The Numerical Solution Of The Radon Transform Along The Circular Curve

The Radon transform is the theoretical basis for the fields of CT technology and reconstruction problems.It is widely applied in science,engineering,and medical diagnostics due to its non-invasive and high-precision reconstruction characteristics.The reconstruction refers to the problem of densifying the internal density or nature of an object from the projection data without destroying the internal structure of the object.One class of reconstruction problem can be deduced to the inversion of the Radon transform along a straight line,such as emission X-ray imaging.Another class of the problem can be attributed to the inversion of the Radon transform along the curve,that is,the inversion of the generalized Radon transform.Taking Compton scattering and spotlight synthetic aperture radar imaging as examples,we can deduce the reconstruction problems to the inversion problems of the Radon transform along the circular curve.This paper studies the inversion of the Radon transform of a bivariate function with compact support and continuous in the support,along the upper semicircle curve centered on the x-axis.We show that when the center and radius of a circular curve change within a certain range,the inversion problem is unique,if the Radon transform along the upper semicircular curve is known.Based on the Fourier transform of the projection function,the inversion problem can be deduced to a solution of the Abel integral equation with weak singularity and oscillatory kernel.We give a numerical method to eliminate this kind of weak singularity.The ill-posed Abel integral equation is discretized as a linear equation with lower triangular coefficient matrix and can be directly solved.Numerical experiments using a bivariate function with compact support and continuous in the support and Shepp-Logan head model,proving the effectiveness of the proposed numerical method.The condition number of the coefficient matrix of a linear equation system reflects the "ill-conditioned" degree of the equation system.When the projection data has a small disturbance,the solution obtained by directly solving the linear equations will produce a large error compared to the original solution.Therefore,considering the noise of the projection data,we give a stable numerical method to improve the condition number of the coefficient matrix by multiple weights,and use the Landweber iterative method to solve the weighted linear equations.Through numerical simulation,it is found that the condition number of the coefficient matrix is significantly improved,and an effective reconstruction result is obtained when the projection data is noisy.