Fund Project:Project supported by the National Natural Science Foundation of China (Grant No. 11674232) and Guangdong Provincial Basic and Applied Basic Research Foundation, China (Grant No. 2019A1515011785)
Received Date:27 October 2020
Accepted Date:11 December 2020
Available Online:09 May 2021
Published Online:20 May 2021
Abstract:Over the last two decades, the grating-based phase-contrast imaging has aroused the interest of a number of researchers. It could provide an access to three complementary signals simultaneously: the conventional absorption contrast, the differential phase contrast related to refraction of incident wave, and the dark-field contrast that relates to ultra small angle scattering in a sample. The grating-based phase-contrast signals have higher contrast sensitivity for some types of soft samples than the absorption signals. Dark-field signals have better diagnostic effects in the detection of lung tumors, pneumothorax and the identification of microcalcifications in breast. There are two main phase retrieval methods in grating-based X-ray phase-contrast imaging, i.e. phase stepping method and Fourier transform method. The phase signals retrieved by phase stepping is high precise and has low noise. But the sample suffers high dose due to at least three exposures. The phase signals retrieved by Fourier transform is low-dose due to the fact that only one image with sample is needed, but it is easily affected by artifacts when the size of the filtering window is too large. However, when the size of the filtering window is too small, the high-frequency information of the phase-contrast image will be lost, and the image will become blurred. A trade-off between definitions of the image and artifacts should be made. Since the phase-contrast signal and the dark-field signal of the sample are modulated by carrier fringes, the frequency spectrum of the detected image consists of many different harmonics. The artifacts in the retrieved signals originate from the spectrum aliasing between primary peak around zero spatial frequency and first-order harmonic peaks. Therefore, the subtraction between two images with phase difference can remove the primary peak, and the artifacts in the phase-contrast signals and dark-field signals will be suppressed. In order to further suppress the artifacts, we increase the frequency of carrier fringes, which results in a larger distance between first-order harmonic peaks in frequency domain. We finally attain artifact-free phase-contrast images and dark-field images while maintaining high definition of the images. The method proposed here is not only applicable to incoherent imaging system, but also to Talbot-Lau interferometer, and it would be useful in fast and low-dose X-ray phase-contrast and dark-field imaging. Keywords:X-ray phase-contrast imaging/ Fourier transform/ dark-field imaging
$G\left( f \right) = A\left( f \right) + C\left( {f - {f_0}} \right) + {C^*}\left( { - f - {f_0}} \right),$
其中$G\left( f \right)$, $A\left( f \right)$和$C\left( f \right)$分别是$I\left( x \right)$, $a\left( x \right)$和$c\left( x \right)$的傅里叶变换. 在条纹的频谱中, $A\left( f \right)$, $C\left( {f - {f_0}} \right)$和${C^*}\left( { - f - {f_0}} \right)$分别代表零级、正一级和负一级频谱, 如图2所示. 图 2 一般情形下载波条纹的频谱图 Figure2. The Fourier spectrum of carrier fringe patterns in general case.
$\frac{1}{2}b\left( x \right)\exp \left[ {{\rm{i}}\phi \left( x \right)} \right]\exp \left( {{\rm{i}}2{\rm{\pi}} {f_0}x} \right).$
对光路中没有样品的条纹图重复上述操作可得
$\frac{1}{2}{b_{\rm{r}}}\left( x \right)\exp \left( {{\rm{i}}2{\rm{\pi}} {f_0}x} \right),$
其中${b_{\rm{r}}}\left( x \right)$表示没有样品时条纹的振幅, 将上述两式相除再取其角度即可得到物体的相位:
${\phi _1}\left( x \right) = {\rm{angle}}\left( {\frac{{b\left( x \right)\exp \left[ {{\rm{i}}\phi \left( x \right)} \right]\exp \left( {{\rm{i}}2{\rm{\pi}} {f_0}x} \right)}}{{{b_{\rm{r}}}\left( x \right)\exp \left( {{\rm{i}}2{\rm{\pi}} {f_0}x} \right)}}} \right),$
其中“angle”表示取角度. 然而, 我们在实验中发现, 由于物体的频谱相当地宽, 不同级次的频谱会不可避免地发生混叠, 如图3所示. 图 3 实际情况下发生的频谱混叠 Figure3. Spectrum aliasing between different harmonic peaks in practice.
