I have had a question for a long time about ideal specular BTDF (BRDF).

Ideal specular BRDF and BTDF are:

- f_r(w_i, w_o) = F(theta_i) * delta(w_o - R(w_i, n)) / |cos(theta_i)|
- f_t(w_i, w_o) = (eta_o / eta_i)^2 * (1 - F(theta_i)) * delta(w_o - T(w_i, n)) / |cos(theta_i)|

Q. Why are they divided by the cosine of angle of incident direction instead of scattered direction?

I think scattered radiance should be divided by the cosine of scattered angle.

If Fresnel term already accounts for cosine of incident angle (that is also in rendering equation) and cosine of scattered angle (in this case Fresnel term represents incident radiance to scattered radiance ratio?), it seems reasonable for BRDF but not for BTDF. Who accounts for cosine of refracted angle for BTDF?

In general BTDF does not obey reciprocity, but for basic radiance (L/eta^2) it restores reciprocity.

I think it appears that the ideal specular BTDF does not obey reciprocity even for basic radiance.

Which is my misunderstanding?

Sorry for my poor English.

Thanks,