Bidirectional reflectance distribution function


The bidirectional reflectance distribution function is a function of four real variables that defines how light is reflected at an opaque surface. It is employed in the optics of real-world light, in computer graphics algorithms, and in computer vision algorithms. The function takes an incoming light direction,, and outgoing direction, , and returns the ratio of reflected radiance exiting along to the irradiance incident on the surface from direction. Each direction is itself parameterized by azimuth angle and zenith angle, therefore the BRDF as a whole is a function of 4 variables. The BRDF has units sr−1, with steradians being a unit of solid angle.

Definition

The BRDF was first defined by Fred Nicodemus around 1965. The definition is:
where is radiance, or power per unit solid-angle-in-the-direction-of-a-ray per unit projected-area-perpendicular-to-the-ray, is irradiance, or power per unit surface area, and is the angle between and the surface normal,. The index indicates incident light, whereas the index 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, which are of no interest for, might illuminate the surface which would unintentionally affect, whereas is only affected by.

Related functions

The Spatially Varying Bidirectional Reflectance Distribution Function is a 6-dimensional function,, where describes a 2D location over an object's surface.
The Bidirectional Texture Function 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, is a further generalized 8-dimensional function 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:. Note that in the typical case where all optical elements are linear, the function will obey except when : that is, it will only emit light at wavelength equal to the incoming light. In this case it can be paramaterized as, with only one wavelength parameter.

Physically based BRDFs

Physically realistic BRDFs have additional properties, including,
The BRDF is a fundamental radiometric concept, and accordingly is used in computer graphics for photorealistic rendering of synthetic scenes, as well as in computer vision for many inverse problems such as object recognition. BRDF has also been used for modeling light trapping in solar cells or low concentration solar photovoltaic systems.
In the context of satellite remote sensing, NASA uses a BRDF model to characterise surface anisotropy. For a given land area, the BRDF is established based on selected multiangular observations of surface reflectance. While single observations depend on view geometry and solar angle, the MODIS BRDF/Albedo product describes intrinsic surface properties in several spectral bands, at a resolution of 500 meters. The BRDF/Albedo product can be used to model surface albedo depending on atmospheric scattering.

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. Matusik et al. found that interpolating between measured samples produced realistic results and was easy to understand.

Some examples

Traditionally, BRDF measurement devices called gonioreflectometers employ one or more goniometric arms to position a light source and a detector at various directions from a flat sample of the material to be measured. To measure a full BRDF, this process must be repeated many times, moving the light source each time to measure a different incidence angle. 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.

BRDF Fabrication

BRDF Fabrication refers to the process of implementing a surface based on the measured or synthesized information of a target BRDF. There exist three ways to perform such a task, but in general, it can be summarized as the following steps:
Many approaches have been proposed for manufacturing the BRDF of the target :