Global Illumination using Photon Mapping
Photon tracing
The purpose of the photon tracing pass is to compute indirect illumination on diffuse surfaces. This is done by emitting photons from the light sources, tracing them through the scene, and storing them at diffuse surfaces
Photon emission
The power ("wattage") of the light source has to be distributed among the photons emitted from it. If the power of the light is ${P_{light}}$ and the number of emitted photons is ${\eta_{e}}$, the power of each emitted photon is
${\LARGE{ P_{photon} = \frac{P_{\textit{light}}}{\eta_{e}} }}$ (1)
Mutiple lights
If the scene contains multiple light sources, photons should be emitted from each light source. More photons should be emitted from brighter ligths than from dim lights
Projection maps
In scenes with sparse geometry, many emitted photons will not hit any objects. Emitting these photons is a waste of time. To optimize the emission, projection maps can be used. A projection map is simply a map of the geometry as seen from the light source.
${\LARGE{ P_{photon} = \frac{P_{\textit{light}}}{\eta_{e}} \frac{cells_{objects}}{cells_{total}} }}$ (2)
Photon tracing
When a photon hits an object, it can either be reflected, transmitted, or absorbed, based on the material parameters of the surface. To decide whether to reflect, transmit or absorbe, Russian roulette is used.
Reflection, transmission, or absorption?
for a monochromatic simulation, given a reflective surface with diffuse flection coefficient ${d}$ and specular reflection coefficient ${s}$ with (with ${d + s \leq 1}$), using a random variable ${ \xi \in [0, 1] }$
${\large{ \xi \in [0, d]} \longrightarrow}$ diffuse reflection
${\large{ \xi \in [d, s + d]} \longrightarrow}$ specular reflection
${\large{ \xi \in [s + d, 1]} \longrightarrow}$ absorption
With more color bands (for example RGB colors), The probabilities for specular and diffuse reflection can be based on the total energy reflected by each type of reflection or on the maximumum energy reflected in any color band. using the maximum energy, probability for reflection is${\large{ P_r = max(d_r + s_r, d_g + s_g, d_b + s_b)}}$
where ${d_r}$, ${d_g}$, and ${d_b}$ are the diffuse reflection coefficients in the red, green, and blue color bands, and ${s_r}$ ${s_g}$, and ${s_b}$ are the specular reflection coefficients in the red, green, and blue color bands. (The probability of absorption is ${P_a = 1 = P_r}$) With this the probability ${P_d} of diffuse reflection is:${\LARGE{P_d = \frac{ d_r + d_g + d_b }{ d_r + d_g + d_b + s_r + s_g + s_b} P_r }}$
Similarly, for the probability ${P_s}$ of specular reflection:${\LARGE{P_s = \frac{ s_r + s_g + s_b }{ d_r + d_g + d_b + s_r + s_g + s_b} P_r = P_r - P_d }}$
With these probabilities, the decision of which type of reflection or absorption should be chosen takes the following form${\large{ \xi \in [0, P_d]} \longrightarrow}$ diffuse reflection
${\large{ \xi \in [P_d, P_s + P_d]} \longrightarrow}$ specular reflection
${\large{ \xi \in [P_s + P_d, 1]} \longrightarrow}$ absorption
The power of the reflected photon needs to be adjusted to account for the probability of survival. specualer reflection example:${\large{ P_{refl,r} = P_{inc,r} \frac{s_r}{P_s} }}$
${\large{ P_{refl,g} = P_{inc,g} \frac{s_g}{P_s} }}$
${\large{ P_{refl,b} = P_{inc,b} \frac{s_b}{P_s} }}$
where ${P_{inc}}$ is the power of the incident photonRadiance estimate
Radiance estimate at a surface
The photon map can be seen as a representation of the incoming flux; to compute radiance we need to integrate this information. This can be done using the expression for reflected radiance:
${\large{ \displaystyle L_{r}(x, \vec{\omega}) = \int_{\Omega_{x}} f_r(x, \vec{\omega}', \vec{\omega}) L_{i}(x, \vec{\omega}') | \vec{n_x} \cdot \vec{\omega}' |d\omega_i' }}$ (3)
where ${L_r}$ is the reflected radiance at ${x}$ in the direction ${\vec{\omega}}$. ${\Omega_x}$ is the (hemi)sphere of the incoming direction, ${f_r}$ is the BRDF at ${x}$ and ${L_i}$ is the incoming radiance. Since the photon map provides information about the incoming flux we need to rewrite this term. This can be done using the relationship between radiance and flux:${\large{ \displaystyle L_i(x, \vec{\omega}') = \frac{d^2 \Phi_i(x, \vec{\omega}')}{ \cos \theta_i d\omega_i' dA_i } } }$ (4)
and we can rewrite the integral as${\large{ \displaystyle L_{r}(x, \vec{\omega}) = \int_{\Omega_{x}} f_r(x, \vec{\omega}', \vec{\omega}) \frac{d^2 \Phi_i(x, \vec{\omega}')}{ \cos \theta_i d\omega_i' dA_i } | \vec{n_x} \cdot \vec{\omega}' |d\omega_i' }}$ (5)
The incoming flux ${\Phi_i}$ is approximated using the photon map by locating the ${n}$ photons that have the shortest distance to x. Each photon ${p}$ has the power ${\displaystyle \Delta \Phi_p(\vec{\omega_p})}$ and by assuming that the photons intersects the surface at x we obtain${\large \displaystyle L_r(x, \vec{\omega}) \approx \Sigma_{p=1}^{n} f_r(x, \vec{\omega_p}, \vec{\omega}) \frac{\Delta \Phi_p(x, \vec{\omega_p})}{\Delta A} }$ (6)
By assuming that the surface is locally flat around x then${\large \displaystyle \Delta A = \pi r^2}$ (7)
where ${r}$ is the radius of the sphere. ie. the largest distance between ${x}$ and each of the photons This results in the following equation for computing the reflected radiance at a surface using the photon map${\large \displaystyle L_r(x, \vec{\omega}) \approx \frac{1}{\pi r^2} \Sigma_{p=1}^{n} f_r(x, \vec{\omega_p}, \vec{\omega}) \Delta \Phi_p(x, \vec{\omega_p}) }$ (7)