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

TL;DR

This paper introduces the Direct Integration Theorem (DIT), a new framework for consistent discrete solutions in inverse Radon problems, improving accuracy and reducing artifacts in computed tomography reconstructions.

Contribution

The paper develops a rigorous DIT-based framework that eliminates ramp-filtering and interpolation issues, enabling quasi-exact, artifact-free image reconstruction in CT.

Findings

Achieves quasi-exact reconstruction constrained by sampling and grid geometry.

Eliminates common artifacts like intensity cupping in CT images.

Outperforms FBP in PSNR, SSIM, and reprojection fidelity.

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.