${\phi _2} ( x ) \!=\! {\rm{angle}} \left[ {\frac{{b ( x )\exp [ {{\rm{i}}\phi ( x )} ]\exp ( {{\rm{i}}2{\rm{\pi }}{f_0}x} )}}{{{b_r} ( x )\exp ( {{\rm{i}}2{\rm{\pi }}{f_0}x} )}}} \right].$
3.实验验证23.1.两幅图像傅里叶变换法抑制伪影的效果 -->
3.1.两幅图像傅里叶变换法抑制伪影的效果
下面通过实验来检验上述两幅图像傅里叶变换法抑制伪影的效果. 实验装置示意图如图1所示, X射线源到探测器的距离为80 cm, 光栅离探测器的距离为45.78 cm, 此时光栅在探测器平面上的投影条纹周期与探测器像素尺寸的比值r = 3. X射线源的电压为40 kVp, 电流为160 μA, 在每个光栅位置采集三幅图像取平均以降低噪声, 每幅图像曝光4 s, 样品为直径5 mm的有机玻璃棒(PMMA), 其离探测器的距离为53 cm. 两幅图像傅里叶变换法和单幅图像傅里叶变换法恢复出来的PMMA相衬像如图6所示. 滤波窗口都为高斯窗, 其标准差$\sigma $等于零频和一频间距的二分之一. 图6(a)是两幅图像傅里叶变换法恢复出的相衬像, 该相衬像中没有出现伪影. 而图6(b)则是单幅图像傅里叶变换法恢复出的相衬像, 该相衬像则有明显的伪影. 从图6的直观比较中可以看出, 两幅图像傅里叶变换法所恢复出的相衬像(图6(a))比单幅图像傅里叶变换法所恢复出的相衬像(图6(b))在抑制伪影方面有明显的改进. 图 6 (a) 两幅图像傅里叶变换法所恢复出的PMMA相衬像; (b)单幅图像傅里叶变换法所恢复出的PMMA相衬像 Figure6. (a) The phase-contrast image of PMMA retrieved by Fourier transform with two images; (b) the phase-contrast image of PMMA retrieved by Fourier transform with one image.
为了进一步比较两种算法对伪影的抑制效果, 需要进行定量的分析. 下面我们截取图6(a)中的白色方框区域, 对其进行按行求平均, 绘制得到图7(a)中的蓝色曲线; 在图6(b)中截取与图6(a)中白色方框所示的相同位置的区域, 同样按行取平均, 得到图7(a)中的红色曲线. 同时取图6(a)中的黑色矩形框区域, 对其进行按行求平均, 绘制得到图7(b)中的蓝色曲线; 对图6(b)取相同位置的区域, 同样按行取平均, 得到图7(b)中的红色曲线. 从图7(a)的红色曲线可以看出, 利用单幅图像傅里叶变换法恢复出的相位值有明显的周期性, 且周期恰好为3个像素宽度, 这和图4(a)的仿真结果吻合地很好. 而利用两幅图像傅里叶变换法求得的相位则无明显的周期性, 且相位值更接近理论值零. 这意味着两幅图像傅里叶变换法不仅对伪影的抑制效果更好, 而且恢复出的相位更准确. 图7(b)中红色曲线的中间部分出现了明显的震荡, 而蓝色曲线的中间部分震荡不明显. 图7(b)中红色曲线的峰峰值较蓝色曲线的峰峰值高, 说明伪影不仅仅会造成背景信号偏高, 也会使得物体的相衬信号偏高. 图 7 (a) 图6中白色方框区域按行取平均后绘制的曲线; (b) 图6中黑色方框区域按行取平均后绘制的曲线 Figure7. (a) Curves from averaging the area of white rectangle in figure 6 by row; (b) curves from averaging the area of black rectangle in Figure 6 by row.
当频谱间隔较小时, 两幅图像傅里叶变换法无法完全抑制频谱混叠产生的伪影. 因此, 选择合适的频谱间隔对于抑制伪影是非常重要的. 下面我们通过实验对比不同频谱间隔对伪影产生的影响. 实验装置示意图如图1所示, X射线源的电压为50 kVp, 电流为160 μA. X射线源离探测器80 cm, 样品鸡翅离探测器12 cm. 采集光栅在两个不同位置的图像, 每个位置采集三幅图像取平均去噪, 每幅图像曝光4 s. 实验结果如图8所示. 图 8 载波条纹周期与探测器像素尺寸比值r = 3时, 鸡翅的频谱 (a)、相衬像(c)和暗场像(e). r = 5时, 鸡翅的频谱(b)、相衬像(d)和暗场像(f) Figure8. Fourier spectrum (a), phase-contrast image (c) and dark-field image (e) of a chicken wing when r = 3. Fourier spectrum (b), phase-contrast image (d) and dark-field image (f) of a chicken wing when r = 5.