PET医学图像重构的积分方程方法

This project is devoted to a development of an integral equation method for PET medical image reconstruction with high-accuracy. As one of the most important means for early diagnosis and treatment guidance of cancer, PET imaging has attracted great attention in medicine, nuclear physics and computational science. The discrete reconstruction model compatible with PET’s data acquisition method has been solely and consistently used in academic research and clinical practice of PET imaging. However, due to its low accuracy, it has become a bottleneck in the development of PET image reconstruction with high-accuracy. In this project, we shall build a semi-continuous, semi-discrete integral equation model for PET image reconstruction based on its physics and data acquisition principal, and develop a fast numerical method for PET image reconstruction with high-accuracy. In order to suppress image noise and improve spatial resolution, a sparse regularization using a multi-scale approximation of the solution of the integral equation, based on the priori information of PET medical image will be proposed for the resulting new ill-posed discrete system derived from the high-accuracy approximation of high order piecewise polynomials. Furthermore, we shall develop an aggregation method for selecting parameters of the multi-parameter regularization. We shall design a fixed-point proximity iteration scheme for solving the resulting large-scale non-smooth non-convex optimization problem for the reconstruction, with preconditioning to overcome its ill-posedness and improve the convergence speed of the iteration. This project will lead to a breakthrough in the mathematical modeling and numerical computation of PET medical imaging reconstruction.

本项目致力于发展PET医学图像高精度重构的积分方程方法。作为癌症早期诊断和指导治疗最重要的技术之一，PET成像技术已成为医学、核物理、计算科学等学科的热点研究方向。学术研究和临床上的PET成像一直沿用与数据采样方式相适应的离散模型。由于其精度的限制，离散模型已成为发展PET高精度成像方法的瓶颈。本项目拟根据PET成像的物理原理和数据采样方式，建立PET成像的半连续-半离散积分方程模型，并发展其高精度快速有效的数值计算方法。为了抑制图像噪声并提高空间分辨率，将基于医学图像的先验特征，对由高阶分片多项式的高精度逼近所导出的新的不适定离散方程，提出多尺度近似稀疏正则化方法，并发展多正则化参数的聚合法选取策略。对于由此得到的大规模非光滑非凸的重构优化问题，拟发展带预处理的不动点迫近迭代算法，以克服重构问题的病态性、提高算法的收敛速度。本项目将在PET医学成像的数学建模和数值计算方面做出原创性突破。

正电子发射型计算机断层成像（PET）是当今医学临床上诊断和指导肿瘤治疗的最佳手段。本项目建立了PET成像系统的连续积分方程模型，并在此基础上采用张量型与插值型的高阶分片多项式多尺度逼近，发展了高精度的离散策略。借助示踪剂分布函数的先验光滑性质和小波函数的消失矩性质，构造了适用于连续模型高精度逼近解的正则化方法。建立了一种基于自适应非结构化网格的正则化图像重建方法以及提出了一种全新的近似稀疏正则化模型。采用不动点方程刻画相应的凸或非凸优化问题的解，并基于该刻画，结合预条件与动量加速技术，发展了快速数值求解算法，最后分析了不动点算法的收敛性以及收敛速度。项目执行期间，成员共发表SCI期刊论文16篇。