基于分数阶导数的图像重建方法及其在工业CT中的应用

CT image reconstruction has important applications in the fields of industry, medical diagnose and nuclear physics. The goal of image reconstruction is to recover the inner structure of objects and obtain quantitative character of materials, such as density and edge position. The time cost of image reconstruction is also important to application fields. This project will do investigation on image reconstruction modeling and algorithm. Instead of integer-order derivatives, the fractional-order derivatives based image reconstruction models will be approached in order to get better recovery of image textures. The order of fractional derivative will be chosen adaptively according to the local structure of image. The Alternating Direction Method of Multipliers will be used to solve the proposed models. The original optimization problem will be divided into a few sub-problems to solve. The challenge work is how to construct the sub-problems, and make sure their convergence to the original problem. The fast algorithm for each sub-problem is going to be designed. Finally, the proposed models and algorithms will be tested by synthetic data and real data.

CT图像重建问题在工业、医学以及核物理诊断中有重要应用。对被测物体内部精细结构的重建和某些物理量（如密度、边界）的定量诊断是图像重建的目的，图像重建速度也是应用领域十分关注的问题。本项目拟在图像重建的模型和算法方面开展工作。建立基于分数阶导数的图像重建模型，用分数阶导数取代整数阶微分算子，根据图像局部结构，给出分数阶导数阶数的自动选取方法，以达到对图像精细结构的重建。算法方面，拟基于交替方向优化法ADMM的思想，将原问题分解为几个子问题分别求解，研究子问题的分解方式、子问题的快速算法以及对原问题的收敛性。利用模拟数据和实验数据对所提的模型和算法进行检验。

基于投影的图像重建问题在医学、工业、核物理等领域都有重要应用，基于在武器物理研究中高能x射线成像问题的需求背景，我们主要研究了具有轴对称性的三维物体的物理量重建。建立了不同噪音类型的图像复原与图像重建数学模型及算法，在模型中引入了分数阶全变分正则化项，给出了基于ADMM及Primal-Dual的数值求解方法，数值结果显示了分数阶模型较整数解模型对纹理图像去噪及细节重构方面的优势。数值试验验证了所提模型和算法的有效性。研究工作展示了分数阶导数在图像处理中的应用前景，发展了图像重建及复原的相关模型和算法，促进了优化问题算法的研究，也将进一步推动分数阶导数的理论研究。