图像恢复与重建中的参数自动选取理论及方法

In mathematics, the problem of image restoration and reconstruction is ill-posed and a regularization approach should be used in the image restoration and reconstruction process. It is very important and challenging to study this problem since the quality of the image restoration and reconstruction depends on the selection of the regularization parameter, moreover, the large scale of the system matrix makes it very difficult to solve the problem efficiently. This project is devoted to studying the selection theory and the methods of regularization parameters and realizing the automation and intelligence of image restoration and reconstruction. Different noise models and the regularization terms such as total variation and low rank matrix are considered in this project, data-driven parameter selection models are proposed to avoid data over-fitting or under-fitting; the methods such as variable splitting and duality are used to solve the numerical difficulty caused by prior constraints and non-convex regularization terms; the theories and methods such as subspace and pre-processing are used to solve the problem of time-consuming large-scale matrix calculation derived by optimization problems; the applications such as cryo-EM are considered. The research results of this project can be applied to other automated parameter selection problems in the fields of medical imaging and structural biology, which have high application value and research significance.

图像恢复与重建在数学上是一个病态问题，正则化理论是解决其病态性的有效方法。正则化方法中参数选取的优劣影响着图像恢复与重建质量，然而系统矩阵的大规模性使得参数的高效求解非常困难，因此正则化参数的选取是一个重要且具有挑战性的问题。本项目致力于研究正则化方法中参数的选取理论及方法，实现图像恢复与重建的自动化和智能化；研究不同噪声模型下采用总变分和矩阵低秩约束等正则项时的参数选取方法，建立数据驱动的参数选取模型，避免数据的过拟合或欠拟合；利用变量分裂和对偶理论等方法，解决先验约束、非凸正则项等所造成的数值计算困难问题；应用子空间和预处理等方法，解决优化问题所导致的大规模矩阵计算耗时问题；实现冷冻电镜等实际问题中图像的自动恢复与重建。本项目的研究成果可以应用于其他如医疗成像、结构生物学等领域的图像自动恢复与重建中，具有很高的应用价值和研究意义。

本项目主要研究图像恢复与重建中参数选取问题的数值方法及理论，以推动图像恢复及重建技术向自动化方向发展。主要成果概括如下：.1）基于偏差原理的正则化参数选取方法。利用偏差上界参数能够通过观察数据的统计特性获得的优点，提出直接求解约束模型的思路，发展了求解约束模型的有效数值计算方法，建立了由偏差上界参数直接计算所对应的正则化参数的显性公式。针对全变差正则化图像去噪问题，提出了采用一致性原理估计偏差上界参数的新方法。针对低秩矩阵正则问题，提出采用随机化奇异值分解以减少运算时间、提高计算速度。研究成果应用于冷冻电镜三维重构问题。.2）基于变分贝叶斯推断的正则化参数选取方法。针对基于凸变Mumford-Shah变分模型的图像分割中的多参数选择问题，提出了基于变分贝叶斯推断的图像分割模型，利用交替迭代方法分别估计模型的解和模型的正则化参数，所有参数都迭代更新，以实现自动调整参数的目的。.3）研究了泊松去噪问题，提出了基于低秩矩阵正则的泊松噪声去噪模型并发展相应的数值计算方法计算模型解，证明了算法的收敛性。针对图像卡通与纹理分解问题，发展了求解Meyer模型的快速算法。.在项目资助期间，项目组在SIAM J. Sci. Comp.等期刊上发表SCI期刊论文9篇；荣获省级自然科学奖二等奖2项；获得国家发明专利授权2项；培养博士生4名、硕士生11名。