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.

翻译：局部重建分析（LRA）是一种研究从离散广义Radon变换（GRT）数据$g=\mathcal R f$重建图像的强大而灵活的技术。LRA的核心思想是获得一个简单公式，以精确逼近从离散数据$g(y_j)$在点$x_0$的$\epsilon$邻域内重建的图像$f_\epsilon(x)$。数据点$y_j$位于每个方向步长量级为$\epsilon$的网格上。本文研究一种迭代重建算法，该算法通过最小化二次代价泛函实现。该代价泛函由数据保真项与Tikhonov正则化项之和构成。待重建函数$f$在光滑曲面$\mathcal S$上存在跳跃间断。固定点$x_0\in\mathcal S$及任意$A>0$，本文主要结果为计算极限$\Delta F_0(\check x;x_0):=\lim_{\epsilon\to0}(f_\epsilon(x_0+\epsilon\check x)-f_\epsilon(x_0))$，其中$f_\epsilon$为最小化问题的解，且$|\check x|\le A$。通过圆形GRT的数值实验表明，$\Delta F_0(\check x;x_0)$能精确逼近通过代价泛函最小化得到的实际重建结果。