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

2025-11-04

AI Summary

This paper addresses the reconstruction problem for locally translation-invariant (LTI) signals in 2D computed tomography (CT), where random sampling is performed *prior* to the Radon transform. Methodologically, it departs from the conventional “Radon-then-sample” paradigm by instead applying random sampling directly in the signal domain, followed by acquisition of its Radon projections. Theoretically, it proves that, with sufficiently large sample size, such random sampling sets satisfy the stability condition on the LTI subspace with high probability, and derives an explicit, closed-form reconstruction formula directly from the Radon data. Key contributions include: (i) the first stability theory for random sampling in the Radon domain, eliminating reliance on uniform angular or radial sampling; and (ii) the first efficient, deterministic reconstruction algorithm tailored to non-uniform random projections, achieving significantly improved accuracy and robustness under sparse sampling conditions.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web

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.

Problem

• Reconstruct signals from Radon random samples in local shift-invariant spaces
• Establish stability of random sampling sets with sufficient sample size
• Develop explicit reconstruction formula using Radon transform data

Innovation

• Random sampling before Radon transform
• Stable sampling set with probability
• Explicit reconstruction formula from samples