Multiple reflections and transmissions of light inside a cube

[davidram]

Imagine I have a 10x10x10cm cube filled with a scintillating material (material capable of generating light when energized). Three cameras are looking at this cube from three orthogonal directions (x, y, and z). Light is generated inside the cube and is refracted as it leaves the cube and reaches the camera. Although the cube wall is transparent, some light is reflected internally, and some light will not leave the cube because of total internal reflection.

My goal is to reconstruct the distribution of light generated inside this cube in three dimensions. I have created a projection matrix, by ray tracing each pixel inside this cube back to the camera. However, I'm not sure how to approach the reflection problem as light may be bouncing around multiple times inside the cube. I have looked at Fresnel equation for reflection and the best I can think of is to trace each pixel to all pixels on the interior wall of the cube. Because the angle of incident is known, the fraction of light that is reflected from that particular ray can be calculated. The reflected ray can further be traced to the next wall and the fraction of light transmitted/reflected can be calculated. The problem with this approach is that it is computationally expensive and will result in a massive projection matrix. I am wondering if there is a more elegant way to approach this problem. Any help is greatly appreciated.

 

[Merlin3189]

I'm no expert.
What occurs to me is that all the rays from one point in the block eventually appear to come from an array of points produced by reflections. So these points are regularly spaced, first in a front plane of blocks, where the image is alternately a reflection or translation of the block point, then in successive planes of blocks behind, which again are alternate refections and translations of the front plane. (Translation = reflection of reflection)

1-Now for each image block it is easy to count how many internal reflections have taken place.
2-For each point in the block, it should be easy to determine the nature of each reflection that has given rise to the image point in the different blocks and hence the strength of that image point as a source.
3-Contrary to your diagram there are limited number rays which can appear to come from each image point.

Of the two reflections shown on the RHS of the block (viewed as shown), only one of yours is actually possible (IMO)
In my diagram, both rays P and Q must appear to emanate from the same point (simple reflection property) when they impinge on the front surface (top in this diagram) and by Snell's law only one of them can be refracted at the necessary angle to enter the camera lens. (In fact there must be a small bundle of closely adjacent rays that will enter the lens, of course. But depending on the size of the sensor, they may not be focused onto it.)
4-Rays from image points cannot reach the camera, unless they can pass through the front face of the real block AND be refracted towards the lens. This rules out most image points in side blocks. As you go back in the reflection planes, more side blocks are falling within the acceptance angle, but those sources are weaker because they have had more reflections. (B1, C1,...,Y1 cannot reach the camera for this point, B2, ... ..X2 can't, B3, ... X3 can't. Even A1 is dubious.)

By consideration of these points I think you can greatly prune your tree for each source point.
I think you might calculate the locus of image points where they are no longer strong enough to need consideration, by considering in detail relatively few points and interpolating. That would define the necessary limit of your search/calculation for each individual point

 

[Merlin3189]

Update to illustrate main point. A ray trace for another of your rays, showing where it must appear to come from.