Photon Maps

Each photon is a small data structure that stores

• power (in r,g,b components with at most 1.0 for each component)
• arrival direction (from which the photon arrives at a surface)
• position (on the surface where the photon lands)

Send photons in random directions from the lights. Use standard raytracing to follow photons through the scene. Upon a photon/surface hit, do only one of

• $s$: reflect specularly
• $d$: reflect diffusely in a random direction
• $t$: refract

The choice of event, from the three events above, is done probabilistically, with $s, d, t$ being the probability of each event. These probabilities could be $$\begin{array}{l} \displaystyle s = \alpha \; { \max( k_s ) \over \max( k_s ) + \max( k_d ) }\\ \displaystyle d = \alpha \; { \max( k_d ) \over \max( k_s ) + \max( k_d ) }\\ \displaystyle t = 1 - \alpha \\ \end{array}$$

where $k_s, k_d, \alpha$ are the material properties at the ray/surface intersection point and "max" is the maximum of the rgb components.

You should, as always, ensure that $\max(k_s) + \max(k_d) \le 1$ in your material properties.

A photon is stored on a surface if

• the photon has made at least one refraction or specular reflection
• and subsequently arrives from above
• at a diffuse surface (i.e. a surface with some component of $k_d$ being non-zero).

Once the photon is stored on a surface, raytracing of that photon stops. Raytracing also stops if the photon has made more than some predetermined number of bounces.

Build a kd-tree to hold the photons that have been collected on each surface. A separate 2D kd-tree can used on each surface, or a single 3D kd-tree can be used for the whole scene.

A kd-tree node represents an axis-aligned box. On a surface, for example, this is a 2D box:

The box is split such that half of its photons go into each child. Choose a median photon to define the split position:

Recurse so that each leaf holds no more than some maximum number of photons. Each level of the hierarchy alternates the split axis from $x$, then to $y$, then to $x$ again, then to $y$ again, and so on:

The kd-tree data structure must also provide a function that returns all photons within a distance, $r$, of a position, $x$.

Perform normal raytracing from the eye. At each bounce, query the kd-tree for photons near the bounce point (i.e. the ray/surface intersection point). Consider each of these photons as coming from an additional light source and incorporate them into the Phong model.

Below, $x$ is the bounce point, $C$ is a circle on the surface of radius $r$ around $x$, and each photon (with an arrow for its arrival direction) is shown in red. A query would return the three photons inside the circle.

The rendering equation states that $$L(x,\omega_o) = \int_\Omega f(\omega_i, \omega_o) \; L(x,\omega_i) \; \cos\theta_i \; d\omega_i$$

where:

• $x$ is the point at which we are calculating the outgoing light;
• $\omega_o$ is the direction of the outgoing light;
• $\omega_i$ is a direction from which light arrives;
• $L(x,\omega_o)$ is the radiance leaving the point $x$ in direction $\omega_o$;
• $L(x,\omega_i)$ is the radiance arriving at $x$ from direction $\omega_i$;
• $\Omega$ is the hemisphere of directions above the surface at $x$ from which light arrives;
• $f(\omega_i,\omega_o)$ is the fraction of light reflected from $\omega_i$ to $\omega_o$, per solid angle, and is often modelled with Phong using $\omega_i$ as the incoming direction and $\omega_o$ as the eye direction; and
• $\cos\theta_i$ is the attenuation of incoming light as it is spread out over the surface at $x$.

Discretizing and approximating results in $$\begin{array}{rl} L_\textrm{out}(x,\omega_o) & = \displaystyle \int_\Omega f(\omega_i, \omega_o) \; L(\omega_i) \; \cos\theta_i \; d\omega_i \\ & \displaystyle \approx \sum_{p_i \in C} f(\omega_i,\omega_0) \; L(p_i) \; \cos\theta_i \; \Delta\omega_i \\ & = \displaystyle \sum_{p_i \in C} f(\omega_i,\omega_0) \; { P_i \over \Delta\omega_i \; \cos\theta_i \; \Delta A} \; \cos\theta_i \; \Delta\omega_i \\ \end{array}$$

where:

• $C$ is the circle of radius $r$ around $x$;
• $p_i$ is one photon in $C$;
• $\Delta A = \pi \: r^2$ is the area around $x$; and
• $P_i$ is the power of photon $p_i$.

Conclusion