Equation 5 of Siddon's ray tracing paper

[Little-Stars520]

I don't understand formula 5 in the Siddon'spaper.
Why should we take the maximum value among x, z, y for the parameter α_min? Doesn't this mean that some points are missed?
In that paper, they first take the minimum parameter values α_min in x, y, and z respectively, and then why do we need to taske the maximum value among them if they are all greater than 0?
In my mind, it should take the smallest one in [α_x,α_y,α_z] first,and then take the max between 0 and the smallest one in [α_x,α_y,α_z] .
i.e. alpha_min = max([0 min([min(aX_1,aX_end) min(aY_1,aY_end) min(aZ_1,aZ_end)])]);

[wahln]

As far as I recall, the problem is that a plane is not bounded by the volume, and as such an intersection with a plane can/will happen outside of the volume (before you hit the actual volume). if you find the maximum value of all first intersections, you will find the actual first intersection with the volume.

Think about a ray coming in about 60° gantry angle in the LPS matRad system (0° would be incoming along the y axis). It will first intersect the outer plane y0, even before it hits the ct. This will be alpha_y_min. Then it will hit the last x plane (x_end) later, which will be the first intersection with the actual volume, which will be alpha_x_min > alpha_y_min.

If you take the formula as it is in the paper, you will do alpha_min = max(0, alpha_y_min, alpha_x_min), where alpha_x_min will be the larger of the two and the correct choice.

If we take your interpretation, we will start tracing voxels outside of the ct (when we first hit a plane at all).

[wahln]

I guess you can try this by debugging through matRad_siddonRayTracer during dose calculation choosing a 60° gantry angle for your plan. Just set a break point at line 134 ;-)

[wahln]

Finally, I will convert this issue to a discussion. Let me know if you have further questions regarding the raytracer.

[Little-Stars520]

Thank you very much for your answer. Since a plane can extend infinitely, there indeed exist intersection points located outside the volume.

[Little-Stars520]

Finally, I will convert this issue to a discussion. Let me know if you have further questions regarding the raytracer.最后我把这个问题转为讨论。如果您对光线追踪器还有其他疑问，请告诉我。

When I try to understand the code further, I have a few questions.
1）For photon beams, why do we need to reconstruct a ray_matrix instead of directly using 2 * ray_position as the target point?
2)

After I run the matRad_siddonRayTracer function, I can get all the indices that the i-th ray passes through when traversing the CT matrix. Then why do I still need to further filter these indices? The i-th ray does pass through all the obtained indices and why can we ignore the indices greater than rayMxSpacing/2?
3)

Why do we need to subtract d/2 after calculating the total distance that the ray has passed through according to Formula 14 in Line 112?