Topic 3 – Implementing the Beer-Lambert Law on GPU using OpenGL

IBSim-4i 2020

Dr Franck P. Vidal

13th Aug 2020

Path Length: Naive Approach

Is finding intersections in the right order important?

Detect every intersection between a ray and the objects;
Sort intersection (Can be handled by GPUs using depth-peeling, a multi-pass rendering technique for semi-transparent polygonal objects without sorting polygons);
Compute path length.

Finding intersections in any order does not matter

Intersection sorting is actually not needed!

By convention normals are outward;
A ray penetrates into an object when the dot product between the view vector (V) and the normal (N_i) at the intersection point is positive;
It leaves an object when the dot product is negative.

L_p=Σ_i - sng(V · N_i) x d_i

i refers to i^th intersection in an arbitrary order;
d_i distance from X-ray source to intersection point;
sgn(V · N_i) stands for the sign of the dot product between V and N_i;
In a shader program, compute:
- sgn(V · N_i);
- d_i the distance between the X-ray source and the intersection;
- Assign -sng(V · N_i) x d_i as the fragment value.
For each pixel, compute L_p thanks to high-dynamic range and OpenGL blending function (pixel values may not be between 0 and 1).

pixel = E x N_out

pixel = E x N_in(E) e^{(-Σ_i μ_i L_p(i))}

Needs 3 FBOs with high-dynamic range capability for off-line rendering:
For each object of the scene:
1. Compute L_p(i);
2. Update results of Σ μ_i L_p(i).
For the final image only:
1. Compute N_out;
2. (Optional when direct display only is needed).

OpenGL pipeline to compute the Beer-Lambert law (monochromatic case).

Take into the different energies within the incident beam;
This is known as beam hardening;
Iterate over several energy channels:
- pixel = Σ_j E_j x N_out(E_j)
- pixel = Σ_j E_j x N_in(E_j) e^{(-Σ_i μ_i(E_j,ρ,Z) d_i)}
Example:

Simulation parameters

Polychromatic beam spectrum for 90kV X-ray tube peak voltage

Intensity profiles, see dash line in image above

Take into account the focal spot of the X-ray source;
Iterate over several point sources within the volume of the focal spot:
- pixel = Σ_k Σ_j E_j x N_in(E_j) e^{(-Σ_i μ_i(E_j,ρ,Z) d_i(k))}
- See blur in the corresponding image below.

Parallel beam	Infinitely small point source	1³mm source

Iterate over several energy channels: Three extra for loops;
Iterate over several point sources within the volume of the focal spot: One extra for loop.

Final OpenGL pipeline