Total absorption through medium along ray

[cronusf]

I have a 3D graphics book, which gives the formula for absorption of radiance along a ray. I am trying to derive the details and would like to see if my derivation is correct.

Let o(p) be the probability density that light is absorbed per unit length at point p.

They give the formula as: exp( -int_0^d(o(p+tv)dt) )

where p is the starting point of the ray entering the medium, and it exits at the point p+dv.

So to set this up, I look at a small section along the ray, to see how much light is absorbed across that small section of length h. I also recast things in terms of t, since the position along the ray is a function of t. Let L(t) denote the light radiance at point t.

L(t+h) = L(t) - o(t)*h*L(t)

That is, the light radiance after passing through a segment of length h equals the incoming light radiance minus the amount absorbed.

[L(t+h) - L(t)] / h + o(t)*L(t) = 0

Taking the limit h-->0

dL(t) / dt + o(t)*L(t) = 0

I multiply by integrating factor exp(int_0^t(o(t)dt))

to get

(exp(int_0^t(o(t)dt)) * L(t))' = 0

Integrating from 0 to d:

exp(int_0^d(o(t)dt))*L(d) - exp(int_0^0(o(t)dt))*L(0) = C

We know L(0) = 0 since no absorption yet.

L(d) = C*exp(-int_0^d(o(t)dt))

I'm not sure how to get rid of the C. I think it should not have occurred since I use definite integral. Also, is my integrating factor correct: exp(int_0^t(o(t)dt))??

 

[torquil]

I have a 3D graphics book, which gives the formula for absorption of radiance along a ray. I am trying to derive the details and would like to see if my derivation is correct.

Let o(p) be the probability density that light is absorbed per unit length at point p.

They give the formula as: exp( -int_0^d(o(p+tv)dt) )

where p is the starting point of the ray entering the medium, and it exits at the point p+dv.

So to set this up, I look at a small section along the ray, to see how much light is absorbed across that small section of length h. I also recast things in terms of t, since the position along the ray is a function of t. Let L(t) denote the light radiance at point t.

L(t+h) = L(t) - o(t)*h*L(t)

If you want to rephrase in terms of t, it should be o(p+tv), not o(t). Also, you meant "the light radiance at time t", not "the light radiance at point t".

That is, the light radiance after passing through a segment of length h equals the incoming light radiance minus the amount absorbed.

[L(t+h) - L(t)] / h + o(t)*L(t) = 0

Taking the limit h-->0

dL(t) / dt + o(t)*L(t) = 0

I multiply by integrating factor exp(int_0^t(o(t)dt))

to get

(exp(int_0^t(o(t)dt)) * L(t))' = 0

Integrating from 0 to d:

exp(int_0^d(o(t)dt))*L(d) - exp(int_0^0(o(t)dt))*L(0) = C

That last one is not correct. When you do a definite integral (with limits) you don't get an integration constant. So you should have 0 on the right hand side (the integral of 0 from a to b is still 0). Also, remember the o(p+tv).

We know L(0) = 0 since no absorption yet.

No, L(0) is your incoming radiance, so it is not zero. Just leave it in, or choose it equal to 1 as they have done in the book. Your equation above, where I have corrected by putting C=0 and the o(p+tv) then leads to:

L(d) = exp(-int_0^d(o(p+tv)dt))

Torquil