# Bidirectional reflectance distribution function

Diagram showing vectors used to define the BRDF. All vectors are unit length. $\omega_{\text{i}}$ points toward the light source. $\omega_{\text{r}}$ points toward the viewer (camera). $n$ is the surface normal.

The bidirectional reflectance distribution function (BRDF; $f_{\text{r}}(\omega_{\text{i}},\, \omega_{\text{r}})$ ) is a four-dimensional function that defines how light is reflected at an opaque surface. The function takes a negative incoming light direction, $\omega_{\text{i}}$, and outgoing direction, $\omega_{\text{r}}$, both defined with respect to the surface normal $\mathbf n$[clarification needed], and returns the ratio of reflected radiance exiting along $\omega_{\text{r}}$ to the irradiance incident on the surface from direction $\omega_{\text{i}}$. Each direction $\omega$ is itself parameterized by azimuth angle $\phi$ and zenith angle $\theta$, 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 Fred Nicodemus around 1965.[1] The definition is:

$f_{\text{r}}(\omega_{\text{i}},\, \omega_{\text{r}}) \,=\, \frac{\operatorname dL_{\text{r}}(\omega_{\text{r}})}{\operatorname dE_{\text{i}}(\omega_{\text{i}})} \,=\, \frac{\operatorname dL_{\text{r}}(\omega_{\text{r}})}{L_{\text{i}}(\omega_{\text{i}})\cos\theta_{\text{i}}\,\operatorname d\omega_{\text{i}}}$

where $L$ is radiance, or power per unit solid-angle-in-the-direction-of-a-ray per unit projected-area-perpendicular-to-the-ray, $E$ is irradiance, or power per unit surface area, and $\theta_{\text{i}}$ is the angle between $\omega_{\text{i}}$ and the surface normal, $\mathbf n$. The index $\text{i}$ indicates incident light, whereas the index $\text{r}$ indicates reflected light.

The reason the function is defined as a quotient of two differentials and not directly as a quotient between the undifferentiated quantities, is because other irradiating light than $\operatorname dE_{\text{i}}(\omega_{\text{i}})$, which are of no interest for $f_{\text{r}}(\omega_{\text{i}},\, \omega_{\text{r}})$, might illuminate the surface which would unintentionally affect $L_{\text{r}}(\omega_{\text{r}})$, whereas $\operatorname dL_{\text{r}}(\omega_{\text{r}})$ is only affected by $\operatorname dE_{\text{i}}(\omega_{\text{i}})$.

## Related functions

The Spatially Varying Bidirectional Reflectance Distribution Function (SVBRDF) is a 6-dimensional function, $f_{\text{r}}(\omega_{\text{i}},\,\omega_{\text{r}},\,\mathbf{x})$, where $\mathbf{x}$ describes a 2D location over an object's surface.

The Bidirectional Texture Function (BTF) is appropriate for modeling non-flat surfaces, and has the same parameterization as the SVBRDF; however in contrast, the BTF includes non-local scattering effects like shadowing, masking, interreflections or subsurface scattering. The functions defined by the BTF at each point on the surface are thus called Apparent BRDFs.

The Bidirectional Surface Scattering Reflectance Distribution Function (BSSRDF), is a further generalized 8-dimensional function $S(\mathbf{x}_{\text{i}},\,\omega_{\text{i}},\,\mathbf{x}_{\text{r}},\,\omega_{\text{r}})$ in which light entering the surface may scatter internally and exit at another location.

In all these cases, the dependence on the wavelength of light has been ignored and binned into RGB channels. In reality, the BRDF is wavelength dependent, and to account for effects such as iridescence or luminescence the dependence on wavelength must be made explicit: $f_{\text{r}}(\lambda_{\text{i}},\,\omega_{\text{i}},\,\lambda_{\text{r}},\,\omega_{\text{r}})$.

