The Numerical Solution of the Radon Transform Along the Circular Curve

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.