 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  .
<fluency>  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  .
 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 associated units. For any 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 .