 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  .
<meaning-changed>  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  .
 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 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 .