基于物理模型的CT重建及快速多尺度配置法

The development of low-dose CT is still in its early stage, which is limited by the poor reconstruction image quality and motivates the study of fast CT reconstruction algorithm based on an accurate physical model. The CT imaging can be modeled as an integral equation. The discrete system model was consistently used in practice for CT reconstruction, which has unavoidable approximate error from the true physical model. This approximate error degraded the resolution of the reconstructed images. The issue that limits the use of integral equation model in practice is the huge computational cost to solve the integral equations arising from CT imaging. We propose to use directly the physical model of CT imaging, exploit the multiscale structure of the medical CT images, and develop fast multiscale collocation method for CT reconstruction, reducing the computational complexity such that it is comparable to Fast Fourier Transform. We will systematically study the integral equation model of CT imaging which includes several system blurring mechanics, construct the multiscale basis and corresponding multiscale collocation functionals with vanishing moment and biorthogonal properties for solving this type of integral equations in polar coordinate system, and develop the appropriate system matrix compression algorithm based on the constructed multiscale basis and collocation functionals. Based on these, we will develop the fast multiscale collocation method for solving integral equations arising from CT imaging and apply the method to accurate model-based low-dose CT reconstruction.

低放射剂量CT成像的研究迫切要求我们发展基于精确物理模型的快速CT重建算法。CT成像的物理模型是一个积分方程模型。传统的CT重建使用的是离散的系统模型，该模型是连续积分方程模型的分片常数逼近，具有不可回避的逼近误差，限制了重建图像的分辨率。实际应用中未曾直接使用积分方程模型是因为求解该积分方程耗费的巨大计算量。本项目计划利用医学CT图像的多尺度结构，建立应用于CT重建的基于积分方程模型的快速多尺度配置法，在保证逼近精度的基础上将CT重建的计算复杂度降低到和快速傅里叶变换可比较的数量级。我们将研究包含多种物理模糊机制的CT成像积分方程模型，构造在极坐标系下数值求解CT成像积分方程所需的具有消失矩和双正交性质的小波基函数和多尺度配置泛函，并利用它们建立适用的离散化系统矩阵压缩算法。在上述工作基础上，我们将建立求解CT成像积分方程的快速多尺度配置法，并将其应用于低放射剂量CT成像的精确快速重建。

本项目针对低放射剂量断层成像的研究，从理论上研究了低剂量计算断层成像(CT)和发射计算断层成像(ECT)基于连续物理模型的快速迭代重构算法。我们采取了连续积分方程模型以描述断层成像过程，为减低连续积分方程模型带来的高计算量，我们采用了非结构化网格上的分片线性基函数进行连续模型的离散化，提出了基于非结构网格的有限元基函数的快速配置法。为了抑制低放射剂量断层成像的高噪声，我们提出了一种定义在非结构化网格上可保留边缘的正则化方法，将断层成像重建建模为一个不光滑最优化问题的求解。我们利用了次微分和迫近(Proximity)算子等数学工具，将原最优化问题化为等价的不动点方程，提出了一种预条件不动点迭代算法，并证明了不动点迭代算法的收敛性。数值实验证明我们的算法在精度上和速度上相比于传统的基于离散模型的算法都有较大的提高。同时，我们建立了基于高斯核的积分方程模型的快速多尺度配置法并应用于图像恢复问题，在保证逼近精度的基础上将图像恢复的计算复杂度降低到和快速傅里叶变换可比较的数量级。

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