Veach - refractive BTDF adjoint

[dawelter]

Hello!

I'm trying to understand Veach's derivation of the adjoint of the specular refractive BTDF. It seems I hit a brick wall. Some of that stuff looks like black magic to me. Specifically, there is a relation for differential solid angle on the following page:

non-symmetry-due-to-refraction.png (90.2 KiB) Viewed 5054 times

What is the underlined term in 5.3 mathematically? What confuses me is that there is a dsigma in both nominator and denominator. So it is unlike a regular derivative nor a Radon-Nikodym derivative?

Further down, we have Eq (5.4). Same thing. I read dsigma as a small patch of solid angle, which is determined by the spherical parameterization (*). So if (*) holds true, then how comes that for different values of (theta, phi) there is mysteriously a ratio of eta_i / eta_t appearing. Sure I get that Veach simply plugged in Snell's law in (*). But I don't get the meaning of it.

I should add that my knowledge of measure theory is practically non existent.