Volume rendering: confusion with front-to-back compositing

Question

In, for example, GPU Gems the front-to-back compositing equation (for colour) is

C'_i = (1 - A'_i-1)C_i + C'_i-1

where C'_i is the output accumulated colour value; A'_i-1 is the accumulated alpha (opacity) value up to the previous voxel; C_i is the colour value of the current voxel; and C'_i-1 is the accumulated colour value up to the previous voxel.

This formulation raises two questions to me:

1. Termination of front-to-back occurs once the accumulated opacity reaches approximately 1. What, then, to do about the colour channels (RGB) that go past the maximum before the opacity limit is reached? Do you just clamp the values between 0..255 (e.g. 500,1000,2000 would become 255,255,255), or look to the ratio between the channels (e.g. 500,1000,2000 would become 64,128,255).

2. The answer to the previous question possibly feeds into this. The colour output of the current voxel depends on one minus the accumulated opacity. What if the accumulated opacity is zero and the current voxel's opacity is zero? - the output is a completely opaque voxel, since (1 - A'_i-1) = 1, even though it is supposedly a transparent voxel!?

Any hints much appreciated.

Answer 1

1. A and C should be in the range 0-1. (If you're using unsigned bytes as the representation, divide by 255.0, but note that for some volume rendering application areas this will give you insufficient control over small alpha/low opacity regions to really be satisfactory. These days it's generally just easier to compute using floats from the outset). It turns out that the alpha and color values can never escape outside this range using the your formulas.

2. The sequence for the ray alpha A' is A'(i) = (1-A'(i-1)).A(i) + A'(i-1) (where A(i) is the voxel alpha), so if your accumulated ray starts with A' zero, and passes through a transparent (zero A) voxel, the ray now has A' = (1-0)*0+0, which is still zero as expected.