Bidirectional reflectance distribution function

The bidirectional reflectance distribution function (BRDF;

) is a four-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

Radiance

Radiance and spectral radiance are radiometric measures that describe the amount of radiation such as light or radiant heat that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. They are used to characterize both emission from...

exiting along

to the irradiance

Irradiance

Irradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...

incident on the surface from direction

. Note that each direction

is itself parameterized by azimuth angle

and zenith angle

, therefore the BRDF as a whole is 4-dimensional. The BRDF has units sr⁻¹, with steradian

Steradian

The steradian is the SI unit of solid angle. It is used to describe two-dimensional angular spans in three-dimensional space, analogous to the way in which the radian describes angles in a plane...

s (sr) being a unit of solid angle

Solid angle

The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...

Definition

The BRDF was first defined by Fred Nicodemus around 1965. The modern definition is:

where

is the radiance

Radiance

is the irradiance

Irradiance

, and

is the angle made between

and the surface normal

Surface normal

A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

Related functions

The Spatially Varying Bidirectional Reflectance Distribution Function (SVBRDF) is a 6-dimensional function,

, where

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

Bidirectional scattering distribution function

The definition of the BSDF is not well standardized. The term was probably introduced in 1991 by Paul Heckbert. Most often it is used to name the general mathematical function which describes the way in which the light is scattered by a surface...

), 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 wavelength has been ignored and binned into RGB channels. In reality, the BRDF is wavelength dependent, and to account for effects such as iridescence

Iridescence

Iridescence is generally known as the property of certain surfaces which appear to change color as the angle of view or the angle of illumination changes...

or luminescence

Luminescence

Luminescence is emission of light by a substance not resulting from heat; it is thus a form of cold body radiation. It can be caused by chemical reactions, electrical energy, subatomic motions, or stress on a crystal. This distinguishes luminescence from incandescence, which is light emitted by a...

the dependence on wavelength must be made explicit:

Physically based BRDFs

Physically based BRDFs have additional properties, including,

positivity:
obeying Helmholtz reciprocity
Helmholtz reciprocity
The Helmholtz reciprocity principle describes how a ray of light and its reverse ray encounter matched optical adventures, such as reflections, refractions, and absorptions in a passive medium, or at an interface....

: .
conserving energy:

Applications

The BRDF is a fundamental radiometric concept, and accordingly is used in computer graphics

Computer graphics

Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

for photorealistic rendering

Rendering (computer graphics)

Rendering is the process of generating an image from a model , by means of computer programs. A scene file contains objects in a strictly defined language or data structure; it would contain geometry, viewpoint, texture, lighting, and shading information as a description of the virtual scene...

of synthetic scenes (see the Rendering equation

Rendering equation

In computer graphics, the rendering equation is an integral equation in which the equilibrium radiance leaving a point is given as the sum of emitted plus reflected radiance under a geometric optics approximation. It was simultaneously introduced into computer graphics by David Immel et al. and...

), as well as in computer vision

Computer vision

Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...

for many inverse problems such as object recognition.

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

Lambertian reflectance

If a surface exhibits Lambertian reflectance, light falling on it is scattered such that the apparent brightness of the surface to an observer is the same regardless of the observer's angle of view. More technically, the surface luminance is isotropic...

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
Fresnel equations
The Fresnel equations , deduced by Augustin-Jean Fresnel , describe the behaviour of light when moving between media of differing refractive indices...

at grazing angles
being well-suited to Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

s.

Wojciech 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
Phong reflection model
The Phong reflection model is an empirical model of the local illumination of points on a surface...

, a phenomenological model akin to plastic-like specularity.
Blinn–Phong model
Blinn–Phong shading model
The Blinn–Phong shading model is a modification to the Phong reflection model developed by Jim Blinn....

, 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's anisotropic model, a specular-microfacet model with a 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 a specific lighting and viewing direction at a time using gonioreflectometer

Gonioreflectometer

A gonioreflectometer is a device for measuring a bidirectional reflectance distribution function .The device consists of a light source illuminating the material to be measured and a sensor that captures light reflected from that material. The light source should be able to illuminate and the...

s. 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

High dynamic range imaging

In image processing, computer graphics, and photography, high dynamic range imaging is a set of techniques that allows a greater dynamic range between the lightest and darkest areas of an image than current standard digital imaging techniques or photographic methods...

. The standard algorithm is to measure the BRDF point cloud from images and optimize it by one of the BRDF models.