的深入数学分析}{19}{}%
\contentsline {subsubsection}{\numberline {4.2.1}峰值信噪比（PSNR）的理论基础}{19}{}%
\contentsline {subsubsection}{\numberline {4.2.2}结构相似性指数（SSIM）的数学模型}{19}{}%
\contentsline {subsubsection}{\numberline {4.2.3}多尺度SSIM（MS-SSIM）}{20}{}%
\contentsline {subsubsection}{\numberline {4.2.4}其他评估指标}{21}{}%
\contentsline {subsection}{\numberline {4.3}实验设计的统计学考虑}{21}{}%
\contentsline {subsubsection}{\numberline {4.3.1}实验重复性与统计显著性}{21}{}%
\contentsline {subsubsection}{\numberline {4.3.2}假设检验设计}{22}{}%
\contentsline {subsubsection}{\numberline {4.3.3}实验因子设计}{22}{}%
\contentsline {subsection}{\numberline {4.4}算法实现的详细规范}{22}{}%
\contentsline {subsubsection}{\numberline {4.4.1}传统滤波方法的参数优化}{22}{}%
\contentsline {subsubsection}{\numberline {4.4.2}PDE方法的数值实现细节}{23}{}%
\contentsline {subsubsection}{\numberline {4.4.3}性能评估的实现细节}{23}{}%
\contentsline {subsection}{\numberline {4.5}实验环境与计算资源}{23}{}%
\contentsline {subsubsection}{\numberline {4.5.1}硬件配置}{23}{}%
\contentsline {subsubsection}{\numberline {4.5.2}软件环境}{24}{}%
\contentsline {subsubsection}{\numberline {4.5.3}代码质量保证}{24}{}%
\contentsline {section}{\numberline {5}实验结果与分析}{24}{}%
\contentsline {subsection}{\numberline {5.1}定量分析结果}{24}{}%
\contentsline {subsection}{\numberline {5.2}TV与ITV方法的对比分析}{25}{}%
\contentsline {subsection}{\numberline {5.3}性能曲线分析}{26}{}%
\contentsline {subsection}{\numberline {5.4}性能热力图分析}{27}{}%
\contentsline {subsection}{\numberline {5.5}计算效率分析}{27}{}%
\contentsline {subsection}{\numberline {5.6}视觉效果对比}{28}{}%
\contentsline {subsubsection}{\numberline {5.6.1}原始图像与噪声图像}{28}{}%
\contentsline {subsubsection}{\numberline {5.6.2}综合视觉效果对比}{29}{}%
\contentsline {subsubsection}{\numberline {5.6.3}轻微噪声($\sigma =0.05$)下的方法对比}{30}{}%
\contentsline {subsubsection}{\numberline {5.6.4}中等噪声($\sigma =0.10$)下的方法对比}{31}{}%
\contentsline {subsubsection}{\numberline {5.6.5}TV与ITV方法的直接对比}{32}{}%
\contentsline {section}{\numberline {6}核心算法实现}{33}{}%
\contentsline {subsection}{\numberline {6.1}TV方法实现}{33}{}%
\contentsline {subsection}{\numberline {6.2}ITV方法实现}{34}{}%
\contentsline {section}{\numberline {7}讨论与分析}{36}{}%
\contentsline {subsection}{\numberline {7.1}方法优势分析}{36}{}%
\contentsline {subsubsection}{\numberline {7.1.1}ITV方法的优势}{36}{}%
\contentsline {subsubsection}{\numberline {7.1.2}传统方法的适用场景}{36}{}%
\contentsline {subsection}{\numberline {7.2}局限性分析}{36}{}%
\contentsline {subsection}{\numberline {7.3}实际应用建议}{36}{}%
\contentsline {section}{\numberline {8}结论}{37}{}%
\contentsline {subsection}{\numberline {8.1}核心发现与主要贡献}{37}{}%
\contentsline {subsubsection}{\numberline {8.1.1}ITV方法的显著优势}{37}{}%
\contentsline {subsubsection}{\numberline {8.1.2}传统方法与PDE方法的性能对比}{38}{}%
\contentsline {subsubsection}{\numberline {8.1.3}计算效率与实用性分析}{39}{}%
\contentsline {subsection}{\numberline {8.2}理论贡献与方法学意义}{39}{}%
\contentsline {subsubsection}{\numberline {8.2.1}数学理论验证}{39}{}%
\contentsline {subsubsection}{\numberline {8.2.2}评估体系的完善}{39}{}%
\contentsline {subsection}{\numberline {8.3}实际应用指导}{40}{}%
\contentsline {subsection}{\numberline {8.4}研究局限性与未来方向}{40}{}%
\contentsline {subsubsection}{\numberline {8.4.1}当前研究的局限性}{40}{}%
\contentsline {subsubsection}{\numberline {8.4.2}未来研究方向}{40}{}%
\contentsline {subsection}{\numberline {8.5}总结}{41}{}%
\contentsline {section}{\numberline {A}附录：完整源代码}{42}{}%
\contentsline {section}{\numberline {B}实现算法的详细描述与分析}{42}{}%
\contentsline {subsection}{\numberline {B.1}传统滤波方法的数学模型与实现}{42}{}%
\contentsline {subsubsection}{\numberline {B.1.1}高斯滤波的深入分析}{42}{}%
\contentsline {subsubsection}{\numberline {B.1.2}中值滤波的鲁棒统计学基础}{43}{}%