强噪声图像复原问题的数值方法研究

Image restoration is an important research topic in image processing. The aim of image restoration is to get an approximate image as close as possible to the original image from the degraded image contaminated by blur and noise. With the increasing applications of images, image restoration corrupted by strong noises becomes a more and more urgent issue in image processing. But the existing methods of image restoration could not remove strong noises efficiently. By utilizing the knowledge of numerical algebra, image processing, optimization and statistics，this project focuses on constructing numerical methods for image restoration corrupted by strong noises and analyzing the corresponding computational complexity. Based on the principles of image restoration, by analyzing and processing the strong noises in the corrupted images, we will construct restoration models and algorithms for image restoration corrupted by strong noises, and we will further discuss and analyze the mathematical properties of the proposed models and algorithms; we will also restore images corrupted by strong noises by applying data transformation and nuclear norm regularization methods; we will devise methods to solve models using nonconvex regularization terms for strong noises removal; by utilizing the sparsity and special structures of the images and blurring matrices, we will construct fast and efficient iterative algorithms, the corresponding convergence results of the proposed algorithms will also be given. Finally, we will implement the codes of the proposed algorithms and apply them in practical applications.

图像复原是从受到模糊和噪声污染的失真图像中获得尽可能和原始图像接近的清晰图像，是图像处理的重要研究内容。随着图像应用的日渐广泛，受到强噪声污染的图像复原问题日益突出，而现有的复原方法对强噪声的去除不太理想。本项目将运用数值代数、图像处理、优化理论和概率统计知识，寻求被强噪声污染图像复原问题的数值方法，并研究相应的计算复杂性理论。基于图像处理的基本原理，对图像中的强噪声进行分析和处理，构造强噪声图像复原模型和求解方法，对这些模型的代数性质及其解的数学性质进行深入讨论和分析；应用数据变换以及核范数正则化方法对强噪声污染的图像进行复原；构造去除强噪声非凸正则化模型的数值方法；利用图像及模糊矩阵的具体结构和稀疏性质构造快速有效的迭代算法，并分析算法的收敛理论。最后，我们将编制相应的实用程序。

随着图像应用的日渐广泛，受到强噪声污染的图像复原问题日益突出，因而对强噪声图像复原问题的研究有着重要的理论意义和很高的实用价值。本项目按原计划展开研究，完成了原计划的研究内容，取得了预期的研究成果。项目组成员构造了被强噪声污染的图像复原问题的数值方法，并研究了相应的计算复杂性理论。应用Box-Cox变量变换法去除强乘性伽玛噪声，设计了选取Box-Cox变换中最优变换参数的最大似然估计方法，构造了去噪模型并设计了数值求解方法；应用图像块匹配矩阵的秩作为正则项去除图像中不同类型的强噪声，针对不同噪声的特性，设计合理的模型，构造模型的求解方法，分析这些方法的数值特性；对同时受到模糊和加性强高斯噪声污染的图像，构造了基于加权核范数最小的复原模型，所提模型在迭代法的求解运算中，避免了一般核范数最小化图像复原中出现的方程组的病态性，提高了运算性能；将图像强噪声的去除方法应用于图像Retinex问题中。基于这些研究结果，在国内外学术刊物上发表论文11篇。项目组成员多次参加国际会议和国内会议及研讨会并作报告，进行多次学术访问。举办国际会议和国际研讨会各1次。培养硕士生10名，5名已毕业；博士生4名，1名已毕业。