SPECT/CT成像与图像重建的数学建模及快速科学计算方法

As an important methodology in medical research, medical imaging provides functional and anatomic bioinformation of patients by means of imaging technique. It is widely applied to clinical diagnosis and treatment, and has become a research focus of medical science, computational science and physics. This project is dedicated to mathematical modeling of specific medical images and development of fast scientific computing methods. Specifically, we will develop continuous integral equation models for SPECT/CT imaging systems. We then discretize the continuous imaging models using high order piecewise polynomial multiscale approximation and propose a high precision discretization strategy. We plan to develop a sparse matrix representation for the involved integral operator. In order to overcome the ill-posedness of image reconstruction problem, suppress random noise and improve spatial resolution, we introduce a novel class of multiscale structure-based sparse regularization methods suitable for piecewise polynomial solutions of the integral equation. We plan to accomplish this task by utilizing the vanishing moment of multiscale wavelet basis. For parameter determination strategy in multi-parameter regularization, we propose to exploit the method of aggregation of regularized solutions. We further plan to characterize the solution of the resulting optimization problem via a system of fixed-point equations and derive other equivalent fixed-point characterizations. We then develop fast preconditioned fixed-point algorithms based on the equivalent characterizations. Finally, we will study the convergence property and convergence rate of the proposed algorithms. This study will lead to high-quality research results, and publish 4 to 5 high-quality papers.

作为医学研究的重要手段，医学影像利用成像技术获得人体功能性和结构性的生物信息。其被广泛应用于临床诊断和治疗，已成为医学、计算科学和物理等学科的热点研究领域。本项目致力于研究特定医学影像数据的数学建模以及相应的快速科学计算方法。具体地，本项目拟建立SPECT/CT成像系统的连续积分方程模型，并在此基础上采用分片多项式多尺度逼近，发展高精度离散策略，建立成像积分算子的稀疏矩阵表示；利用多尺度小波的消失矩性质，构造适用于连续模型高精度逼近解的多尺度稀疏正则化方法，以克服重建问题的不适定性、压制噪声并提高空间分辨率；提出正则解聚合法作为多参数正则化的参数选取策略。本项目拟采用不动点方程刻画图像重建优化问题的最优解，对所得不动点刻画进行适当的等价变换，并基于该等价刻画发展快速的预处理不动点迭代算法，最后分析不动点算法的收敛性以及收敛速度。预计本项目将产生高质量的研究成果，发表一流的学术论文4至5篇。

发射型计算机断层成像（ECT）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SPECT与PET两种ECT成像系统的连续积分方程模型，并在此基础上采用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SPECT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7篇。

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