# Understanding Incident/Exitant Radiance

Reading "Physically Based Rendering", I'm trying to understand what the meaning of the incident and exitant radiance functions.

I understand that radiance $L(p,\omega) =\frac{d^2\phi}{d\omega dA^{\perp}}$ where $\phi$ is the Flux, $\omega$ is the direction of the light coming towards the surface and $A^{\perp}$ is the surface perpendicular to $\omega$ . So, what I'm effectively trying to measure is the "brightness" of the light at this direction $\omega$ . This is where the incident and exitant radiance come in: $L_{i}(p,w)$ is described as the radiance arriving at the point p and $L_{o}(p,w)$ as the outgoing reflected radiance from the surface.

I don't understand this concept at all. Isn't $L_{i}(p,w)$ what $L(p,w)$ is in the first place? Is it the case that $L(p,w) = L_{i}(p,w) + L_{o}(p,w)$ since the "brightness" of a ray can be described as the radiance from all the lightsources in that direction + the radiance from emitted from the surface in that direction as well? Can someone please explain this concept more intuitively, as I'm trying to understand it for Computer Graphics?

## 1 Answer

Since $L_i(p,\omega)$ and $L_o(p,\omega)$ are specific kinds of radiance, it is meaningless to compare them to $L(p,\omega)$.

In a vacuum, provided that the $\omega$ vectors point outward from the surface, it is the case that $$L_i(p \leftarrow \omega) = L_o(p \rightarrow -\omega).$$

For more details, read Section 2.2.3 of Wojciech Jarosz's thesis.

Moreover, the reflection equation holds: $$L_o(p,\omega_o) = L_e(p,\omega_o) + \int_{H^2} f_r(p, \omega_i \to \omega_o) L_i(p,\omega_i) \cos(\theta_i) d \omega_i,$$ where $H^2$ is the hemisphere, $f_r$ is the BRDF, and $L_e$ is the emitted radiance. This relation relates the outgoing radiance at a point to the incoming radiance, BRDF of the surface, and emitted radiance.