What is difficult to understand in Radiance theory


I am reading a paper to learn about the theory of Radiance,. A Ray Tracing Solution for Diffuse Interreflection, but there are some things I don’t understand, so I would like to know two things.

In Equation 2, the upper limit of sigma addition is n, 2n. Why is this? If we think of it as a transformation from Equation 1 to Equation 2, wouldn’t it be n, 4n?

I have derived the inequality in equation (3a) using the first order Taylor expansion, how can this be derived?
I tried to prove that this inequality holds from Taylor’s theorem, but I could not derive it.
If you don’t mind, I’d like to know the process of deriving the equation.

Please help me.

I think the 2n2 in equation 2 is just the number of samples… I see it as a Montecarlo integration, we are calculating the average of a bunch of samples. If you see the Sums, you have 1..n and 1..2n; meaning n*2n = 2n2 samples. These samples are distributed in the hemisphere. In the Rendering with Radiance book that equation is different, I think, as it does not assume n an 2n samples; but n and m; and m*n = ambient_divitions

I have not studied Equation 3 at all…

PS: I am afraid Greg will destroy this answer, but I hope you find it useful for a couple of minutes before that.

Not to destroy anything, but at some point, there is no need to discuss the details of the 1988 paper. We can actually divide phi and theta however we like in the sum – we chose this as a reasonable compromise for the division cell aspect ratios, though it’s still pretty bad. (See Figure 12.7 from “Rendering with Radiance,” reproduced below.)


Likewise, we don’t use the split sphere estimate for calculation spacing anymore. We have supplanted this method in the current version with a more accurate Hessian calculation due to Schwarzhaupt, Wann Jensen and Jarosz:

“Practical Hessian-Based Error Control for Irradiance Caching”.
Jorge Schwarzhaupt, Henrik Wann Jensen, and Wojciech Jarosz
ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia 2012), Singapore, November 2012

They use the Shirley-Chiu hemisphere subdivision, which does a much better job and simplifies neighbor relationships. We also fixed an issue they had with locally shaded regions, as described in my 2014 workshop talk.


By the way, Greg, is there any reason for sampling randomly WITHIN bins as opposed to just sampling randomly within the whole hemisphere?

Yes – it’s called “stratified sampling” and is a common Monte Carlo variance reduction technique that helps with most well-behaved integrals.

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Greg, German, thanks for the heads up.

It gave me a good understanding of the question I wanted to ask ①.
Now, what about question ②?

Eq. (3a) isn’t derived from Taylor’s expansion – it is the first-order Taylor’s expansion for any two-variate function. Eq. (3b) is the Taylor’s expansion for the split sphere condition, and is computed by differentiating the formula for projected area as a function of position and orientation from the test point.

As I said, none of this is currently used by Radiance, so is really only of historical interest.

Greg, thanks.

My question has been answered.
I was asking the question out of sheer intellectual curiosity to know the history.

Thank you very much.

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