John Equation Constraints for the 3D X-ray Transform under a Cylindrical-Spherical Mixed Parameterization: Theoretical Derivation, Experimental Validation, and Application Analysis

TL;DR

This paper derives the John equation for the 3D X-ray transform using a novel mixed cylindrical-spherical parameterization, providing mathematical tools for data verification, calibration, and reconstruction in CT imaging.

Contribution

It introduces a new geometric parameterization scheme and systematically derives the corresponding John equation, bridging mathematical theory and practical CT imaging applications.

Findings

Derived the John equation under mixed parameterization.

Simplified constraint equations under specific geometric configurations.

Provided mathematical tools for data consistency and calibration in 3D CT.

Abstract

The John equation serves as the mathematical foundation of the X-ray transform, describing the intrinsic compatibility conditions that projection data must satisfy. In this paper, within three-dimensional (3D) Euclidean space, an innovative mixed parameterization scheme is adopted: the source point is represented using cylindrical coordinates a=(s cos{\theta},s sin{\theta},z_0), and the ray direction is represented using spherical coordinates d=\{rho}(-cos\{beta}sin{\alpha},cos\{beta}cos{\alpha},sin\{beta}). The specific form of the John equation under this geometric parameterization is systematically derived. Through detailed partial differential operator transformations, application of -1 homogeneity, and algebraic simplification, a complete system of constraint equations is obtained. In particular, under the special configurations where the ray direction is perpendicular to the radial direction of the source point in the horizontal plane (i.e., the so-called alignment condition:{\alpha} = {\theta}) and the ray has no tilt (\{beta} = 0), the constraint equations simplify to differential relations with clear physical meanings. This paper not only establishes a bridge between abstract mathematical theory and concrete imaging geometry, but also provides rigorous mathematical tools for data consistency verification, geometric parameter calibration, and incomplete-data reconstruction in 3D Computed Tomography (CT) systems. The research results are of great significance for advancing the mathematical theory and practical applications of CT imaging.