An Algorithm to Compute the X-ray Transform
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The X-ray transform models a forward projection operator of image formation, which has been widely used for tomographic image reconstruction. We propose a new algorithm to compute that transform of an image represented by unit (pixel/voxel) basis functions, for various two-dimensional (2D)/three-dimensional (3D) scanning geometries, such as 2D/3D parallel beam, 2D fan beam, 3D circular/helical cone beam, etc. Since the transform is acting as a line integral, the fundamental task is to calculate this integral of the unit basis functions, which is equivalently the intersection length of the ray with the associated unit. For a given ray, using the support of unit basis function, we derive the sufficient and necessary condition of non-vanishing intersectability, and obtain the analytic formula for the intersection length, which can be used to distinguish the units that produce valid intersections with the given ray, and then perform simple calculations only for those units. The algorithm is easy to be implemented, and the computational cost is optimal. Moreover, we discuss the intrinsic ambiguities of the problem itself that perhaps happen, and present a solution. The algorithm not only possesses the adaptability with regard to the center position, scale and size of the image, and the scanning geometry as well, but also is quite suited to parallelize with optimality. The resulting projection matrix can be sparsely stored and output if needed, and the adjoint of X-ray transform can be also computed by the algorithm. Finally, we validate the correctness of the algorithm by the aforementioned scanning geometries.
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