Bidirectional reflectance distribution function

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Diagram showing vectors used to define the BRDF. All vectors are unit length. ωi points toward the light source. ωo points toward the viewer (camera). n is the surface normal.
Diagram showing vectors used to define the BRDF. All vectors are unit length. ωi points toward the light source. ωo points toward the viewer (camera). n is the surface normal.

The bidirectional reflectance distribution function (BRDF; {f_r(\omega_i , \omega_o)\ }) is a 4-dimensional function that defines how light is reflected at an opaque surface. The function takes an incoming light direction, \omega_i\ , and outgoing direction, \omega_o\ , both defined with respect to the surface normal n\ , and returns the ratio of reflected radiance exiting along \omega_o\ to the irradiance incident on the surface from direction \omega_i\ . Note that each direction \omega\ is itself parameterized by azimuth angle θ and elevation φ, therefore the BRDF as a whole is 4-dimensional. The BRDF has units sr-1, with steradians (sr) being a unit of solid angle.

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[edit] Definition

The BRDF was first defined by Edward Nicodemus in the mid-sixties[1]. The modern definition is:

f_r(\omega_i, \omega_o)=\frac{dL_r(\omega_o)}{dE_i(\omega_i)}=\frac{dL_r(\omega_o)}{L_i(\omega_i)\cos(\theta_i)d\omega_i}


where L is the radiance, E is the irradiance, and θi is the angle made between ωi and the surface normal, n.

[edit] Physically based BRDFs

Physically based BRDFs have additional properties, including,

  • obeying Helmholtz reciprocity: f_r(\omega_i , \omega_o) = f_r(\omega_o , \omega_i)\ .
  • conserving energy: \forall \omega_i, \int_\Omega f_r(\omega_i, \omega_o)\,d\omega_o \le 1

[edit] Applications

The BRDF is a fundamental radiometric concept, and accordingly is used in computer graphics for photorealistic rendering of synthetic scenes (see the Rendering equation), as well as in computer vision for many inverse problems such as object recognition.

[edit] Models

BRDFs can be measured directly from real objects using calibrated cameras and lightsources[2]; however, many phenomenological and analytic models have been proposed including the Lambertian reflectance model frequently assumed in computer graphics. Some useful features of recent models include:

[edit] Some examples

[edit] Acquisition

Traditionally, BRDF measurements were taken for a specific lighting and viewing direction at a time using gonioreflectometers. Unfortunately, using such a device to densely measure the BRDF is very time consuming. One of the first improvements on these techniques used a half-silvered mirror and a digital camera to take many BRDF samples of a planar target at once[3]. Since this work, many researchers have developed other devices for efficiently acquiring BRDFs from real world samples, and it remains an active area of research.

[edit] See also

[edit] Further Reading

[edit] References

  1. ^ Nicodemus, Fred. "Directional reflectance and emissivity of an opaque surface" (abstract). Applied Optics 4 (7): 767-775. 
  2. ^ Rusinkiewicz, S.. A Survey of BRDF Representation for Computer Graphics. Retrieved on 2007-09-05.
  3. ^ a b Ward, Gregory J. (1992). "Measuring and modeling anisotropic reflection". Proceedings of SIGGRAPH: 265–272. doi:10.1145/133994.134078. Retrieved on 2008-02-03. 
  4. ^ K. Torrance and E. Sparrow. Theory for Off-Specular Reflection from Roughened Surfaces. J. Optical Soc. America, vol. 57. 1976. pp. 1105-1114.