基于块组低秩稀疏结构的图像处理变分模型与算法研究

Low rank and sparse are very popular concepts in image processing, compressed sensing, machine learning and data mining in recent years. Low rank and sparse are very popular concepts in image processing, compressed sensing, machine learning and data mining in recent years. The matrix recovery methods based on low rank and sparse structures are widely studied and applied in image processing fields. Generally, as matrix, image does not have low rank and sparse property. However, we can get nice low rank and sparse structures by extracting patch group which is contained by many similar patches in the image. So far, the image denoising methods based on patch group low rank and sparse structures achieve the state-of-the-art results, but both the speed and the theory are limited. Therefore, the study of extensive image processing models, fast numerical methods and theories based on low rank and sparse patch group has becoming the most perspective research area. In this project, we plan to extend the concepts of low rank and sparse and apply them to more image processing topics. We will propose some new variational models and numerical algorithms in order to enhance the quality, speed and accuracy of image processing. In this project, we will focus on the following four contents: Firstly, we will study the variational models for image processing based on patch group low rank and sparse structures in the image. Secondly, we will design fast numerical methods to solve the models. Thirdly, we will study the automatic parameter selection schemes. Finally, we will establish the theories of the related models and algorithms.

低秩稀疏是近年来图像处理、压缩感知、机器学习和数据挖掘等领域中非常热门的概念。基于低秩稀疏结构的矩阵恢复方法正被广泛研究并逐步应用到图像处理领域。图像本身作为矩阵一般并不具有低秩稀疏性，但通过抽取图像中的若干相似子块构成块组则可以获得很好的低秩稀疏结构。目前为止，基于块组低秩稀疏结构的图像去噪方法在效果上最为显著，但求解速度较慢、理论匮乏。所以研究基于块组低秩稀疏结构的更广泛的图像处理模型和快速求解方法及其理论是极具有潜力的研究方向。本项目将拓展块组低秩稀疏结构的概念，建立新型的图像处理变分模型和算法，运用到更多的图像处理具体问题，致力于提高图像处理质量，速度和精度。拟研究的主要内容包含如下四个方面：（1）研究基于图像块组低秩稀疏结构的新型图像处理模型；（2）研究设计求解模型的快速算法；（3）研究模型参数的自动选择问题；（4）研究相关模型和算法的理论问题。

图像处理问题广泛存在于卫星遥感、医疗、通信工程、军事等各个领域，研究开发高效高精度的图像处理技术具有重要意义和实用价值。非局部自相似性是图像数据所具有的重要特性，充分利用这一先验信息有助于提升图像质量。本项目深入挖掘这一性质，研究了基于块组低秩稀疏结构的图像先验和正则化理论框架，并进行了理论和应用方面的拓展。针对图像去噪、图像去模糊、图像重建、图像着色、图像分割等具体问题，分别建立了新型的图像处理模型，设计了求解非凸优化模型的快速算法，研究了部分模型中参数的自动选择问题，理论上对模型解的性质和算法的收敛性进行了分析。从数值实验看，与现有算法相比，所提出的算法能够有效提高图像处理质量、速度和精度，具有实际应用价值。在项目资助下，发表相关学术论文23篇，其中SCI检索论文22篇，CCF B类会议论文1篇。