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CT image reconstruction method for small amount of projection data and differential equation solution method by neural network

少量投影数据的CT图像重建方法及神经网络微分方程求解方法

基本信息

批准号：
11650065
负责人：
TAKEDA Tatsuoki
金额：
$ 2.24万
依托单位：
The University of Electro-Communications
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

By using squared residuals of a differential equation for the object function of a neural network we can solve the differential equation as the network itself becomes the solution after training. We investigated this method by applying it to Navier-Stokes equation, Poisson equation, Lorenz equation and so on, and obtained satisfactory results. We found that similar method is applied to CT image reconstruction problem with small amount of projection data. We applied it to model CT image reconstruction problem where an integral equation is solved and obtained satisfactory results. By generalizing the form of the object function this method can be applied to very wide range of problems. Defining the object function by the squared residuals of differential equations, integral equations, and algebraic equations multiplied by some appropriate penalty coefficients various kinds of problems can be solved owing to the excellent expressivity of the multi-layer neural network comparatively easily. One of the important application is the solution of data assimilation problem which is very important in the field of the meteorological and oceanological numerical simulations. We have also performed model numerical experiment of the data assimilation and obtained satisfactory results. This method is also applicable to the generalized Abel inversion which appears in various scientific and engineering researches.For comparing the new method with the conventional standard methods we studied these methods. As the mathematical basis of the CT image reconstruction we studied the inverse Radon transform. As for the decay Radon transform we derived an inverse formula, which we proved mathematically and numerically. We studied the Poisson equation and the Helmholtz equation in the infinite domain. We introduced an artificial boundary condition and studied the validity of the condition mathematically and numerically.

利用神经网络目标函数的微分方程的残差平方，可以在神经网络本身成为训练后的解时解微分方程。将该方法应用于Navier-Stokes方程、Poisson方程、Lorenz方程等，得到了满意的结果。我们发现类似的方法也适用于少量投影数据的CT图像重建问题。将其应用于模型CT图像重建问题，求解了一个积分方程，得到了满意的结果。通过推广目标函数的形式，这种方法可以应用于非常广泛的问题。通过微分方程、积分方程和代数方程的残差平方乘以适当的惩罚系数来定义目标函数，由于多层神经网络具有较好的表达能力，可以比较容易地解决各种问题。其中一个重要的应用是解决在气象和海洋数值模拟领域中非常重要的数据同化问题。我们还进行了数据同化的模式数值实验，取得了满意的结果。该方法也适用于各种科学和工程研究中出现的广义阿贝尔反演。为了与传统的标准方法进行比较，我们对这些方法进行了研究。作为CT图像重建的数学基础，我们研究了Radon逆变换。对于衰变Radon变换，我们推导了一个反公式，并进行了数学和数值证明。我们研究了无限域中的泊松方程和亥姆霍兹方程。引入了一个人工边界条件，并从数学和数值上研究了该条件的有效性。