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.