A question about BSSRDF generalized to rough surface

[shocker_0x15]

Hi,

Now I try to implement SSS feature in my renderer using BSSRDF.
I have started with "A Practical Model for Subsurface Light Transport" and Donner's thesis "Towards Realistic Image Synthesis of Scattering Materials".
They say that the diffuse component of BSSRDF for smooth surface ( = ideal specular BTDF?) is

Sd_smooth.png

(Screen shot from the former material)

In Donner's thesis they generalize enclosing surface to rough surface, that is

Sd_rough.png

(Screen shot from the latter material)

I'm confused by the symbols used in the latter material.
So, I tried to rewrite what they say by myself.

Sd_rough_my_understanding.png

Here, integration domain H^2- means lower hemisphere (inside material) and I write the diffuse transmittance term by using BSDF(BTDF) instead of BRDF they used.

My understanding for this generalized BSSRDF process is
1. The first integration term represents a total energy entering a material over the interface due to a light coming from wi direction at xi
2. The term R converts the energy to radiant exitance at xo
3. By multiplying 1 /pi to the radiant exitance, we get radiance reaching to xo from any direction in lower hemisphere
4. The second integration term converts the radiance from lower hemisphere to outgoing radiance from xo in direction wo

Is something wrong in my understanding?

Thanks,

[shocker_0x15]

Conventions for symbols I used are

convention.png

I got the following if I substitute an ideal specular BTDF to the equation I posted.

substitute_ideal_specular.png

However this slightly differs from the original one.
(The latter Fresnel term is the same to F(eta, wo), therefore the problem is the fraction term contains cosines)

[citadel]

Have yet to read that paper. But PBRT has a good explanation on that matter.
BTW, 1/pi is a like a normalization factor as in BRDF.
And the diffuse Fresnel transmittance is no more than approximation to specular
Fresnel transmittance from all directions. From my experience it may give darker results
if you don't rather use specular Fresnel transmittance for each light sample.

[shocker_0x15]

Thank you for your reply.

And the diffuse Fresnel transmittance is no more than approximation to specular
Fresnel transmittance from all directions.

The original BSSRDF seems to use "directional" Fresnel transmittance not diffuse one.
Is it natural that substituting the ideal specular BTDF produces different BSSRDF compared to the original one?

Of course, there is a high probability I made incorrect BSSRDF expresssion that is generalized to rough surface or I have some misunderstanding about ideal specular BxDF.