Local reconstruction analysis of inverting the Radon transform in the plane from noisy discrete data

TL;DR

The paper addresses how discretization and random noise in 2D CT data affect local image reconstruction. It shows that the noise-driven reconstruction error $N_{\epsilon}^{rec}$ admits a distributional limit $N^{rec}(\check x;x_0)$ as the data step $\epsilon\to0$, and that this limit is a zero-mean Gaussian random field with an explicit covariance, derived from the noise variance $\sigma^{2}$ and the reconstruction kernel. The results are established via Lyapunov-type CLTs and tightness arguments, yielding both finite-dimensional Gaussian limits and a continuous Gaussian random field on rectangles, with a detailed covariance formula. Numerical experiments corroborate the theory, demonstrating accurate pointwise Gaussian behavior and matching covariance predictions at native resolution scales. This provides a rigorous, local, high-resolution statistical description of reconstruction noise in discrete, noisy CT data, enabling inference without additional smoothing away from the data scale.

Abstract

In this paper, we investigate the reconstruction error, $N_\e^{\text{rec}}(x)$, when a linear, filtered back-projection (FBP) algorithm is applied to noisy, discrete Radon transform data with sampling step size $ε$ in two-dimensions. Specifically, we analyze $N_\e^{\text{rec}}(x)$ for $x$ in small, $O(\e)$-sized neighborhoods around a generic fixed point, $x_0$, in the plane, where the measurement noise values, $η_{k,j}$ (i.e., the errors in the sinogram space), are random variables. The latter are independent, but not necessarily identically distributed. We show, under suitable assumptions on the first three moments of the $η_{k,j}$, that the following limit exists: $N^{\text{rec}}(\chx;x_0) = \lim_{\e\to0}N_\e^{\text{rec}}(x_0+\e\chx)$, for $\check x$ in a bounded domain. Here, $N_\e^{\text{rec}}$ and $ N^{\text{rec}}$ are viewed as continuous random variables, and the limit is understood in the sense of distributions. Once the limit is established, we prove that $N^{\text{rec}}$ is a zero mean Gaussian random field and compute explicitly its covariance. In addition, we validate our theory using numerical simulations and pseudo random noise.

Figures (4)

• Figure 1: Illustration of the quantity $l_\star$, the set $I_l$, one of the two intervals that make up $I_\star$, namely the interval $\{|\alpha|\le\pi: \kappa\phi^{\prime}(\alpha)\ge l_\star+(1/2)\}$. The other interval $\{|\alpha|\le\pi: \kappa\phi^{\prime}(\alpha)\le -(l_\star+(1/2))\}$ is not visible.
• Figure 2: Example image reconstructions from uniform random noise with $j_m = 10^3$ on the full scale (a), and within an $\epsilon$ neighborhood of zero (b).
• Figure 3: Observed (a) and predicted (b) pdf functions for $x_0 = \frac{1}{4}(\sqrt{2},\sqrt{3})$ and $x_1 = x_0 + \frac{\epsilon}{2\sqrt{2}}(1,1)$, with $j_m = 10^3$ and $n = 10^4$ samples. (c) - covariance error as a function of $\epsilon$; $n = 10^4$ samples were used to calculate the covariance error for all values of $\epsilon$ in the plot in (c).
• Figure 4: Observed (a) and predicted (b) pdf functions for $x_0 = \frac{1}{4}(\sqrt{2},\sqrt{3})$ and $x_1 = x_0 + \frac{5\epsilon}{\sqrt{2}}(1,1)$, with $j_m = 10^3$ and $n = 10^4$ samples.

Theorems & Definitions (17)

• Definition 2.3
• Theorem 2.5
• Corollary 2.6
• Theorem 2.7
• Corollary 2.8
• Theorem 2.9
• Definition 3.1: Khoshnevisan2002
• Theorem 3.2: Khoshnevisan2002
• Lemma 4.1
• Lemma 4.2