基于稀疏优化的有限角度CT采样条件及重建算法研究

Compared to traditional iterative reconstruction algorithms, the sparse optimization-based methods has acquired significant improvements in limited views computed tomography(CT), which not only broaden the geometrical limitation of scanning, but also lay the theoretical foundation for new imaging system. However, due to the lack of mathematical and physics characterization for deficiency of continuous angular data, there is almost no theoretically analytic results of exact reconstruction sampling conditions for limited views problem. Based on the theory of sparse optimization, we attempt to conduct study on sampling conditions and reconstruction algorithms for limited view problems: (1) The method to estimate the sample angle bound of projection data will be investigated by analyzing system matrix, distributed system matrix, and transform matrix for spatial. And digital phantoms will be designed to estimate the performance of reconstruction algorithms by comparing the sample angle bound computed by the estimating method. (2) To restrain the effects caused by data deficiency of continuous angle, modification strategy incorporated with segmentation and curvature regulation will be studied; (3) Improving the efficiency of the methods for sampling condition estimating and iterative reconstruction by applying the distributed strategy of sparse optimization. The study of the project will promote the understanding and cognition to the singularity resulting from the deficiency of continuous angular data. Besides, it will help to narrow the gap between theoretical analysis and real reconstruction performance, and offer theoretical analysis and reference for limited-view CT problem in practice.

基于稀疏优化重建算法已经在有限角度CT成像实际应用中取得了显著优于传统迭代算法的效果，放宽了扫描的几何限制，也为各种新型成像系统的设计奠定了理论基础。然而，由于缺乏对连续角度范围数据缺失造成影响的数学物理刻画，现阶段对于有限角度精确重建采样条件几乎没有理论分析结果。本项目以稀疏优化理论为基础，从采样条件和重建算法两方面展开研究：（1）对有限角度下的系统矩阵、分布式矩阵及频域变换矩阵性质进行分析，研究精确重建采样角度范围的估计方法，并设计专门用于有限角度重建算法性能测试的数字体模；（2）针对连续扫描角度缺失对重建的影响，探索分割结合迭代及曲率正则化的修正策略；（3）结合分布式稀疏优化策略，改善采样条件分析及迭代重建的算法效率。本项目的研究，将直接推动对连续角度范围数据缺失所产生奇异性的理解和认识，缩小理论分析与实际重建性能的差距，也将为有限角度新型成像系统的设计提供理论分析和参考。

基于稀疏优化的重建算法已经在有限角度CT成像实际应用中取得了显著优于传统迭代算法的效果，放宽了扫描的几何限制，也为各种新型成像系统的设计奠定了理论基础。然而，针对有限角度重建问题，仍然缺乏对连续角度范围数据缺失造成影响的数学物理刻画，因此缺少针对有限角度问题特性的精确重建理论分析研究。本项目以稀疏优化理论为基础，基于TV最小化模型分别对空域和频域采样矩阵研究了有限角度精确重建采样条件，量化分析了不同扫描角度范围下精确重建所需的投影数量。针对连续角度范围数据缺失对重建图像边缘和曲率信息的破坏，设计了Euler弹性正则化重建模型；利用有限角度采样分析理论，提出了基于支集补信息的重建策略；同时考虑投影域图像域奇异性的刻画，设计了数据驱动双域联合正则化重建模型；此外还提出了分割结合迭代重建策略及基于深度网络的投影补全策略等一系列重建算法。为了改善计算效率，利用交替方向法设计了求解更高效的有限角度精确重建采样条件分析算法，同时提出了基于有序子集的交替方向重建加速算法设计框架。通过本项目研究，推进了对连续角度范围数据缺失所产生奇异性的理解和认识，缩小了理论分析与实际重建性能的差距，也将为有限角度条件下的新型成像系统设计提供理论参考。