# Bidirectional reflectance distribution function

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File:BRDF Diagram.svg
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 function of four real variables that defines how light is reflected at an opaque surface. It is employed both in the optics of real-world light, in computer graphics algorithms, and in computer vision algorithms. The function takes an incoming light direction, $\omega_{\text{i}}$, and outgoing direction, $\omega_{\text{r}}$ (taken in a coordinate system where the surface normal $\mathbf n$ lies along the z-axis), 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 a function of 4 variables. 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. 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 realistic BRDFs have additional properties, 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: $\forall \omega_{\text{r}},\, \int_\Omega f_{\text{r}}(\omega_{\text{i}},\, \omega_{\text{r}})\,\cos{\theta_{\text{i}}} d\omega_{\text{i}} \le 1$

## 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. BRDF has also being used for modeling low concentration solar photovoltaic systems.

## Models

BRDFs can be measured directly from real objects using calibrated cameras and lightsources; 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.

### 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.
• Blinn–Phong model, resembling Phong, but allowing for certain quantities to be interpolated, reducing computational overhead.
• Torrance–Sparrow model, a general model representing surfaces as distributions of perfectly specular microfacets.
• Cook–Torrance model, a specular-microfacet model (Torrance–Sparrow) accounting for wavelength and thus color shifting.
• Ward model, a specular-microfacet model with an elliptical-Gaussian distribution function dependent on surface tangent orientation (in addition to surface normal).
• Oren–Nayar model, a "directed-diffuse" microfacet model, with perfectly diffuse (rather than specular) microfacets.
• Ashikhmin-Shirley model, allowing for anisotropic reflectance, along with a diffuse substrate under a specular surface.
• HTSG (He,Torrance,Sillion,Greenberg), a comprehensive physically based model.
• Fitted Lafortune model, a generalization of Phong with multiple specular lobes, and intended for parametric fits of measured data.
• Lebedev model for analytical-grid BRDF approximation.

## Acquisition

Traditionally, BRDF measurements were taken for one 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.