The Direct Integration Theorem: A Rigorous Framework for Consistent Discrete Solutions of the Inverse Radon Problem

2026-05-09

AI Summary

This work addresses the zero-frequency singularity, spectral distortion, and discretization errors inherent in conventional filtered back-projection (FBP) for CT reconstruction, which arise from frequency-domain interpolation and ramp filtering. To overcome these limitations, the authors propose a rigorous continuous–discrete unified mathematical framework grounded in the Direct Integration Theorem (DIT), enabling an inverse Radon transform that eliminates the need for both frequency-domain interpolation and ramp filtering. By incorporating an exact discrete inverse Radon transform and precise grid-based geometric modeling, the method effectively suppresses intensity cupping artifacts and achieves superior performance over FBP in terms of peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), and reprojection fidelity, while accurately preserving the statistical properties of the original image and enabling quasi-exact reconstruction.

Abstract

This paper presents a novel Direct Integration Theorem (DIT), derived as a non-trivial corollary of the classical Central Slice Theorem (CST). The DIT provides a mathematically consistent transition from the continuous to the discrete domain - a fundamental challenge in computed tomography - thereby eliminating the need for frequency-domain interpolation without resorting to conventional ramp-filtering. The proposed approach circumvents two principal limitations inherent in traditional methods: (i) the zero-frequency singularity and spectral distortions introduced by the mandatory ramp-filtering step, and (ii) discretization inaccuracies associated with frequency-domain interpolation. Based on the DIT, we develop a rigorous framework for consistent discrete solutions of the inverse Radon problem. Mathematical modeling demonstrates that this approach achieves quasi-exact reconstruction, with errors constrained solely by sampling parameters and grid geometry. Furthermore, while Filtered Back Projection (FBP) inherently distorts the variance of the reconstructed image, the DIT-based algorithm preserves it. Comparative simulations confirm that the proposed method eliminates common artifacts, such as intensity cupping, and consistently outperforms FBP in terms of PSNR, SSIM, and reprojection fidelity, faithfully restoring the original image's statistical characteristics.

Problem

• inverse Radon problem
• discrete reconstruction
• computed tomography
• frequency-domain interpolation
• ramp filtering

Innovation

• Direct Integration Theorem
• Inverse Radon Problem
• Discrete Consistency
• Ramp-filter-free Reconstruction
• Quasi-exact Tomographic Reconstruction