An efficient algorithm to compute the X-ray transform

Abstract

We propose a new algorithm to compute the X-ray transform of an image represented by unit (pixel/voxel) basis functions. The fundamental task is equivalently calculating the intersection lengths of the ray with associated units. For the given ray, we derive the sufficient and necessary condition for non-vanishing intersectability. By this condition, we can distinguish the units that produce valid intersections with the ray. Only for those units, we calculate the intersection lengths by the obtained analytic formula. The proposed algorithm is adapted to various two-dimensional (2D)/three-dimensional (3D) scanning geometries, and its several issues are also discussed, including the intrinsic ambiguity, flexibility, computational cost and parallelization. The proposed method is fast and easy to implement, more complete and flexible than the existing alternatives with respect to different scanning geometries and different basis functions. Finally, we validate the correctness of the algorithm.

Keywords:

• X-ray transform
• intersection length
• projection matrix
• non-vanishing intersectability
• ambiguity and flexibility
• tomographic image reconstruction

2010 AMS Subject Classifications:

• 65R32
• 68U10
• 92C55
• 94A08

Acknowledgments

The authors would like to express their sincere gratitude to the anonymous reviewers for their review and helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work of Chen was partially funded by the Beijing Natural Science Foundation under grant Z180002. The work of Öktem was supported by the Swedish Foundation of Strategic Research grant AM13-0049.