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The bidirectional reflectance distribution function (BRDF; $f_\left(\omega_,\, \omega_\right)$ ) is a function of four real variables that defines how light is reflected at an
opaque Opacity or opaque may refer to: * Impediments to (especially, visible) light: ** Opacities, absorption coefficients ** Opacity (optics), property or degree of blocking the transmission of light * Metaphors derived from literal optics: ** Opaque con ...
surface. It is employed in the
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...

of real-world light, in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great dea ...

algorithms, and in
computer vision Computer vision is an interdisciplinary scientific field that deals with how computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern computers can perform ge ...
algorithms. The function takes an incoming light direction, $\omega_$, and outgoing direction, $\omega_$ (taken in a coordinate system where the
surface normal In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
$\mathbf n$ lies along the ''z''-axis), and returns the ratio of reflected
radiance In radiometry Radiometry is a set of techniques for measuring Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are ...
exiting along $\omega_$ to the
irradiance In radiometry Radiometry is a set of techniques for measuring ' Measurement is the numerical quantification of the attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measure ...

incident on the surface from direction $\omega_$. Each direction $\omega$ is itself azimuth angle $\phi$ and
zenith angle The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere In astronomy and navigation, the celestial sphere is an abstraction, abstract sphere that has an arbitrarily large radius and is concen ...
$\theta$, therefore the BRDF as a whole is a function of 4 variables. The BRDF has units sr−1, with
steradian The steradian (symbol: sr) or square radian is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sys ...

s (sr) being a unit of
solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poi ...

.

# Definition

The BRDF was first defined by Fred Nicodemus around 1965. The definition is: $f_\left(\omega_,\, \omega_\right) \,=\, \frac \,=\, \frac$ where $L$ is
radiance In radiometry Radiometry is a set of techniques for measuring Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are ...
, or
power Power typically refers to: * Power (physics) In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, p ...
per unit -in-the-direction-of-a-ray per unit projected-area-perpendicular-to-the-ray, $E$ is
irradiance In radiometry Radiometry is a set of techniques for measuring ' Measurement is the numerical quantification of the attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measure ...

, or power per unit ''surface area'', and $\theta_$ is the angle between $\omega_$ and the
surface normal In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

, $\mathbf n$. The index $\text$ indicates incident light, whereas the index $\text$ 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 $\operatorname dE_\left(\omega_\right)$, which are of no interest for $f_\left(\omega_,\, \omega_\right)$, might illuminate the surface which would unintentionally affect $L_\left(\omega_\right)$, whereas $\operatorname dL_\left(\omega_\right)$ is only affected by $\operatorname dE_\left(\omega_\right)$.

# Related functions

The Spatially Varying Bidirectional Reflectance Distribution Function (SVBRDF) is a 6-dimensional function, $f_\left(\omega_,\,\omega_,\,\mathbf\right)$, where $\mathbf$ 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 Subsurface scattering (SSS), also known as subsurface light transport (SSLT), is a mechanism of light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that can be visual perception, per ...

. 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 $S\left(\mathbf_,\,\omega_,\,\mathbf_,\,\omega_\right)$ in which light entering the surface may scatter internally and exit at another location. In all these cases, the dependence on the
wavelength In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

of light has been ignored. In reality, the BRDF is wavelength dependent, and to account for effects such as
iridescence Iridescence (also known as goniochromism) is the phenomenon of certain surfaces that appear to gradually change color Color (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States Englis ...
or
luminescence Luminescence is spontaneous emission of light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that can be visual perception, perceived by the human eye. Visible light is usually defined ...

the dependence on wavelength must be made explicit: $f_\left(\lambda_,\,\omega_,\,\lambda_,\,\omega_\right)$. Note that in the typical case where all optical elements are
linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...
, the function will obey $f_\left(\lambda_,\,\omega_,\,\lambda_,\,\omega_\right) = 0$ except when $\lambda_ = \lambda_$: that is, it will only emit light at wavelength equal to the incoming light. In this case it can be paramaterized as $f_\left(\lambda,\,\omega_,\,\omega_\right)$, with only one wavelength parameter.

# Physically based BRDFs

Physically realistic BRDFs have additional properties, including, * positivity: $f_\left(\omega_,\, \omega_\right) \ge 0$ * obeying
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. It does not apply to moving, non- ...
: $f_\left(\omega_,\, \omega_\right) = f_\left(\omega_,\, \omega_\right)$ * conserving energy: $\forall \omega_,\, \int_\Omega f_\left(\omega_,\, \omega_\right)\,\cos d\omega_ \le 1$

# Applications

The BRDF is a fundamental
radiometric Radiometry is a set of techniques for measuring Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are dependent on the ...
concept, and accordingly is used in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great dea ...

for
photorealistic rendering __NOTOC__ 250px, The Indigo Renderer is unbiased. This 2009 render is of a German country road. Within the field of computer graphics, unbiased rendering refers to any rendering technique that does not introduce systematic error, or bias, int ...
of synthetic scenes (see the
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 ...
), as well as in
computer vision Computer vision is an interdisciplinary scientific field that deals with how computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern computers can perform ge ...
for many
inverse problem An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, sound source reconstruction, source reconstruction in aco ...

