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A Fast and Adaptive Algorithm to Compute the X-ray Transform

Chong Chen, Runqian Wang, Chandrajit Bajaj, Ozan Öktem

TL;DR

The paper tackles the problem of fast forward projection in tomography by developing an adaptive X-ray transform algorithm for images represented by unit basis functions. It derives a sufficient and necessary condition for non-vanishing ray–unit intersections and provides analytic formulas to compute intersection lengths, enabling a ray-by-ray, unit-by-unit efficient computation. The method extends from the basic 2D parallel beam to 2D/3D fan beams and further to 3D circular and helical cone-beam geometries through explicit transformation formulas, while supporting polygonal/polyhedral bases and sparse projection matrices. Computationally, it achieves $O(N)$ per ray and $O(NM)$ overall, with strong parallelization potential and explicit handling of ambiguities, validated across multiple geometry test suites. The approach offers a flexible, scalable solution for forward projection and adjoint computations in tomographic reconstruction, potentially benefiting low-dose and sparse-view imaging regimes.

Abstract

We propose a new algorithm to compute the X-ray transform of an image represented by unit (pixel/voxel) basis functions. The fundamental issue is equivalently calculating the intersection lengths of the ray with associated units. For any given ray, we first derive the sufficient and necessary condition for non-vanishing intersectability. By this condition, we then distinguish the units that produce valid intersections with the ray. Only for those units rather than all the individuals, we calculate the intersection lengths by the obtained analytic formula. The proposed algorithm is adapted to 2D/3D parallel beam and 2D fan beam. Particularly, we derive the transformation formulas and generalize the algorithm to 3D circular and helical cone beams. Moreover, we discuss the intrinsic ambiguities of the problem itself, and present a solution. The algorithm not only possesses the adaptability with regard to the center position, scale and size of the image, but also is suited to parallelize with optimality. The comparison study demonstrates the proposed algorithm is fast, more complete, and is more flexible with respect to different scanning geometries and different basis functions. Finally, we validate the correctness of the algorithm by the aforementioned scanning geometries.

A Fast and Adaptive Algorithm to Compute the X-ray Transform

TL;DR

The paper tackles the problem of fast forward projection in tomography by developing an adaptive X-ray transform algorithm for images represented by unit basis functions. It derives a sufficient and necessary condition for non-vanishing ray–unit intersections and provides analytic formulas to compute intersection lengths, enabling a ray-by-ray, unit-by-unit efficient computation. The method extends from the basic 2D parallel beam to 2D/3D fan beams and further to 3D circular and helical cone-beam geometries through explicit transformation formulas, while supporting polygonal/polyhedral bases and sparse projection matrices. Computationally, it achieves per ray and overall, with strong parallelization potential and explicit handling of ambiguities, validated across multiple geometry test suites. The approach offers a flexible, scalable solution for forward projection and adjoint computations in tomographic reconstruction, potentially benefiting low-dose and sparse-view imaging regimes.

Abstract

We propose a new algorithm to compute the X-ray transform of an image represented by unit (pixel/voxel) basis functions. The fundamental issue is equivalently calculating the intersection lengths of the ray with associated units. For any given ray, we first derive the sufficient and necessary condition for non-vanishing intersectability. By this condition, we then distinguish the units that produce valid intersections with the ray. Only for those units rather than all the individuals, we calculate the intersection lengths by the obtained analytic formula. The proposed algorithm is adapted to 2D/3D parallel beam and 2D fan beam. Particularly, we derive the transformation formulas and generalize the algorithm to 3D circular and helical cone beams. Moreover, we discuss the intrinsic ambiguities of the problem itself, and present a solution. The algorithm not only possesses the adaptability with regard to the center position, scale and size of the image, but also is suited to parallelize with optimality. The comparison study demonstrates the proposed algorithm is fast, more complete, and is more flexible with respect to different scanning geometries and different basis functions. Finally, we validate the correctness of the algorithm by the aforementioned scanning geometries.

Paper Structure

This paper contains 47 sections, 61 equations, 4 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Outline.
  4. Algorithm for 2D scanning geometries
  5. Preliminaries
  6. 2D scanning geometries.
  7. 2D imaging coordinate system.
  8. Pixel indexes.
  9. Derivation of intersection length
  10. 2D parallel beam
  11. Case 1: $\phi \in (0, \pi/2)$.
  12. Case 2: $\phi = 0$.
  13. Fan beam
  14. Equiangular fan beam.
  15. Equispaced fan beam.
  16. ...and 32 more sections

Figures (4)

  • Figure 1: (a) 2D parallel beam; (b) Fan beam
  • Figure 2: (a) Pixel index in 1D form; (b) Pixel index in 2D form
  • Figure 3: (a) 3D parallel beam; (b) Circular cone beam; (c) Helical cone beam
  • Figure 4: (a) Voxel index in 1D form; (b) Voxel index in 3D form

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6