总广义变差在图像恢复中的建模理论及其算法研究

Total variation and partial differential equation methods have been widely applied to image processing. However, the methods of total variation and high order partial defferential equation have two problems: one is that total variation can lead to staircase effects, the other is that high order partial diffenential equation methods can make some edges and details lost, so the restored image looks blurry. For solving the two problems above, in this project, we plan to construct the adaptive models and nonlocal models based on total generalized variation. Differt from total variation, the total generalized variation is a novel concept, and it is the research hotspot of image processing and applied mathematics. The novelty of this project lies in the following several aspects: firstly, the new adaptive total generalized variation based image restoration models will be constructed; secondly, the nonconvex total generalized variation regularization models with sparse representation low-rank constraint for image restoration will be introduced; lastly, in order to remove the nosie of the texture images effectively, a nonlocal total generalized variation based model will be proposed. In addition, the end of the project will give the application of new methods, which makes the results of image restoration more efficient.

总变差和偏微分方程在图像处理中已经得到广泛应用，然而总变差和高阶偏微分方程方法存在下面两个问题：（1）总变差易导致阶梯效应；（2）高阶偏微分方程方法易导致图像的边缘和细节丢失，从而引起图像模糊。本项目计划通过建立自适应的总广义变差模型和非局部总广义变差模型等解决上述两个基本问题。总广义变差不同于总变差，它是一个全新的数学概念，是当前图像处理和应用数学研究的热点。本项目将对总广义变差正则方法进行深入、全面地研究，创新之处在于：（1）建立新的自适应总广义变差图像恢复模型；（2）建立非凸总广义变差和稀疏正则相结合的图像恢复模型；（3）建立非局部总广义变差图像恢复模型。另外，我们将给出新方法在图像恢复（去噪、去模糊、修复、分解等）问题上的应用。本课题预期在理论上有突破，方法和技术上有创新，为该方法的实际应用奠定基础。

变分理论和偏微分方程是目前研究图像处理的主要数学方法，两个领域的最新理论表明，总变差和高阶偏微分方程在图像恢复过程中易引起阶梯效应和边缘模糊，为克服此缺陷，本项目主要对最新出现的总广义变差在图像恢复中的建模理论及其算法进行深入、全面地研究，主要内容包括：1. 利用总广义变差建立自适应的图像恢复模型，消除了阶梯效应，并保护了图像的边缘等；2. 利用最新的稀疏表示理论，建立稀疏和总广义变差相结合的正则化模型；3. 建立了非凸正则化模型，结合迭代重加权方法和原始-对偶方法给出了快速算法，克服了传统模型计算困难的问题；4. 利用小波理论的最新进展，建立新的空域和变换域相结合的图像恢复模型；5. 针对上述提出的模型，给出了相应的快速求解方法。 将上述模型和算法应用于图像去噪、分解、修复等，推动了本项目成果的应用。