Iterative regularization methods for phase retrieval in the space domain based on TIE equation
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摘要:
利用同步辐射X射线同轴显微层析（CT）方法对页岩进行扫描成像是一种无损的，对研究页岩孔裂隙大小、形态、三维结构及连通性等微观结构特征有重要意义的方法.同步辐射的引入将在物理上为提高页岩成像的分辨率提供了可能，在相位-吸收二重性假设下利用光强传递TIE（transport-of-intensity）方程可以较好地抑制由于相位信息带来的“边缘增强”效应，但该问题本质上是不适定的反演问题.本文根据实际问题构造模型，提出了一种与传统基于频域方法不同的，基于空间域的相位恢复迭代算法，并采用迭代Tikhonov正则化在数值上解决了噪声干扰下的不稳定性问题.研究结果表明，新方法的残差仅为频域方法的1%左右，该方法可用于未来实际数据的处理.
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关键词:
- X射线层析成像 /
- CT重建 /
- 相位恢复 /
- Tikhonov正则化
Abstract:X-Ray propagation-based computerized tomography (CT) is a non-destructive and useful method to study the size, shape, 3D pore structures and interconnections of pores in shale. However, it remains a challenging task to adequately calculate the location and size of pores. Synchrotron radiation makes it possible to realize reconstruction of the shale structure on multiple scales. Conventionally, computerized tomography relies on the absorption contrast of the sample. This process can be formulated in Radon transform. When we only consider absorption contrast, we may face a phenomenon of the "edge enhancement" effect. It is caused by the effect of phase shift and we need to correct it with phase retrieval. Under the assumption of phase-attenuation duality, the process of phase retrieval can be described in the transport-of-intensity equation (TIE).But this is an ill-posed inverse problem essentially. The existing methods usually focus on how to deal with phase retrieval by filtration in the frequency domain. The method we propose in this paper is a novel approach, trying to solve this problem in the space domain.
First, we study a model with the size of 512×512 pixels based on real shale.We give each pixel of the model a numerical linear absorption index. Second, we execute the process of Radon transform and make the deviation caused by phase to simulate the actual projection data. Then we use three methods to retrieval the phase: the filter method in the frequency domain, direct method in the space domain, and iterative Tikhonov regularization method in the space domain. After we finish the process of phase retrieval, we use the standard filtered back-projection (FBP) method to present the outcome. By analyzing the results from different methods, the effects of different methods can be shown.
Numerical simulations demonstrate that the results calculated by the method proposed in this paper are stable and with less artifacts. In the case of noiseless data, the direct method and the iterative Tikhonov regularization method perform much better than the filter method in the frequency domain. It is obvious that there exist artifacts around the pixels with larger index calculated by the frequency-domain method. When we add 1% Gaussian noise, the direct method performs bad because of the ill-posed property of the linear system.The iterative Tikhonov regularization method and the frequency-domain method can eliminate the noise disturbance. However, we still find the artifact around the pixels with larger index from the detail of the results calculated by the frequency-domain method.And this phenomenon does not exist in our new method.
Our proposed iterative Tikhonov regularization method can tackle the "edge enhancement" effect. With appropriate choice of the regularization parameter, we can get more stable and accurate results under the interference of different level of noise. Numerical results show that the residual of the new method is nearly 1% of the traditional method in the frequency domain, and hence the new method is promising for processing of real data.
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图 1
待测物体坐标系和观测平面
Figure 1.
Coordinate system for the measured object and the plane of observation
图 7
噪声1%下线性吸收系数反演结果
Figure 7.
Inversion results of linear absorption coefficients with noise 1%
表 1
模型成分在30 keV能量下β的理论值
Table 1.
Theoretical β values for materials in the model at energy E=30 keV
序号 成分 百分比 β 1 石英 75.04% 7.4×10-10 2 高岭石 3.20% 6.5×10-10 3 伊蒙间层 3.20% 7.2×10-10 4 斜长石 3.20% 9.0×10-10 5 白云石 2.92% 11.5×10-10 6 伊利石 3.02% 12.4×10-10 7 方解石 3.10% 16.0×10-10 8 绿泥石 3.04% 20.5×10-10 9 孔隙 3.27% 0 
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