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Bidirectional reflectance distribution function
 
 
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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
Radiance

Radiance and spectral radiance are radiometry measures that describe the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction....
 exiting along to the irradiance
Irradiance

Irradiance, radiant emittance, and radiant exitance are radiometry terms for the power of electromagnetic radiation at a surface, per unit area....
 incident on the surface from direction . Note that each direction is itself parameterized by azimuth angle and elevation , therefore the BRDF as a whole is 4-dimensional.






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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
Radiance

Radiance and spectral radiance are radiometry measures that describe the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction....
 exiting along to the irradiance
Irradiance

Irradiance, radiant emittance, and radiant exitance are radiometry terms for the power of electromagnetic radiation at a surface, per unit area....
 incident on the surface from direction . Note that each direction is itself parameterized by azimuth angle and elevation , 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, O, is the angle in three-dimensional space that an object subtends at a point. It is a measure of how big that object appears to an observer looking from that point....
.

Definition


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

where is the radiance
Radiance

Radiance and spectral radiance are radiometry measures that describe the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction....
, is the irradiance
Irradiance

Irradiance, radiant emittance, and radiant exitance are radiometry terms for the power of electromagnetic radiation at a surface, per unit area....
, and is the angle made between and the surface normal
Surface normal

A surface normal, or simply normal, to a Flatness is a vector which 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 space to that surface at P....
, .

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
Computer graphics

Computer graphics are graphics created by computers and, more generally, the representation and manipulation of pictorial data by a computer....
 for photorealistic rendering
Rendering (computer graphics)

Rendering is the process of generating an image from a 3D model, by means of computer programs. The model is a description of three-dimensional objects in a strictly defined language or data structure....
 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....
), as well as in computer vision
Computer vision

Computer vision is the science and technology of machines that see. As a scientific discipline, computer vision is concerned with the theory for building artificial systems that obtain information from images....
 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....
 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 medium of differing refractive index....
     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 when computer simulation physics and mathematics systems....
    s.


Some examples


  • Lambertian model, representing perfectly diffuse (matte) surfaces by a constant BRDF.
  • Phong reflectance model
    Phong shading

    Phong shading refers to a set of techniques in 3D computer graphics. Phong shading includes a model for the reflection of light from surfaces and a compatible method of estimating pixel colors by interpolation surface normals across rasterized polygons....
    , 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.Blinn-Phong is the default shading model used in OpenGL and Direct3D's fixed-function pipeline , and is carried out on each vertex as it passes down the graphics pipeline; pixel values between vertices are interpolated by Gouraud shading by...
    , 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.


Acquisition

Traditionally, BRDF measurements were taken for a specific lighting and viewing direction at a time using gonioreflectometer
Gonioreflectometer

The 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....
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.

See also

  • BSDF
    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....
  • Radiometry
    Radiometry

    In optics, radiometry is the field that studies the measurement of electromagnetic radiation, including visible light. Note that light is also measured using the techniques of photometry_, which deal with brightness as perceived by the human eye, rather than absolute power....
  • Photometry (astronomy)
    Photometry (astronomy)

    Photometry is a technique of astronomy concerned with measurement the flux, or intensity of an astronomical object's electromagnetic radiation....
  • Reflectance
  • Albedo
    Albedo

    The albedo of an object is the extent to which it diffusely reflects light from the Sun. It is therefore a more specific form of the term reflectivity....
  • Opposition spike
  • Gonioreflectometer
    Gonioreflectometer

    The 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....
  • Specular highlight
    Specular highlight

    A specular highlight is the bright spot of light that appears on shiny objects when illuminated . Specular highlights are important in 3D computer graphics, as they provide a strong visual cue for the shape of an object and its location with respect to light sources in the scene....
  • Schlick's_approximation
    Schlick's approximation

    In 3D computer graphics, Schlick's approximation is a formula for approximating the BRDF of metallic surfaces. It was proposed by Christophe Schlick to approximate the contributions of Fresnel terms in the specular reflection of light from conducting surfaces....


Further reading