针对计算机断层成像(computed tomography,简称CT)中投影数据与图像重建关系,综述了CT在投影策略方面对重建质量的影响.对不同采样策略获取的不完全投影数据,应用迭代类算法对投影数据进行重建,研究了均匀采样和非均匀采样情况下不同数据结构对重建图像质量的影响.对仿真数据和实际数据重建结果进行分析,同时对不同策略下的投影数据结合其数据分布特点探讨了重建质量优劣的原因.可以为CT重建领域的研究工作者提供全面的采样方法梳理和总结,为当前不完全投影数据获取方式对应的算法改进提供思路,最后对当前研究重点和未来发展加以展望.
Computed tomography (CT) is an imaging technique which produces cross sectional map of object from its projections. Image reconstruction algorithms require collection of projections covering the whole measurement range. Incomplete projection is still a hot research topic. This paper reviews the relationship between projection data and image reconstruction in computed tomography, and summarizes the effect of computed tomography on reconstruction quality. For the incomplete projection data acquired by different sampling strategies, the iterative algorithm is used to reconstruct the projection data. The effects of different data structures on the reconstructed image quality under uniform sampling and non-uniform sampling are studied, and the results are compared and analyzed. Meanwhile, the reasons of the reconstruction quality of the pros and cons are discussed in conjunction with the projection data distribution with different strategies. This paper provides a comprehensive sampling method for researchers in the field of CT reconstruction, and offers some ideas for the improvement of the corresponding algorithm for incomplete projection data. Furthermore, it also points out current focus of the study and research direction in future.
计算机断层成像(computed tomography, 简称CT)技术是通过对物体进行不同角度下的射线投影测量进而获取物体截面信息的成像技术[
Zou等人[
针对投影数据不足的重建图像质量问题, 现阶段的解决思路有两种:一是发展新的重建算法, 添加先验约束, 如单材质、低密度等到迭代过程中影响图像质量; 二是从投影数据本身入手研究投影幅数与重建质量之间的关系, 从数据完备性上影响图像.因此, 研究不完全投影基于迭代算法框架下的不同采样策略对重建图像质量的影响, 对不完备数据重建如何通过合理采样获取高质量重建图像有一定指导意义.
工业CT的物理原理与医用CT基本一致, 医用CT通常采用低能射线.射线源电压最高可达80KV, 而工业CT所用的射线能量范围则大得多, 高达450KV.当一定能量的射线穿越物体时, 由于吸收和散射, 射线将产生衰减.研究指出, 射线穿过物质并与之发生作用后, 射线强度将受到射线路径上物质的吸收而衰减, 衰减规律由比尔定律确定[
CT投影示意图
Schematic diagram of CT projection
且路径
对于给定的二维平面区域
其中,
对沿着投影方向上的每条投影线的密度函数进行积分, 就得到了该射线上物体衰减的投影值[
采样投影与积分
Sampling projection and integration
Radon为CT技术建立了数学理论的基础, 认为:任何物体线性衰减系数分布可由其所有积分集合确定, 模型为
由此可知, 只要获取所有积分信息, 即可去求取反映物体断层内部结构和组成的数值信息.对单色窄束X射线而言, 物体中每个体素值将由唯一的衰减系数确定, 获取断层衰减系数分布是断层图像重建的最终目的.若己知
CT投影采样、重建流程示意图
CT projection sampling, reconstruction flow chart
传统CT重建算法需要完备的投影数据[
不完全投影数据采样方式正弦图分布
Incomplete projection data sinograms distribution
目前, 由投影重建图像问题主要有:解析法和迭代法, 其中解析法在完全投影的条件下可获得高质量的重建图像, 不完全投影条件下重建切片图像的质量较差, 然而迭代法可用于不完全投影条件下图像重建.
