基于稀疏优化和低秩矩阵分解的高维图像重建研究

Image reconstruction is one of the essential topics in image processing, and the related problems are ill-posed inverse problems. Due to the ill-posedness, an effective strategy to obtain a reliable solution is to use some regularization techniques. According to the problem type, a regularization model is constructed incorporating some prior knowledge of the original image. With the improvement of imaging equipment, the data of the image (image size and image dimension) is getting bigger and bigger. Therefore, more reasonable models and more efficient algorithms are required. This project focuses on high dimensional image reconstruction problems based on sparse optimization and low-rank matrix decomposition. Finally, new optimization models or methods with high extendibility will be proposed for the high dimensional image.This project mainly aims at addressing this fundamental problem in three aspects: (1) design new image reconstruction models in image processing, and study the corresponding solving algorithm; (2) design approximation functions for tensor rank, and establish the corresponding model for high dimensional image restoration via tensor completion; (3) expand our research to study medical image restoration problems and spectral unmixing problems of hyperspectral images.

图像重建是图像处理研究中的一个重要领域，其问题通常是不适定的反问题，求解此问题的常用方法是基于真实图像的先验信息引入正则项，并构建相应的正则化模型。随着成像设备的提升，我们获取的图像规模（图像大小以及图像维度）越来越大，合理的模型，高效算法的研究势在必行。本项目围绕基于稀疏优化和低秩矩阵分解的高维图像重建问题这一前沿课题展开，并最终提出适用性较强并具有可扩展性的新型优化模型和方法。本项目拟主要研究如下三个方面：(1) 设计图像处理中的新型图像重建模型，并设计相应快速求解算法；(2) 设计张量秩的新型近似函数，拟建立相应张量完备模型，研究高维图像修复问题；(3) 扩展研究医学图像修复和高光谱图像解混问题。

项目组通过努力，基本完成了项目了研究目标，取得了一定的研究成果，成果主要围绕数学基础与图像处理两方面进行研究。在数学基础领域，根据图像处理、视频修复的张量运算通常需要展开成矩阵计算，从而会产生矩阵方程，项目组研究了矩阵方程的相关算法与求解，发表了相应的学术论文；在图像处理领域，项目组围绕图像重建中的图像去雾问题进行研究，结合现今热门的深度学术技术，对现有的图像去雾方法进行了改进，一定程度上提高了雾天图像重建的质量，发表了相应的学术论文。成果多为交叉应用，有较好的应用前景和科学意义。