X射线微分相位衬度CT重建算法研究
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摘要
对于弱吸收物质,X射线相衬成像方法使用折射信息,而不是衰减信息作为成像信号,可以提供更加丰富的关于物体内部结构的信息。到现在为止,X射线相衬成像技术已经发展出四大类方法：干涉成像法,相位传播法,衍射增强成像法和光栅成像法。前三种方法依赖于具有空间相干性或时间相干性的X射线源。然而,此类射线源往往造价昂贵且体型巨大,这是相位衬度成像方法走向实际应用的主要障碍。光栅成像法可以采用常规x射线管实现相位衬度成像,与其他方法相比,更具有实际应用的可能。干涉成像法可以测量X射线的相位分布,相位传播法可以测量X射线相位的二阶梯度分布,衍射增强成像法和光栅成像法可以测量x射线相位的一阶梯度分布。因此,衍射增强成像法和光栅成像法常被统称为微分相位衬度(Differential Phase-Contrast, DPC)成像法。基于这两种成像方法的计算机断层成像(Computed-Tomography, CT)技术可被称为DPC-CT。
通常,相位衬度CT需要在每个角度下多次曝光,进而导致了相当长的曝光时间和较大的辐射剂量。因此,如何在数据量较少的条件下获得更高质量的重建结果是DPC-CT中一个值得研究的问题。在基于同步辐射源的DPC-CT的实验中,射线束具有极高的平行度,CT重建问题可通过平行束算法解决。在基于常规X射线管的DPC-CT的实验中,射线源发射出球面波,扇形或锥形束算法更加合适。虽然锥形束可以覆盖物体的若干断层,但要重建柱状物体,还必须交替执行平移操作和旋转操作。相比之下,螺旋扫描模式允许平移操作和旋转操作同时执行,具有较高的重建效率。针对螺旋DPC-CT的研究尚不多见。针对少数据量重建和螺旋DPC-CT重建这两个问题,我们研究了如下三个方面的内容：
(1)将压缩感知理论引入DPC-CT重建中用于解决少数据量重建问题。本文测试了经典的压缩感知算法如OMP算法、GPSR算法、近似点(Proximal Point, PP)算法和Bregman分离执行(Bregman Operator Splitting, BOS)算法在DPC-CT中的重建效果。在此之后,本文将经典的ART算法融入近似点算法和BOS算法中,提出了适合相位项梯度重建的ART-PP算法和ART-BOS算法。随后,本文又将ART-BOS算法推广至相位项直接重建中。该算法拥有比经典的ART-POCS-TV算法更快的收敛速度。
(2)将“正反投影法”融入螺旋DPC-CT重建当中,提出了螺旋DPC-CT的近似重建方法。该方法的基本思想是根据物体临近断层之间的相关性,首先将螺旋扫描条件下采集到的数据插值到某个断层上,估计出该断层的正弦图,再根据“正反投影法”提取折射角,最后通过扇形束重建算法完成重建。该方法的优点是仅需一次扫描即可重建三维物体,缺点是缺乏严格的理论支持。
(3)提出了一种螺旋锥束近似重建算法和一种精确重建算法,这两种算法都是基于PI线的。近似重建算法的基本思想是将锥形束等效为若干倾斜扇束,再用扇形束算法进行重建。该算法适用于一维光栅成像的情形,在小锥角条件下可以取得良好的重建效果。精确重建算法将二维微分相位衬度投影与相位衬度沿X射线传播方向的积分值之间的关系表达为一个多元偏微分方程的形式,再将该方程融入吸收衬度螺旋锥束CT的Katsevich算法中,理论上适用于任意锥角情形。由于精确算法在数学上的严密性,它可对DPC-CT算法的应用和研究起到一定的指导作用。
For weakly absorbing objects, x-ray phase-contrast-imaging methods use the phase shift rather than the absorption as the imaging signal and offer more details regarding the internal structure. Until now, several x-ray phase-contrast-imaging methods have been proposed, including interferometer, propagation, analyzer-crystal and analyzer-grating methods. The first three methods rely on the spatial coherence or temporal coherence of the x-ray source. However, such x-ray sources are often expensive, which is the main obstacle for the phase-contrast-imaging methods to application. The grating-based methods seem more promising than others in realizing practical phase-contrast imaging systems by using conventional x-ray tube. Unlike other methods, DEI-and grating-based imaging methods measure the gradient of the phase distribution, which is called differential phase-contrast (DPC) imaging technology. The Computed-Tomography using DEI and/or grating-based imaging methods are usually called differential phase-contrast CT (DPC-CT).
