单光子发射计算机断层成像（SPECT）重建算法研究

Single photon emission computed tomography (SPECT) is one of the important molecular medicine imaging techniques. The complex physical process of SPECT imaging causes the projection data is seriously noised, and often the partial data problem theoretically. The reconstruction techniques of SPECT image can be classified in analytical method and algebraic method, and the qualities of SPECT image are directly affected by mathematical method of reconstruction. The analytical methods are not able to handle the partial data problem, and the reconstructed image qualities are very low. Though the algebraic method can give better image, the computational cost is very high, and the convergent rate of the existing algorithms is low. Due to the ability of the image qualities improvement, the investigation of regularized approach and numerical algorithm is a new trend in SPECT image reconstruction field. In this project, we focus on the theories、models and numerical algorithms of SPECT image reconstruction. Firstly, integrating the characteristics of projection data、(high) total variance norm、 norm, we will design the regularized models for SPECT image reconstruction and the simultaneously regularized model, which poses regularization in image domain and transform domain. Secondly, we investigate the generation of desired perturbation and termination conditions of superiorized iteration, and apply the superiorization algorithm to implement the designed regularized models. Furthermore, we will use different mesh size for smooth and edge regions in the reconstructed image domain, which can decrease the computational cost dramatically, and modify the existing EM algorithm to accelerate the convergent rate. Lastly, we exert ourselves to explore the theories and algorithms of analytical reconstruction method for partial data problem.

单光子发射计算机断层成像（SPECT）是重要的分子医学成像技术之一。SPECT成像复杂的物理过程导致投影数据质量较差及在理论上的不完全性。SPECT重建的方法包括解析方法和代数方法，重建方法的选择直接影响图像重建质量和计算量。解析方法通常不能处理不完全数据的重建问题，且重建图像差；代数方法的重建图像质量较高，但该方法的计算量大，且没有高效算法。由于能有效提高图像质量，正则化重建模型和算法是当前研究的趋势。本项目主要研究SPECT重建的理论、模型和算法：研究建立结合SPECT投影数据特点、（高阶）全变差范数、 范数的正则化重建模型，以及在图像域和变换域同时正则化的模型；研究模型的superiorization求解算法，以及扰动生成和终止条件；改进代数重建模型，在不同区域使用大小不同的离散网格，降低代数重建问题的规模；改进现有的迭代算法，加快算法的收敛速度。研究不完全数据的解析重建理论和算法。

根据项目计划书，我们主要围绕图像重建的正则化模型、算法和应用展开。模型方面：为了避免全变差正则化方法的阶梯效应，研究了基于二阶全变差的重建模型和本原对偶算法；基于l1范数极小化的CT（SPECT）图像重建模型和梯度下降法；研究了基于正则化伴随状态的SPECT识别问题的反演方法；自适应加权的并行MRI重建模型。算法方面：除了研究以上模型的求解算法外，我们提出了改进的superiorization算法以及speriorized 迭代算法扰动参数的计算方法，并作了详细的理论分析，得到了两个重要定理。在具体应用方面：提出分块存储成像系统矩阵的方法以节约计算资源；根据投影数据的对称性给出CT投影数据的矫正算法；使用TV正则化方法重建页岩微观结构。这方面的研究将对我国的页岩气得勘探和开采具有潜在应用价值。