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

TL;DR

This paper analyzes local image reconstruction from discrete generalized Radon transform data using an iterative quadratic minimization approach, providing explicit formulas for the reconstruction limit near discontinuities and validating with numerical experiments.

Contribution

It introduces a local analysis method for iterative GRT reconstruction, deriving explicit formulas for the limit behavior near discontinuities, and demonstrates accuracy through numerical experiments.

Findings

Explicit formula for the limit of reconstructed image near discontinuities.

Validation of the formula with numerical experiments on circular GRT data.

The iterative method effectively captures local image features in the presence of jumps.

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, $f_\epsilon(x)$, reconstructed from discrete data $g(y_j)$ in an $\epsilon$-neighborhood of a point, $x_0$. The points $y_j$ lie on a grid with step size of order $\epsilon$ 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 $\Delta F_0(\check x;x_0):=\lim_{\epsilon\to0}(f_\epsilon(x_0+\epsilon\check x)-f_\epsilon(x_0))$, where $f_\epsilon$ is the solution to the minimization problem and $|\check x|\le A$. A numerical experiment with a circular GRT demonstrates that $\Delta F_0(\check x;x_0)$ accurately approximates the actual reconstruction obtained by the cost functional minimization.