迭代类算法可以解决解析算法不能处理的离散数据问题, 其对不完全数据重建有着不可替代的优势[
其中,
由于
式中,
利用
其中,
对于模型中参数
由于不同的迭代算法模型各异, 性能不同, 重建结果也不同, 本文主要探讨投影采样策略对应的不同数据类型对重建质量的影响, 在算法本身对重建图像质量产生的影响方面未作比较.下文中不同采样模式的投影重建均采用基于CS构架的TVAL3重建算法[
根据短扫描方式下重建条件, 只要保证扫描角度为180+
圆轨迹下[0, 180°]范围内投影数据采集
Projection data acquisition in the range of [0, 180°] under circular locus
不同采样策略下圆盘投影信息分布.(a)~(e)为多孔圆盘对应的1, 2, 3, 4, 5这5种模式
The projection information distribution of phantom and disc. (a)~(e) are the porous disc for modle 1, 2, 3, 4, 5
按照上述采集方式得到的投影数据进行重建, 仿真参数如下:射线源到探测器距离600mm, 射线源到旋转中心距离500mm, 采样范围[0, 240°], 重建图像分辨率为256×256.根据真实的扫描投影背景区域的幅值变化, 仿真投影过程中增加
稀疏采样策略下的重建结果
Reconstructed results under the sparse sampling strategy
均匀和非均匀稀疏采样圆盘投影的重建图像沿AB线灰度分布
The gray-scale distributions of reconstructed images along the AB line for the uniform and non-uniformly sampled disc
均匀与非均匀采样重建结果对比.(a)~(c)不同数目均匀、非均匀采样重建对比.a1, 2, 3为均匀采样90幅重建结果; b1, 2, 3为非均匀采样120幅重建结果; c1, 2, 3为非均匀采样150幅重建结果
为了验证投影方式对重建切片影响规律的一致性, 对真实的铝质工件#A1获取投影.实验数据的CBCT系统为德国Yxlon公司的Y.TU 450-D02, 探测器为Varian公司的PaxScan 2520, 投影采集过程中扫描电压为200KV, 曝光量0.4mA·s, DSO为925.041 7mm, DOD为286.925 1mm, 锥形束CT的锥角为5°, 圆周扫描采集360幅投影, 在[0, 240°]范围内均匀选取60幅、30幅投影和非均匀的90幅、60幅投影进行重建, 重建图像分辨率为512×512.
均匀与非均匀实际投影重建结果对比
Comparison of uniform and non-uniform projection reconstruction results
虽然迭代类算法可以有效对投影数据进行重建, 但基于算法本身的属性, 需要更多的投影数据补充到迭代过程中来修正重建带来的伪影对图像质量的影响.圆轨迹采样奇偶数对重建结果也存在影响, 合理的采样策略对重建结果至关重要.对于[0, 180°]范围内的采样幅数来说, 无论奇数投影数还是偶数投影数, 对应在[0, 360°]范围内均为偶数情况, 所以仅讨论0~360°范围内的情况.这里仅对含噪声投影图像进行讨论, 噪声幅度与前面的设置一致.
圆轨迹下[0, 360°]范围内奇偶策略投影数据采样
Parity projection data sampling in the range of [0, 360°] under circular trajectory
对本节奇偶投影采样投影进行重建,
奇偶数采样角度下的重建结果
Reconstructed results with odd and even sampling angles
不同稀疏采样重建质量对比.(a)~(e)分别为偶数与奇数采样圆盘沿CD线投影重建图像的灰度分布; (d)~(f)分别为偶数与奇数采样Phantom头模型沿EF线投影重建图像的灰度分布
对重建结果进行整体评价, 评价方法采用均方误差(mean square error, 简称MSE)[
图像质量数值比较
Comparison of image quality values
评价内容 | 偶数幅采样 | 奇数幅采样 | |||||
60 | 30 | 10 | 59 | 29 | 9 | ||
Phantom头模型 | MSE | 399.72 | 2 067.6 | 5 369 | 268.936 | 1 943.4 | 4 300.6 |
PSNR | 22.113 0 | 15.146 7 | 10.831 9 | 23.834 3 | 15.245 3 | 11.795 5 | |
NAAD | 0.125 5 | 0.274 1 | 0.459 0 | 0.107 2 | 0.263 2 | 0.403 1 | |
NMSD | 0.125 9 | 0.285 7 | 0.461 3 | 0.103 2 | 0.277 5 | 0.412 9 | |
圆盘 | MSE | 617.95 | 5 321.1 | 20 664 | 418.81 | 5 295.4 | 18 609 |
PSNR | 20.221 2 | 14.473 6 | 4.978 6 | 21.910 6 | 15.891 8 | 5.433 5 | |
NAAD | 0.057 0 | 0.185 3 | 0.335 1 | 0.049 2 | 0.169 5 | 0.322 1 | |
NMSD | 0.061 7 | 0.189 6 | 0.357 0 | 0.050 8 | 0.180 7 | 0.338 7 |
由
同样,
奇偶数真实投影重建结果对比
Comparison of odd-even projection reconstruction results
从真实投影重建结果来看, 根据
基于工业上零件的高密度性局部角度射线无法穿透的问题以及医学上人体辐射的低剂量问题, 通常导致信息获取时局部范围采样信息缺失, 继而出现投影信息分段.分段连续信息采样往往是物体投影无法获取情况下的数据结构[
分段连续采样
Segmented continuous sampling
均匀分段连续投影的过程是采样点和非采样点均匀连续分布的叠加, 如
分段连续采样策略设置
Continuous sampling strategy setting
策略 | 设置 | 采样数据 | 投影总数(幅) |
均匀分段连续 | 模式1 | [0, 360°]范围内间隔15°进行连续采样35幅(每幅/3°) | 105 |
模式2 | [0, 360°]范围内间隔15°进行连续采样25幅(每幅/3°) | 100 | |
非均匀分段连续 | 模式3 | 0°~120°、180°~275°、295°~360°等不同范围内进行连续采样(每幅/3°) | 94 |
模式4 | 30°~170°、190°~210°、255°~360°等不同范围内进行连续采样(每幅/3°) | 89 |
仿真投影sinogram图像分布
The distribution of simulation projection
对获得的叶片和心脏模型分段连续投影进行重建,
分段连续采样投影重建图像
Reconstructed images with successive sampling projections
上述分段连续投影采样策略重建结果均在缺失投影较少情况下伪影不是特别明显, 但是局部有偏差; 当缺失投影较多时, 重建结果均较差.由于缺失角度对应位置无投影, 该角度位置的约束力弱, 其仅由其他采样位置的射线来补偿, 即采样位置的数据反馈给缺失位置体素的信息过少, 因此局部轮廓位置重建结果较差.
