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Radon random sampling and reconstruction in local shift-invariant signal space

Zhanpeng Deng, Jiao Li, Jun Xian

TL;DR

This work tackles reconstruction of $f$ in a local shift-invariant space $\mathcal{S}_{N,K}(\varphi)$ from Radon data collected at random pre-transform locations. It develops a probabilistic stability analysis via Matrix Bernstein to show that Radon random sampling is stable with high probability for sufficiently many samples and proves that the Radon data uniquely determine $f$, providing an explicit linear reconstruction using the sampling matrix $U$. A high-probability reconstruction theorem (with explicit reconstruction functions $\{\Upsilon_j\}$) shows that all $f \in \mathcal{S}_{N,K}(\varphi)$ can be recovered from Radon random samples with probability at least $1-\epsilon_{\mathcal{Q}}$. Numerical experiments using a box-spline generator demonstrate accurate recovery from a modest number of random Radon samples, underscoring the approach’s potential for CT-like imaging in a localized signal-model setting.

Abstract

In this paper, we deal with the problem of reconstruction from Radon random samples in local shift-invariant signal space. Different from sampling after Radon transform, we consider sampling before Radon transform, where the sample set is randomly selected from a square domain with a general probability distribution. First, we prove that the sampling set is stable with high probability under a sufficiently large sample size. Second, we address the problem of signal reconstruction in two-dimensional computed tomography. We demonstrate that the sample values used for this reconstruction process can be determined completely from its Radon transform data. Consequently, we develop an explicit formula to reconstruct the signal using Radon random samples.

Radon random sampling and reconstruction in local shift-invariant signal space

TL;DR

This work tackles reconstruction of in a local shift-invariant space from Radon data collected at random pre-transform locations. It develops a probabilistic stability analysis via Matrix Bernstein to show that Radon random sampling is stable with high probability for sufficiently many samples and proves that the Radon data uniquely determine , providing an explicit linear reconstruction using the sampling matrix . A high-probability reconstruction theorem (with explicit reconstruction functions ) shows that all can be recovered from Radon random samples with probability at least . Numerical experiments using a box-spline generator demonstrate accurate recovery from a modest number of random Radon samples, underscoring the approach’s potential for CT-like imaging in a localized signal-model setting.

Abstract

In this paper, we deal with the problem of reconstruction from Radon random samples in local shift-invariant signal space. Different from sampling after Radon transform, we consider sampling before Radon transform, where the sample set is randomly selected from a square domain with a general probability distribution. First, we prove that the sampling set is stable with high probability under a sufficiently large sample size. Second, we address the problem of signal reconstruction in two-dimensional computed tomography. We demonstrate that the sample values used for this reconstruction process can be determined completely from its Radon transform data. Consequently, we develop an explicit formula to reconstruct the signal using Radon random samples.

Paper Structure

This paper contains 6 sections, 7 theorems, 73 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Random sampling inequalities for the Radon transform in $\mathcal{S} _{N,K}(\varphi)$
  4. Reconstruction from Radon random sampling in $\mathcal{S} _{N,K}\left(\varphi\right)$
  5. Numerical Test
  6. Conclusion

Key Result

Theorem 2.1

Suppose that $\varphi\in L^2\left(\mathbb{R}^2\right)$ satisfying $(\mathbf{A.1})$ and $\left\{\varphi\left(\cdot-\mathbf{k}\right):\mathbf{k}\in\mathbb{Z}^2 \right\}$ is linearly independent. Let the direction vector be $\mathbf{p}=\left(\cos\theta,\sin\theta\right)$ such that $\mathcal{R}_\mathbf{ Then for sampling set $X=\left\{\mathbf{x}_{j}\right\}_{j=1}^{n}\subseteq \mathcal{E}_{K}$, $f\in \

Figures (4)

  • Figure 1: Radon transform: $\mathcal{R}_{\mathbf{p}}f\left(t\right)$ is the projection of $f$ along the X-ray at angle $\theta$.
  • Figure 2: Left: the plot of $\varphi\left(\mathbf{x}-\mathbf{k}\right)$ with $\mathbf{k}=\left(1,-1\right)$. Right: the plot of $\mathcal{R}_\mathbf{p}\left(\varphi\left(\mathbf{x}-\mathbf{k}\right)\right)$ with $\mathbf{p}$ = [$\frac{5}{13}$, $\frac{12}{13}$].
  • Figure 3: Top left: the plot of function $f$. Top right: Radon transform $\mathcal{R}_{\mathbf{p}}f$ with $\mathbf{p}=\left[\cos\left(\frac{5}{13}\pi\right),\sin\left(\frac{12}{13}\pi\right)\right]$. Bottom left: reconstruction version $\widetilde{f}$ of $f$. Bottom right: the plot of $f-\widetilde{f}$
  • Figure 4: Error analysis of varied sampling points

Theorems & Definitions (14)

  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2: Matrix Bernstein inequality WOS:000306433100001
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Remark 3.5
  • ...and 4 more