Regularizing the method of reconstructing a function on the basis of its spherical Radon transform
Let \(f\in C_0^{\infty}(U)\), where \(U\) is the unit disc in \({\mathbb R}^2\). The spherical Radon transform is determined as \(Rf(p,r)=(1/2\pi) \int_{S^1} f(p+r\theta) d\theta\), \(S^1=(\cos\varphi,\sin\varphi)\), \(\varphi\in [0,2\pi)\), \(r>0\). Recently, D. Finch and M. Haltmeier have obtained the following inversion formula: \[ f(x)=(1/2\pi) \Delta_x \int_{S^1} \int_0^2 r Rf(p,r) \log | r^2-| x-p|^2| dr dp. \] The author suggests a regularized version of the formula applicable to the case where spherical projections are registered with a noise for a finite number of points \(p\). Error estimates for the reconstruction procedure are given.
scientific article; zbMATH DE number 6108659