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

The bidirectional reflectance distribution function (BRDF; ) is a 4-dimensional function that defines how light is reflected at an opaque surface. The function takes an incoming light direction, , and outgoing direction, , both defined with respect to the surface normal , and returns the ratio of reflected radiance exiting along to the irradiance incident on the surface from direction . Note that each direction 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.

Definition

The BRDF was first defined by Edward Nicodemus in the mid-sixties¹. The modern definition is:

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

Physically based BRDFs

Physically based BRDFs have additional properties, including,

• obeying Helmholtz reciprocity: .
• conserving energy:

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.

Models

BRDFs can be measured directly from real objects using calibrated cameras and lightsources²; 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:

• accommodating anisotropic reflection
• editable using a small number of intuitive parameters
• accounting for Fresnel effects at grazing angles
• being well-suited to Monte Carlo methods.

Some examples

• Lambertian model, representing perfectly diffuse (matte) surfaces by a constant BRDF.
• Phong reflectance model, a phenomenological model akin to plastic-like specularity.³
• Blinn-Phong model, resembling Phong, but allowing for certain quantities to be interpolated, reducing computational overhead.⁴
• Torrance-Sparrow model, a general model representing surfaces as distributions of perfectly-specular microfacets.⁵
• Cook-Torrance model, a specular-microfacet model (Torrance-Sparrow) accounting for wavelength and thus color shifting.⁶
• Ward's anisotropic model, a specular-microfacet model with a elliptical-Gaussian distribution function dependent on surface tangent orientation (in addition to surface normal).⁷
• Oren–Nayar model, a "directed-diffuse" microfacet model, with perfectly-diffuse (rather than specular) microfacets.⁸
• Ashikhmin-Shirley model, allowing for anisotropic reflectance, along with a diffuse substrate under a specular surface.⁹
• HTSG (He,Torrance,Sillion,Greenberg), a comprehensive physically-based model.¹⁰
• Fitted Lafortune model, a generalization of Phong with multiple specular lobes, and intended for parametric fits of measured data.¹¹

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 once⁷. 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.

See also

Computer graphics portal

• BSDF
• Radiometry
• Photometry (astronomy)
• Reflectance
• Albedo
• Opposition spike
• Gonioreflectometer
• Specular highlight

Further reading

• Lubin, Dan; Robert Massom (2006-02-10). Polar Remote Sensing: Volume I: Atmosphere and Oceans (1 ed.), Springer. pp. 756. ISBN 3540430970. 
• Matt, Pharr; Greg Humphreys (2004). Physically Based Rendering (1 ed.), Morgan Kauffmann. pp. 1019. ISBN 012553180X. 
• Schaepman-Strub, G.; M.E. Schaepman, T.H. Painter, S. Dangel, J.V. Martonchik (2006-07-15). "Reflectance quantities in optical remote sensing--definitions and case studies". Remote Sensing of Environment 103 (1): 27–42. doi:10.1016/j.rse.2006.03.002, http://www.sciencedirect.com/science/article/B6V6V-4K427VX-1/2/d8f9855bc59ae8233e2ee9b111252701. Retrieved on 18 October 2007. 

References

1. ^ Nicodemus, Fred. "Directional reflectance and emissivity of an opaque surface" (abstract). Applied Optics 4 (7): 767–775, http://ao.osa.org/abstract.cfm?id=13818. 
2. ^ Rusinkiewicz, S.. "A Survey of BRDF Representation for Computer Graphics". Retrieved on 2007-09-05.
3. ^ B. T. Phong, Illumination for computer generated pictures, Communications of ACM 18 (1975), no. 6, 311–317.
4. ^ James F. Blinn (1977). "Models of light reflection for computer synthesized pictures". Proc. 4th annual conference on computer graphics and interactive techniques: 192. doi:10.1145/563858.563893, http://portal.acm.org/citation.cfm?doid=563858.563893. 
5. ^ K. Torrance and E. Sparrow. Theory for Off-Specular Reflection from Roughened Surfaces. J. Optical Soc. America, vol. 57. 1976. pp. 1105–1114.
6. ^ R. Cook and K. Torrance. "A reflectance model for computer graphics". Computer Graphics (SIGGRAPH '81 Proceedings), Vol. 15, No. 3, July 1981, pp. 301–316.
7. ^ ^{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. 
8. ^ S.K. Nayar and M. Oren, "Generalization of the Lambertian Model and Implications for Machine Vision". International Journal on Computer Vision, Vol. 14, No. 3, pp. 227–251, Apr, 1995
9. ^ Michael Ashikhmin, Peter Shirley, An Anisotropic Phong BRDF Model, Journal of Graphics Tools 2000
10. ^ X. He, K. Torrance, F. Sillon, and D. Greenberg, A comprehensive physical model for light reflection, Computer Graphics 25 (1991), no. Annual Conference Series, 175–186.
11. ^ E. Lafortune, S. Foo, K. Torrance, and D. Greenberg, Non-linear approximation of reflectance functions. In Turner Whitted, editor, SIGGRAPH 97 Conference Proceedings, Annual Conference Series, pp. 117–126. ACM SIGGRAPH, Addison Wesley, August 1997.

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