迭代重建算法相比解析算法最主要的区别就是将重建图像模型化.迭代算法中, 重建图像被离散化, 因此迭代重建算法的实质就是解线性方程组, 其将真实的成像几何结构与成像物理效应模型化.通过对上述不同类型的采样策略下获得的投影信息应用迭代算法进行重建, 得到了不同质量的重建图像.
从
分析认为, 数据采样过程实际上是射线与重建体素相交的过程, 射线的重建体素之间的观测关系可以由一系列线性方程组表示, 如
射线与重建体素相交模型
The intersection of ray and reconstruction voxel
由于迭代算法中投影方法与重建模型中的权系数
整个影响过程可以由迭代算法的思想来反映:对于给定的投影数据, 将所得方程组看成是
投影次序超平面示意图
Projection plane diagram
从表达式(12)可以看出, 第i条射线遍历的体素迭代的过程以残差最小化的方式在逼近.对于没有进行投影的位置, 没有残差更新.而非均匀采样情况下, 不同位置体素更新的次数不同, 缺失投影位置体素迭代更新较少, 所以相对采样点来说, 缺失投影位置伪影比较严重.
含两个变量方程组解的过程
Solution of equation solution with two variables
其中,
根据上述
不同采样类型相关性
Correlation of different sampling types
基于上述分析, 针对不同采样策略获得的投影信息必须发展新的算法, 进而改进边缘轮廓和完成内部灰度更新.从而使得在相同投影数据结构下进行重构, 实现数据挖掘.
本文主要针对有限角度类型下的不完全投影数据重建结果探讨采样策略对其质量的影响, 研究了均匀采样和非均匀采样情况下投影数据的特点.分别对仿真投影数据和真实投影数据按照不同采样策略进行获取, 并结合迭代类算法的优势对不同采样策略下的离散投影进行重建, 分析了不同投影类型对重建质量的影响.结果表明, 圆轨迹扫描下的均匀采样策略较非均匀采样策略重建质量要好, 且少量均匀数据比多非均匀数据结果更优; 再者就是对于满足重建要求的投影采集来说, 奇数幅投影相对偶数幅投影重建结果要好一些.因此, 对于数据采样, 可以得出以下总结:圆周采样时, 应优先选择均匀采样策略; 对于无法实施均匀采样策略的应尽可能地使采样角度逼近均匀分布; 优先选择奇数投影数进行重建.同时, 为了减小被访问投影角度的相关性, 采样的投影角度应该在视角范围内尽可能均匀分布, 且投影角度避免出现聚集.实验结果包括了不同结构类型的仿真对象和真实铝质工件#A1、钛质工件#A2扫描对象, 结果和分析表明了该结论的可靠性和针对不同对象具有的广泛适用性.所以, 在现有的大量文献中, 学者们研究着重于低剂量稀疏采样重建[
基于工业对象异形弯曲等特点, 某些角度范围内X射线无法有效穿透物体导致该位置投影缺失, 出现投影数据分段, 所以研究有效算法、降低由于投影数据结构引入的伪影影响, 无疑有着更广的应用价值.当然, 投影信息的保真度是后续高质量重建的保障, 例如, 重建之前对投影的去散射[
通过对稀疏采样、奇偶性采样、分段连续采样等不同采样策略下的重建结果进行分析探讨, 为相关研究学者建立了圆轨迹投影扫描策略的整体框架, 方便相关技术人员和学者结合具体的检测对象选择合适的采样方法, 并为其算法研究提供依据.借鉴压缩感知框架下的数据压缩与挖掘思想, 为不完全投影数据重建高质量图像等研究提供突破点, 充分利用迭代类算法的优势, 推动数据恢复算法与采样方式的进程.
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