A Novel Interpretation of the Radon Transform's Ray- and Pixel-Driven Discretizations under Balanced Resolutions

TL;DR

This paper provides a new theoretical interpretation of common Radon transform discretization methods in tomography, analyzing their approximation errors and convergence properties under balanced resolutions, supported by numerical experiments.

Contribution

It introduces a novel convolutional interpretation of ray- and pixel-driven discretizations and discusses their convergence under balanced resolutions.

Findings

Ray-driven and pixel-driven methods are similar under convolutional interpretation.

Convergence of these methods is established under balanced resolution conditions.

Numerical experiments support the theoretical analysis.

Abstract

Tomographic investigations are a central tool in medical applications, allowing doctors to image the interior of patients. The corresponding measurement process is commonly modeled by the Radon transform. In practice, the solution of the tomographic problem requires discretization of the Radon transform and its adjoint (called the backprojection). There are various discretization schemes; often structured around three discretization parameters: spatial-, detector-, and angular resolutions. The most widespread approach uses the ray-driven Radon transform and the pixel-driven backprojection in a balanced resolution setting, i.e., the spatial resolution roughly equals the detector resolution. The use of these particular discretization approaches is based on anecdotal reports of their approximation performance, but there is little rigorous analysis of these methods' approximation errors. This paper presents a novel interpretation of ray-driven and pixel-driven methods as convolutional discretizations, illustrating that from an abstract perspective these methods are similar. Moreover, we announce statements concerning the convergence of the ray-driven Radon transform and the pixel-driven backprojection under balanced resolutions. Our considerations are supported by numerical experiments highlighting aspects of the discussed methods.