DPC-CT reconstruction usually requires the refraction angle at each view should be extracted from several raw measurements captured at the identical view, which leads to unacceptably long exposure time and huge X-ray doses. In synchrotron radiation-based DPC-CT experiments, the radiation beams can be well approximated as parallel beams, and the reconstruction problem can be solved by the use of parallel-beam algorithms. In conventional x-ray tube based experiments, the x-ray tube emits spherical wave, the fan or cone-beam algorithms are more suitable. For rod-shaped samples, due to the limitation of the grating size, the fan or cone beams cannot recover the whole object at the same time. Thus, if a rod-shaped sample is required to be reconstructed by above algorithms, one should alternately perform translation and rotation on this sample. Helical cone-beam CT allows the translation and rotation operations to be performed simultaneously, which may be more efficient than other algorithms in the case of rod-shaped samples. However, few researches on helical DPC-CT were reported. In order to solve the fewer data reconstruction and helical DPC-CT problems, we carried out our search along the following three aspects:
First, we introduce the Compressed Sensing theory into DPC-CT reconstruction to solve the fewer data reconstruction problem. We compare several classic algorithms such as Orthogonal Matching Pursuit (OMP) algorithm, GPSR algorithm, Proximal-Point algorithm and Bregman Operator Splitting (BOS) algorithm in DPC-CT reconstruction. After that, we introduce the Proximal-Point algorithm and Bregman Operator Splitting (BOS) algorithm into DPC-CT reconstruction and propose two alternating iteration algorithms, ART-PP and ART-BOS algorithm to reconstruct the gradient of refractive index decrement, for sparse angular DPC-CT. The ART-BOS algorithm is also extended to reconstruct the refractive index decrement, whose convergence speed is faster than that of ART-POCS-TV algorithm. The numerical simulation and experiment results show that the proposed algorithm can provide higher quality reconstructions in sparse angular condition.
Second, we propose an approximate strategy for helical fan-beam DPC-CT reconstruction by combining the "Forward-Backward projection method" and the helical scanning mode. The main idea of this strategy is approximating the helical fan/cone-beam geometry by single/multiple parallel fan-beams such that the2D sinogram of a given slice can be synthesized from the neighboring fan-beams by interpolation methods. Once the sinogram has been synthesized, the given slice can be reconstructed using any2D reconstruction methods. The main advantage of this algorithm is that only one helical scanning is required. However, as approximate one, this strategy suffers the problem of strict theoretical support.
Third, an approximate algorithm and a theoretically exact algorithm for helical cone-beam DPC-CT are proposed, which are both based on the concept of PI-line. The approximate algorithm treats cone-beam as a combination of many oblique fan-beams, and the refractive index decrements are directly reconstructed by modifying fan-beam algorithm. The exact algorithm establishes a general relationship between the derivative data required by Katsevich algorithm and the two-dimension refraction angle data. Thus, the Katsevich algorithm, which is for absorption-based helical cone-beam CT, can be implemented to solve the helical cone-beam DPC-CT reconstruction problem. Due to the theoretical support, the exact algorithm may play a guiding role in the research and application of helical cone-beam DPC-CT.
通常,相位衬度CT需要在每个角度下多次曝光,进而导致了相当长的曝光时间和较大的辐射剂量。因此,如何在数据量较少的条件下获得更高质量的重建结果是DPC-CT中一个值得研究的问题。在基于同步辐射源的DPC-CT的实验中,射线束具有极高的平行度,CT重建问题可通过平行束算法解决。在基于常规X射线管的DPC-CT的实验中,射线源发射出球面波,扇形或锥形束算法更加合适。虽然锥形束可以覆盖物体的若干断层,但要重建柱状物体,还必须交替执行平移操作和旋转操作。相比之下,螺旋扫描模式允许平移操作和旋转操作同时执行,具有较高的重建效率。针对螺旋DPC-CT的研究尚不多见。针对少数据量重建和螺旋DPC-CT重建这两个问题,我们研究了如下三个方面的内容：
(1)将压缩感知理论引入DPC-CT重建中用于解决少数据量重建问题。本文测试了经典的压缩感知算法如OMP算法、GPSR算法、近似点(Proximal Point, PP)算法和Bregman分离执行(Bregman Operator Splitting, BOS)算法在DPC-CT中的重建效果。在此之后,本文将经典的ART算法融入近似点算法和BOS算法中,提出了适合相位项梯度重建的ART-PP算法和ART-BOS算法。随后,本文又将ART-BOS算法推广至相位项直接重建中。该算法拥有比经典的ART-POCS-TV算法更快的收敛速度。
(2)将“正反投影法”融入螺旋DPC-CT重建当中,提出了螺旋DPC-CT的近似重建方法。该方法的基本思想是根据物体临近断层之间的相关性,首先将螺旋扫描条件下采集到的数据插值到某个断层上,估计出该断层的正弦图,再根据“正反投影法”提取折射角,最后通过扇形束重建算法完成重建。该方法的优点是仅需一次扫描即可重建三维物体,缺点是缺乏严格的理论支持。
(3)提出了一种螺旋锥束近似重建算法和一种精确重建算法,这两种算法都是基于PI线的。近似重建算法的基本思想是将锥形束等效为若干倾斜扇束,再用扇形束算法进行重建。该算法适用于一维光栅成像的情形,在小锥角条件下可以取得良好的重建效果。精确重建算法将二维微分相位衬度投影与相位衬度沿X射线传播方向的积分值之间的关系表达为一个多元偏微分方程的形式,再将该方程融入吸收衬度螺旋锥束CT的Katsevich算法中,理论上适用于任意锥角情形。由于精确算法在数学上的严密性,它可对DPC-CT算法的应用和研究起到一定的指导作用。
