Local analysis of iterative reconstruction from discrete generalized Radon transform data in the plane

Abstract

Local reconstruction analysis (LRA) is a powerful and flexible technique to study images reconstructed from discrete generalized Radon transform (GRT) data, $g=\mathcal{R}f$. The main idea of LRA is to obtain a simple formula to accurately approximate an image, $f_{\hat{I}}\mu (x)$, reconstructed from discrete data $g(y_j)$ in an $\hat{I}\mu$-neighborhood of a point, $x_0$. The points $y_j$ lie on a grid with step size of order $\hat{I}\mu$ in each direction. In this paper we study an iterative reconstruction algorithm, which consists of minimizing a quadratic cost functional. The cost functional is the sum of a data fidelity term and a Tikhonov regularization term. The function $f$ to be reconstructed has a jump discontinuity across a smooth surface $\mathcal{S}$. Fix a point $x_0\in \mathcal{S}$ and any $A>0$. The main result of the paper is the computation of the limit $\hat{I}”F_0(\check{x};x_0):={\lim }_{\hat{I}\mu \to 0}(f_{\hat{I}}\mu (x_0+\hat{I}\mu \check{x})-f_{\hat{I}}\mu (x_0))$, where $f_{\hat{I}}\mu$ is the solution to the minimization problem and $|\check{x}|\le A$. A numerical experiment with a circular GRT demonstrates that $\hat{I}”F_0(\check{x};x_0)$ accurately approximates the actual reconstruction obtained by the cost functional minimization.