BRDF

**BRDF (bidirectional reflectance distribution Function** The **bidirectional reflection distribution function** , which is used to describe the incident radiation in a given incident direction and the radiation distribution in the direction of reflection, provides a relatively accurate calculation method for BRDF.

, the incident emissivity in the point \ ({p}\) ({d \omega}\) is \ ({de_i (P, \omega_i)}\), and the emission irradiance on the reflection direction \ ({\omega_o}\) is \ ({dl_o (P, \omega_o)}\).

BRDF follows the principle of conservation of energy, and the incident emissivity and emission emissivity should be positively proportional, and the ({Dl_o (P, \omega_o)}\) increases with the increase of ({de_i (P, \omega_i)}\). Can be expressed as:

\ ({dl_o (P, \omega_o)} \propto {de_i (P, \omega_i)}\)

If the **BRDF scale factor** is represented by \ ({f_r (P, \omega_i, \omega_o)}\):

\ ({dl_o (P, \omega_o)} = {F_r (p, \omega_i, \omega_o)}{de_i (p, \omega_i)}\)

In the previous article, we know the incident emissivity \ ({de_i (P, \omega_i)}={l_i (p, \omega_i) \, \cos \theta_i \, D-\omega_i}\), substituting:

\ ({dl_o (P, \omega_o)} = {F_r (p, \omega_i, \omega_o)}{l_i (p, \omega_i) \, \cos \theta_i \, D \omega_i}\)

So BRDF's scale factor \ ({f_r (P, \omega_i, \omega_o)}=\frac{dlo (p,\omega_o)}{l_i (p,\omega_i) \, \cos \theta_i \, d\omega_i)}\)

Reflectance equation of radiation

By the above formula, it is known that the **equation of reflection emissivity** on the stereoscopic angle \ ({\omega_i}\) is:

\ ({l_o (p,\omega_o)}=\int_{\omega_i}{f_r (p, \omega_i, \omega_o)}\, {l_i (P, \omega_i)}\, {\cos \theta_i}\, {d\omega_i}\ )

Characteristics of BRDFs

- Meet the exchange rate: if the interchange \ (\omega_i\) and \ (\omega_o\), the final BRDF value remains unchanged. That is, if you change the direction of light propagation, the radiation remains the same.
- Satisfies the linear characteristic: the total reflection radiation of a point on the surface of the object is equal to the sum of the BRDF reflection radiation.
- follows the conservation of energy: there is no material in reality that can completely reflect the incident light on the surface of an object, and some of the energy is absorbed by the surface of the object and reflected again in other forms. Therefore, the anisotropic reflection emissivity on the surface surface of the object ({da}\) is less than the total absorbed energy.

PBR Step by Step (iii) BRDFS