For weakly absorbing objects, x-ray phase-contrast-imaging methods use the phase shift rather than the absorption as the imaging signal and offer more details regarding the internal structure. Until now, several x-ray phase-contrast-imaging methods have been proposed, including interferometer, propagation, analyzer-crystal and analyzer-grating methods. The first three methods rely on the spatial coherence or temporal coherence of the x-ray source. However, such x-ray sources are often expensive, which is the main obstacle for the phase-contrast-imaging methods to application. The grating-based methods seem more promising than others in realizing practical phase-contrast imaging systems by using conventional x-ray tube. Unlike other methods, DEI-and grating-based imaging methods measure the gradient of the phase distribution, which is called differential phase-contrast (DPC) imaging technology. The Computed-Tomography using DEI and/or grating-based imaging methods are usually called differential phase-contrast CT (DPC-CT).
DPC-CT reconstruction usually requires the refraction angle at each view should be extracted from several raw measurements captured at the identical view, which leads to unacceptably long exposure time and huge X-ray doses. In synchrotron radiation-based DPC-CT experiments, the radiation beams can be well approximated as parallel beams, and the reconstruction problem can be solved by the use of parallel-beam algorithms. In conventional x-ray tube based experiments, the x-ray tube emits spherical wave, the fan or cone-beam algorithms are more suitable. For rod-shaped samples, due to the limitation of the grating size, the fan or cone beams cannot recover the whole object at the same time. Thus, if a rod-shaped sample is required to be reconstructed by above algorithms, one should alternately perform translation and rotation on this sample. Helical cone-beam CT allows the translation and rotation operations to be performed simultaneously, which may be more efficient than other algorithms in the case of rod-shaped samples. However, few researches on helical DPC-CT were reported. In order to solve the fewer data reconstruction and helical DPC-CT problems, we carried out our search along the following three aspects:
First, we introduce the Compressed Sensing theory into DPC-CT reconstruction to solve the fewer data reconstruction problem. We compare several classic algorithms such as Orthogonal Matching Pursuit (OMP) algorithm, GPSR algorithm, Proximal-Point algorithm and Bregman Operator Splitting (BOS) algorithm in DPC-CT reconstruction. After that, we introduce the Proximal-Point algorithm and Bregman Operator Splitting (BOS) algorithm into DPC-CT reconstruction and propose two alternating iteration algorithms, ART-PP and ART-BOS algorithm to reconstruct the gradient of refractive index decrement, for sparse angular DPC-CT. The ART-BOS algorithm is also extended to reconstruct the refractive index decrement, whose convergence speed is faster than that of ART-POCS-TV algorithm. The numerical simulation and experiment results show that the proposed algorithm can provide higher quality reconstructions in sparse angular condition.
Second, we propose an approximate strategy for helical fan-beam DPC-CT reconstruction by combining the "Forward-Backward projection method" and the helical scanning mode. The main idea of this strategy is approximating the helical fan/cone-beam geometry by single/multiple parallel fan-beams such that the2D sinogram of a given slice can be synthesized from the neighboring fan-beams by interpolation methods. Once the sinogram has been synthesized, the given slice can be reconstructed using any2D reconstruction methods. The main advantage of this algorithm is that only one helical scanning is required. However, as approximate one, this strategy suffers the problem of strict theoretical support.
Third, an approximate algorithm and a theoretically exact algorithm for helical cone-beam DPC-CT are proposed, which are both based on the concept of PI-line. The approximate algorithm treats cone-beam as a combination of many oblique fan-beams, and the refractive index decrements are directly reconstructed by modifying fan-beam algorithm. The exact algorithm establishes a general relationship between the derivative data required by Katsevich algorithm and the two-dimension refraction angle data. Thus, the Katsevich algorithm, which is for absorption-based helical cone-beam CT, can be implemented to solve the helical cone-beam DPC-CT reconstruction problem. Due to the theoretical support, the exact algorithm may play a guiding role in the research and application of helical cone-beam DPC-CT.
引文
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