Bidirectional reflection distribution function

A bidirectional reflectance distribution function ( English Bidirectional Reflectance Distribution Function , BRDF ) provides a function for the reflection behavior is of surfaces of a material under arbitrary angles of incidence. It provides for each of the material impinging light beam having a given entry angle the ratio of irradiance and irradiance for each emergent light beam. BRDFs are used, among other things, in realistic 3D computer graphics , where they represent part of the fundamental rendering equation and serve to represent surfaces as realistically and physically correct as possible. A generalization of the BRDF on textures , the BTF ( Bidirectional Texturing Function ) is.

View of the geometric elements used when defining the BRDF

A BRDF is very complex, depending on the selected accuracy: It can be determined, for example, for all angles of incidence / reflection (4-dimensional), for every wavelength (5-dimensional) or for every point on the surface (7-dimensional). Because of this complexity, simple models are mostly used as BRDFs in computer graphics.

Formula: ${\displaystyle f_{\text{r}}(\omega _{\text{i}},\omega _{\text{o}})\equiv {\frac {dL_{\text{o}}(\omega _{\text{o}})}{L_{\text{i}}(\omega _{\text{i}})\cos(\theta _{\text{i}})d\omega _{\text{i}}}}}$

There are basically 2 approaches to represent the BRDF values:

• explicit storage of the ordered measured values ​​or simulation results ( e.g. obtained by a gonioreflectometer ),
  • high storage space requirements with simple storage in finely divided samples,
  • high expenditure of time and hardware for data acquisition,
  • poorly suited for importance sampling ,
  • important for verification tasks,
• Approximation through analytical functions ( local lighting models ).

Physically correct BRDF

The BRDF of a real object must also have the following properties:

• Positivity: ${\displaystyle f_{\text{r}}(\omega _{\text{i}},\,\omega _{\text{o}})\geq 0}$
• fulfills the Helmholtz reciprocity: ${\displaystyle f_{\text{r}}(\omega _{\text{i}},\,\omega _{\text{o}})=f_{\text{r}}(\omega _{\text{o}},\,\omega _{\text{i}})}$
• Energy conservation: ${\displaystyle \forall \omega _{\text{i}}:\,\int _{\Omega }f_{\text{r}}(\omega _{\text{i}},\,\omega _{\text{o}})\,\cos {\theta _{\text{o}}}d\omega _{\text{o}}\leq 1}$

See also

• BTDF , BSSRDF , SSS

Web links

• FE Nicodemus, JC Richmond, JJ Hsia, IW Ginsberg, T. Limperis: Geometrical Considerations and Nomenclature for Reflectance. US Dept. of Commerce (1977) - original definition. (PDF; 5.33 MB)
• N. Gebhardt: Some BRDF models. (PDF; 1.27 MB)