Rendering Equation Derivation

Introduction

This derivation shows how the rendering equation emerges naturally from the radiometric definitions we established in the lecture series.

Starting from Radiance

Recall from the lecture series that radiance eq.radiance is defined as:

L(\mathbf{x}, \omega) = \frac{d^2\Phi}{dA \, d\omega \, \cos\theta}

eq.1.1

This represents the power per unit area per unit solid angle in a given direction.

The Integral Formulation

By integrating over all incoming directions at a surface point, we arrive at the rendering equation:

L_o(\mathbf{x}, \omega_o) = L_e(\mathbf{x}, \omega_o) + \int_{\Omega} f_r(\mathbf{x}, \omega_i, \omega_o) L_i(\mathbf{x}, \omega_i) \cos\theta_i \, d\omega_i

eq.1.2

Where:

$L_o$ is the outgoing radiance
$L_e$ is the emitted radiance
$f_r$ is the BRDF (bidirectional reflectance distribution function)
$L_i$ is the incoming radiance
$\Omega$ is the hemisphere of incoming directions

Key Properties

Table 1.1: Key properties of the rendering equation

Property	Description
Linearity	The equation is linear in radiance
Recursion	Incoming radiance depends on other outgoing radiance
Energy Conservation	$\int_{\Omega} f_r(\omega_i, \omega_o) \cos\theta_i \, d\omega_i \leq 1$

As shown in tab.properties, the rendering equation exhibits several important mathematical properties that inform our solution strategies.

Conclusion

This derivation establishes the theoretical foundation for all physically based rendering algorithms. See fig.radiometry-diagram for the geometric interpretation.