s such as
object recognition The following outline is provided as an overview of and topical guide to object recognition: Object recognition – technology in the field of computer vision for finding and identifying objects in an image or video sequence. Humans recogn ...
. BRDF has also been used for modeling light trapping in
solar cell A solar cell, or photovoltaic cell, is an electrical device that converts the energy of light directly into electricity by the photovoltaic effect, which is a physics, physical and Chemical substance, chemical phenomenon.
s (e.g. using the OPTOS formalism) or low concentration
solar photovoltaic A photovoltaic system, also PV system or solar power system, is a power system designed to supply usable solar power Solar power is the energy transformation, conversion of energy from sunlight into electricity, either directly using photov ...

systems. In the context of satellite remote sensing, NASA uses a BRDF model to characterise surface reflectance 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 Albedo (prounounced ; la, albedo, meaning 'whiteness') is the measure of the diffuse reflection Diffuse reflection is the reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a comm ...

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 Lambertian reflectance is the property that defines an ideal "matte" or diffusely reflecting surface. The apparent brightness of a Lambertian surface to an observer is the same regardless of the observer's angle of view. More technically, the surf ...
model frequently assumed in computer graphics. Some useful features of recent models include: * accommodating
anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions as opposed to isotropy Isotropy is uniformity in all orientations; it is derived from the Greek ''isos'' (ἴσος, ...
reflection * editable using a small number of intuitive parameters * accounting for at grazing angles * being well-suited to
Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
s. W. Matusik 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.K. Torrance and E. Sparrow
Theory for Off-Specular Reflection from Roughened Surfaces
J. Optical Soc. America, vol. 57. 1967. pp. 1105–1114.
* 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 measurement devices called
gonioreflectometer Image:Gantry2.jpg, 300px, The University of Virginia spherical gantry, an example of a modern image-based gonioreflectometer A gonioreflectometer is a device for measuring a bidirectional reflectance distribution function (BRDF). The device consist ...
s 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: * Measuring or synthesizing the target BRDF distribution. * Sample this distribution to discretize it and make the fabrication feasible. * Design a geometry that produces this distribution (with microfacet,
halftoning Halftone is the reprographicReprography is the reproduction of graphics through mechanical or electrical means, such as photography or xerography. Reprography is commonly used in catalogs and archives, as well as in the architectural reprography ...
). * Optimize the continuity and smoothness of the surface with respect to the manufacturing procedure. Many approaches have been proposed for manufacturing the BRDF of the target : *
Milling Milling may refer to: * Milling (grinding), breaking solid materials into smaller pieces by grinding, crushing, or cutting in a mill * Milling (machining), a process of using rotary cutters to remove material from a workpiece * Milling (military tra ...
the BRDF: This procedure starts with sampling the BRDF distribution and generating it with microfacet geometry then the surfaced is optimized in terms of smoothness and continuity to meet the limitations of the milling machine. The final BRDF distribution is the convolution of the substrate and the geometry of the milled surface. * Printing the BRDF: In order to generate spatially varying BRDF (svBRDF) it has been proposed to use gamut mapping and
halftoning Halftone is the reprographicReprography is the reproduction of graphics through mechanical or electrical means, such as photography or xerography. Reprography is commonly used in catalogs and archives, as well as in the architectural reprography ...
to achieve the targeted BRDF. Given a set of metallic inks with known BRDF an algorithm proposed to linearly combine them to produce the targeted distribution.  So far printing only means gray-scale or color printing but real-world surfaces can exhibit different amounts of specularity that affects their final appearance, as a result this novel method can help us print images even more realistically.   *Combination of Ink and Geometry: In addition to color and specularity, real-world objects also contain texture. A 3D printer can be used to manufacture the geometry and cover the surface with a suitable ink; by optimally creating the facets and choosing the ink combination, this method can give us a higher degree of freedom in design and more accurate BRDF fabrication.

# See also

*
Albedo Albedo (prounounced ; la, albedo, meaning 'whiteness') is the measure of the diffuse reflection Diffuse reflection is the reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a comm ...

* *
Gonioreflectometer Image:Gantry2.jpg, 300px, The University of Virginia spherical gantry, an example of a modern image-based gonioreflectometer A gonioreflectometer is a device for measuring a bidirectional reflectance distribution function (BRDF). The device consist ...
* Opposition spike *
Photometry (astronomy) Photometry, from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approxim ...
*
Radiometry Radiometry is a set of techniques for measuring Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are dependent on the ...
*
Reflectance The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic s ...

*
Schlick's approximationIn 3D computer graphics, Schlick's approximation, named after Christophe Schlick, is a formula for approximating the contribution of the Fresnel equation, Fresnel factor in the specular reflection of light from a non-conducting interface (surface) be ...
*
Specular highlight A specular highlight is the bright spot of light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that is visual perception, perceived by the human eye. Visible light is usually define ...

# Further reading

* * *
An intuitive introduction to the concept of reflection model and BRDF.
{{DEFAULTSORT:Bidirectional Reflectance Distribution Function 3D rendering Astrophysics Optics Radiometry Remote sensing