The Numerical Solution of the Radon Transform Along the Circular Curve
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摘要: 研究具有紧支集且在支集内连续的二元函数沿上半圆曲线的Radon变换反演问题。基于对投影函数的Fourier变换，反演问题可以归结为具有弱奇性及震荡核的Abel积分方程的求解。我们证明了当圆曲线中心及半径在一定范围内变化时，在已知沿上半圆曲线的Radon变换情况下，这个积分方程的解具有唯一性，并给出了消除Abel积分方程弱奇性的数值方法。在考虑投影数据噪声的情况下，给出了多次加权改善系数矩阵条件数稳定的数值方法，并通过数值模拟验证所提出方法的有效性。Abstract: In this paper, we study the inverse problem of the Radon transform of a continuous bivariate function along the upper semicircle curve with a compact support set. Based on the Fourier transform of the projection function, the inverse problem can be deduced to the solution of the Abel integral equation with weak singularity and oscillatory kernel. We prove that when the center and radius of the upper semicircle curve change within a certain range, if the Radon transform along the upper semicircle is known, the solution of the Abel integral equation is unique, and we give a numerical method to eliminate this weak singularity. Considering projection data with noise, a stable numerical method for improving the condition number of the coefficient matrix with multiple weighting is proposed, and the validity of the proposed method is verified by numerical simulation.
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Keywords:
- Radon transform /
- Abel integral equation /
- weak singularity /
- multiple weighting
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