Microfacet Rough Surface Reflection

We first assume the surface does not emit light, we want to derive the BRDF \(f\) in the rendering equation. We also assume the BRDF only depends on the surface normal so we can drop the dependency on position \(x\). Now the rendering equation becomes

\[L_o(o) = \int_\Omega f(i, o, n)L_i(i)(n\cdot i)d\omega_i \]

Here we also compute the total energy (power) over the surface. By definition, it is

\[P = \int_A \int_\Omega L_o(o) d\omega_o dA = \int_A \int_\Omega \int_\Omega f(i, o, n)L_i(i)(n\cdot i) d\omega_i (n\cdot o) d\omega_o dA\]

Clean up \(P\),

\[P = \int_A \int_\Omega \int_\Omega f(i, o, n)L_i(i)(n\cdot i)(n\cdot o) d\omega_i d\omega_o dA\]

From another perspective, we can derive \(P\) from another perspective. We derive \(P^{m}\) for each microfacet and add (integrate) them together.

First for easy understanding, let's assume there is a finite number of microfacet directions, so \(D(m)\) is a "mass function" instead of a "density function". Now \(D(m)\) is the ratio between area of microfacet \(m\) and the total surface area \(A\). Let \(\delta D(m)\) denote the total area of microfacet \(m\). We have

\[\delta A^{(m)} = \int_A D(m)dA\]

Let's check \(D(m)\) satisfies the requirements. Rule 1 is trivial since area ratio is certainly non-negative.

\[\sum_m D(m) (m\cdot n) = \frac{\int_A \sum_m D(m) (m\cdot n) dA}{A} = \frac{\sum_m \delta A(m)}{A} = \frac{A}{A} = 1 \]

so it satisfies the requirements described in previous sections.

For a microfacet with normal \(m\), given a differential input solid angle \(\omega_i\) and the associated input direction \(i\), its contribution to output is \(dL_o^{(m)}\) at direction \(o\) is

\[dL_o^{(m)}(o) = f^{(m)}(i,o,m)L_i(i)(m\cdot i) d\omega_i\]

Next, part of the input and output radiance is blocked (shadowing-masking). We notice scaling \(dL_o(o)\) and scaling \(L_i(i)\) has the same effect, so this effect can be captured by introducing function \(G(i,o,m)\), which is the shadowing-masking function described above.

\[dL_o^{(m)}(o) = G(i,o,m)f^{(m)}(i,o,m)L_i(i)(m\cdot i) d\omega_i\]

Now we take the integral over the unit sphere to get

\[L_o^{(m)}(o) = \int_\Omega G(i,o,m)f^{(m)}(i,o,m)L_i(i)(m\cdot i) d\omega_i\]

Now we have the radiance formula for a microfacet \(m\), we can integrate it to find its light energy (power).

\[P^{(m)} = \int_{\delta A^{(m)}} \int_\Omega \int_\Omega G(i,o,m)f^{(m)}(i,o,m)L_i(i)(m\cdot i) d\omega_i (m\cdot o) d\omega_o dA \]

Finally we sum over all microfacets to get the total power

\[P = \sum_m P^{(m)} = \sum_m \int_{\delta A^{(m)}} \int_\Omega \int_\Omega G(i,o,m)f^{(m)}(i,o,m)L_i(i)(m\cdot i) d\omega_i (m\cdot o) d\omega_o dA \]

Next, we go back and take \(D(m)\) to be a distribution function, which is (by definition of integration) equivalent to taking infinitely many possible directions for the facets. We can repeat the steps above and find

\[P = \sum_m P^{(m)} = \int_\Omega \int_{A} \int_\Omega \int_\Omega G(i,o,m)f^{(m)}(i,o,m)L_i(i)(m\cdot i) d\omega_i (m\cdot o) d\omega_o d A^{(m)} \omega_m \]

where

\[dA^{(m)} = D(m)dA\]

Clean up \(P\) we get

\[P = \int_A\int_\Omega\int_\Omega\int_\Omega D(m)G(i,o,m)f^{(m)}(i,o,m) L_i(i)(m\cdot i)(m\cdot o)d\omega_m d\omega_i d\omega_o dA\]

Notice the \(P\) computed from 2 perspective above share some common integrals, so we can differentiate both w.r.t \(\omega_i\), \(\omega_o\), and \(A\) and set them equal. We get

\[f(i,o,n)(n\cdot i)(n\cdot o) = \int_\Omega D(m)G(i,o,m)f^{(m)}(i,o,m)(m\cdot i)(m\cdot o) d\omega_m\]

We finally find out the surface BRDF

\[f(i,o,n) = \int_\Omega D(m)G(i,o,m) \frac{(n\cdot i)(n\cdot o)}{(m\cdot i)(m\cdot o)} f^{(m)}(i,o,m) d\omega_m\]