正则化Kaczmarz算法在页岩纳米CT重构中的应用

利用X射线计算机断层成像（CT）方法对页岩的扫描成像是一种无损的，对研究页岩微纳孔隙结构有重要意义的方法.传统的CT重构通常使用的是显式的滤波反投影（Filtered Back Projection，FBP）方法，该算法具有较快的成像速度，但常伴随有伪影或不稳定等情况.对于纳米CT而言，可以通过迭代优化的方法对投影数据进行成像，传统的迭代成像有收敛速度慢导致的计算时间长等缺点.Kaczmarz算法作为一种重要的代数重建技术（ART），由于其几何意义明显，操作容易等优点，在CT重构中起着重要的作用，我们可以通过块状迭代或随机迭代的方式对其收敛速度进行改进.对于所求解问题的不适定性，代数重建过程中需要引入正则化的技巧来改善解的稳定性.本文根据实际问题的需要，使用页岩数值模型，验证了正则化Kaczmarz方法的有效性，并对重庆漆辽龙马溪组页岩样品的实际数据进行了处理，得到了较好的效果.

Application of regularizing Kaczmarz algorithm in shale Nano-CT reconstruction

X-Ray computerized tomography (CT) is a non-destructive and effective method to study the size, shape, 3D pore structures and interconnections of pores in shale. The main methods of CT reconstruction are FBP (Filtered Back Projection), ART (Algebraic Reconstruction Technique) and methods based on statistic models. FBP is most widely used in industry. The results obtained by this method contains artifact which is difficult to remove. Kaczmarz method, which has explicit geometric meaning, is one kind of ART methods. This method project the solution to the hyperplanes determined by the rows of matrix in a fixed order. Obviously the convergence rate depends on the order of the matrix's row, and it is not so efficient. If we project the solution in each step to the center of the projection points from different hyperplanes, which is known as block projection, we can improve the efficiency of computing greatly. Because the coefficient matrix is highly sparse and the observations are not only incomplete but also contaminated, the problem is not well-posed in the sense of Hadamard. Regularization method is an effective method to solve ill-posed problem. Many forms of regularization methods can be used to solve this problem. We transform the regularized normal equation to the equivalent augmented regularized normal system of equations to solve large dimensionality ill-posed problem. At the same time, we can find some constrains when we deal with the real problem like the absorption of pixels around the sample should equal to zero, the absorption of pixels should not be negative and so on. If we add these kind of constrains, we will get a better result. Our proposed data-constrained regularized Kaczmarz algorithm can perform better than the FBP methods and other projection methods in the numerical test and practical data test, hence the new method is promising for the future practical data processing.