## Physically based BRDFs

Physically based BRDFs have additional properties,[2] including,

• positivity: $f_{\text{r}}(\omega_{\text{i}},\, \omega_{\text{r}}) \ge 0$
• obeying Helmholtz reciprocity: $f_{\text{r}}(\omega_{\text{i}},\, \omega_{\text{r}}) = f_{\text{r}}(\omega_{\text{r}},\, \omega_{\text{i}})$
• conserving energy: $\forall \omega_{\text{i}},\, \int_\Omega f_{\text{r}}(\omega_{\text{i}},\, \omega_{\text{r}})\,\cos{\theta_{\text{r}}} d\omega_{\text{r}} \le 1$

## 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 lightsources;[3] 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:

W. Matusiak et al. found that interpolating between measured samples produced realistic results and was easy to understand.[4]

### Some examples

• Lambertian model, representing perfectly diffuse (matte) surfaces by a constant BRDF.
• Lommel–Seeliger, lunar and Martian reflection.
• Phong reflectance model, a phenomenological model akin to plastic-like specularity.[5]
• Blinn–Phong model, resembling Phong, but allowing for certain quantities to be interpolated, reducing computational overhead.[6]
• Torrance–Sparrow model, a general model representing surfaces as distributions of perfectly specular microfacets.[7]
• Cook–Torrance model, a specular-microfacet model (Torrance–Sparrow) accounting for wavelength and thus color shifting.[8]
• Ward model, a specular-microfacet model with an elliptical-Gaussian distribution function dependent on surface tangent orientation (in addition to surface normal).[9]
• Oren–Nayar model, a "directed-diffuse" microfacet model, with perfectly diffuse (rather than specular) microfacets.[10]
• Ashikhmin-Shirley model, allowing for anisotropic reflectance, along with a diffuse substrate under a specular surface.[11]
• HTSG (He,Torrance,Sillion,Greenberg), a comprehensive physically based model.[12]
• Fitted Lafortune model, a generalization of Phong with multiple specular lobes, and intended for parametric fits of measured data.[13]
• Lebedev model for analytical-grid BRDF approximation.[14]

## 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. 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.

There is an alternative way to measure BRDF based on HDR images. The standard algorithm is to measure the BRDF point cloud from images and optimize it by one of the BRDF models.[15]

## References

1. ^ Nicodemus, Fred (1965). "Directional reflectance and emissivity of an opaque surface" (abstract). Applied Optics 4 (7): 767–775. Bibcode:1965ApOpt...4..767N. doi:10.1364/AO.4.000767.
2. ^ Duvenhage, Bernardt (2013). "Numerical verification of bidirectional reflectance distribution functions for physical plausibility". Proceedings of the South African Institute for Computer Scientists and Information Technologists Conference. pp. 200–208.
3. ^ Rusinkiewicz, S. "A Survey of BRDF Representation for Computer Graphics". Retrieved 2007-09-05.
4. ^ Wojciech Matusik, Hanspeter Pfister, Matt Brand, and Leonard McMillan. A Data-Driven Reflectance Model. ACM Transactions on Graphics. 22(3) 2002.
5. ^ B. T. Phong, Illumination for computer generated pictures, Communications of ACM 18 (1975), no. 6, 311–317.
6. ^ 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.
7. ^ K. Torrance and E. Sparrow. Theory for Off-Specular Reflection from Roughened Surfaces. J. Optical Soc. America, vol. 57. 1967. pp. 1105–1114.
8. ^ 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.
9. ^ Ward, Gregory J. (1992). "Measuring and modeling anisotropic reflection". Proceedings of SIGGRAPH. pp. 265–272. doi:10.1145/133994.134078.
10. ^ 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
11. ^ Michael Ashikhmin, Peter Shirley, An Anisotropic Phong BRDF Model, Journal of Graphics Tools 2000
12. ^ 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.
13. ^ 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.
14. ^ Ilyin A., Lebedev A., Sinyavsky V., Ignatenko, A., Image-based modelling of material reflective properties of flat objects (In Russian). In: GraphiCon'2009.; 2009. p. 198-201.
15. ^ BRDFRecon project