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摘要

Mojette变换是一种最小冗余采样的离散Radon变换, 能够用较少角度的投影数据进行精确的计算层析(computed tomography, CT)重建, 为少量投影角度CT技术的实现提供了一种新思路. 投影角度的空间布局决定了层析重建最少所需投影的数量. 为了获得Mojette变换层析技术中的最优投影空间角度布局方案, 本文对三维Mojette变换数学模型及其精确重建条件进行了研究. 以此为基础, 在考虑实际探测器像素数目受限的条件下, 提出了确定最优投影角度的方法. 研究结果表明: 所有探测器围绕被测物体在同一水平面内进行平行投影采集是最优的投影角度布局方案, 此时投影模型为二维Mojette变换, 所需的投影角度和探测器像素数最少, 投影角度范围最小; 若在实际的测量中该投影条件无法满足, 则投影矢量中| p_i|和| q_i|的值越小越好. 该研究可为实际层析系统的建立提供理论基础.

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Abstract

Computed tomography (CT) is an effective tool for three-dimensional (3D) imaging by using optical detectors to capture the two-dimensional (2D) projections of tested parameters from multiple views and realizing 3D reconstruction through various algorithms. However, for practical applications, typically only a few detectors can be applied due to their high expense and the limited optical access of the test environment. The realization of high precision reconstruction with a few projections is of great significance for promoting the development and application of CT technology. The spatial arrangement of the detectors determines the amount of useful information collected by the system, which greatly affects the quality of CT reconstruction. Therefore, in this work we study the optimization method of projection arrangement based on the 3D Mojette transform theory. Mojette transform is a special discrete form of Radon transform, which can realize projection sampling with minimum redundancy and accurate tomographic reconstruction from less projection angles. It provides a new way to realize the CT technology with fewer projections. However, the existing researches mainly focus on the reconstruction theories of 2D Mojette transform, which is used for realizing the 2D slice tomography. In order to realize the real 3D tomographic reconstruction, in this work we establish a mathematical model of 3D Mojette transform, and study its accurate reconstruction condition. The results show that the 3D Mojette transform is a combination of twice 2D Mojette transform in two directions. The accurate reconstruction condition of 3D Mojette transform is that the sum of the absolute values of projection vectors’ components in x, y, and zdirections is greater than the number of discrete grids in each direction. The correctness of the mathematical model and the accurate reconstruction condition are verified by numerical simulations. Considering the limitation of the pixels in the practical detectors, the method to determine the optimal arrangement of projection angles is proposed. The results indicate that the optimal arrangement is that all detectors are located in the same horizontal plane around the tested object, where the projection model is reduced to 2D Mojette transform. In this case, the minimum projection angles and pixels are required and the projection angles can be positioned in a smaller spatial range. If the condition cannot be satisfied in practice, projection vectors with smaller | p_i| and | q_i| should be chosen. This research provides the theoretical basis for establishing the actual CT system.

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作者及机构信息

通信作者:王佳,wangjiar1001@126.com

基金项目:国家自然科学基金(批准号: 61701385)、装备预研领域基金(批准号: 61406190501, 61406190301)和陕西省科技计划(批准号: 2019CGXNX-037)资助的课题

Authors and contacts

Corresponding author:Wang Jia,wangjiar1001@126.com

Funds:Project supported by the National Natural Science Foundation of China (Grant No. 61701385), the Foundation of Equipment Pre-research Area of China (Grant Nos. 61406190501, 61406190301), and the Science and Technology Project of Shaanxi Province, China (Grant No. 2019CGXNX-037)

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参考文献

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下载: 全尺寸图片幻灯片

条件	投影矢量	(7)式是否满足	是否能精确重建
Case 1	(15, 1, 1), (13, 1, 1), (9, 1, 1), (7, 1, 1), (4, 1, 1)	√	是
Case 2	(15, 1, 1), (13, 1, 1), (9, 1, 1), (7, 1, 1), (3, 1, 1)	×	否
Case 3	(1, 15, 1), (1, 13, 1), (1, 9, 1), (1, 7, 1), (1, 4, 1)	√	是
Case 4	(1, 15, 1), (1, 13, 1), (1, 9, 1), (1, 7, 1), (1, 4, 1)	×	否
Case 5	(1, 1, 15), (1, 1, 13), (1, 1, 9), (1, 1, 7), (1, 1, 5)	√	是
Case 6	(1, 1, 15), (1, 1, 13), (1, 1, 9), (1, 1, 7), (1, 1, 3)	×	否

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