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| [[File:BRDF Diagram.svg|thumb|right|300px|Diagram showing vectors used to define the BRDF. All vectors are unit length. <math>\omega_{\text{i}}</math> points toward the light source. <math>\omega_{\text{r}}</math> points toward the viewer (camera). <math>n</math> is the surface normal.]]
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| The '''bidirectional reflectance distribution function''' ('''BRDF'''; <math>f_{\text{r}}(\omega_{\text{i}},\, \omega_{\text{r}})</math> ) is a four-dimensional function that defines how light is reflected at an opaque surface. The function takes a negative incoming light direction, <math>\omega_{\text{i}}</math>, and outgoing direction, <math>\omega_{\text{r}}</math>, both defined with respect to the [[Normal (geometry)|surface normal]] <math>\mathbf n</math>{{clarify|date=November 2012|reason=In which way are ω_i and ω_r defined with respect to n?}}, and returns the ratio of reflected [[radiance]] exiting along <math>\omega_{\text{r}}</math> to the [[irradiance]] incident on the surface from direction <math>\omega_{\text{i}}</math>. Each direction <math>\omega</math> is itself [[Spherical coordinate system|parameterized by]] [[azimuth angle]] <math>\phi</math> and [[zenith angle]] <math>\theta</math>, therefore the BRDF as a whole is 4-dimensional. The BRDF has units sr<sup>−1</sup>, with [[steradian]]s (sr) being a unit of [[solid angle]].
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| ==Definition==
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| The BRDF was first defined by Fred Nicodemus around 1965.<ref name='nicodemus_1965'>
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| {{Cite journal
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| | volume = 4
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| | issue = 7
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| | pages = 767–775
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| | last = Nicodemus
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| | first = Fred
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| | title = Directional reflectance and emissivity of an opaque surface
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| | journal = Applied Optics
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| | url = http://ao.osa.org/abstract.cfm?id=13818
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| | format = abstract
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| | doi = 10.1364/AO.4.000767
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| | year = 1965
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| |bibcode = 1965ApOpt...4..767N }}</ref> The definition is:
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| <math>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}}}</math>
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| where <math>L</math> is [[radiance]], or [[power (physics)|power]] per unit [[solid angle|solid-angle]]-in-the-direction-of-a-ray per unit [[projected area|projected-area]]-perpendicular-to-the-ray, <math>E</math> is [[irradiance]], or power per unit ''surface area'', and <math>\theta_{\text{i}}</math> is the angle between <math>\omega_{\text{i}}</math> and the [[surface normal]], <math>\mathbf n</math>. The index <math>\text{i}</math> indicates incident light, whereas the index <math>\text{r}</math> indicates reflected light.
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| The reason the function is defined as a quotient of two [[Differential of a function|differential]]s and not directly as a quotient between the undifferentiated quantities, is because other irradiating light than <math>\operatorname dE_{\text{i}}(\omega_{\text{i}})</math>, which are of no interest for <math>f_{\text{r}}(\omega_{\text{i}},\, \omega_{\text{r}})</math>, might illuminate the surface which would unintentionally affect <math>L_{\text{r}}(\omega_{\text{r}})</math>, whereas <math>\operatorname dL_{\text{r}}(\omega_{\text{r}})</math> is only affected by <math>\operatorname dE_{\text{i}}(\omega_{\text{i}})</math>.
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| == Related functions ==
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| The '''Spatially Varying Bidirectional Reflectance Distribution Function''' (SVBRDF) is a 6-dimensional function, <math>f_{\text{r}}(\omega_{\text{i}},\,\omega_{\text{r}},\,\mathbf{x})</math>, where <math>\mathbf{x}</math> describes a 2D location over an object's surface.
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| The '''Bidirectional Texture Function''' ([[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'''.
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| The '''Bidirectional Surface Scattering Reflectance Distribution Function''' ([[Bidirectional scattering distribution function|BSSRDF]]), is a further generalized 8-dimensional function <math>S(\mathbf{x}_{\text{i}},\,\omega_{\text{i}},\,\mathbf{x}_{\text{r}},\,\omega_{\text{r}})</math> in which light entering the surface may scatter internally and exit at another location.
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| 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: <math>f_{\text{r}}(\lambda_{\text{i}},\,\omega_{\text{i}},\,\lambda_{\text{r}},\,\omega_{\text{r}})</math>.
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| == Physically based BRDFs ==
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| Physically based BRDFs have additional properties, including,
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| * positivity: <math>f_{\text{r}}(\omega_{\text{i}},\, \omega_{\text{r}}) \ge 0 </math>
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| * obeying [[Helmholtz reciprocity]]: <math>f_{\text{r}}(\omega_{\text{i}},\, \omega_{\text{r}}) = f_{\text{r}}(\omega_{\text{r}},\, \omega_{\text{i}})</math>
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| * conserving energy: <math>\forall \omega_{\text{i}},\, \int_\Omega f_{\text{r}}(\omega_{\text{i}},\, \omega_{\text{r}})\,\cos{\theta_{\text{r}}} d\omega_{\text{r}} \le 1</math>
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| ==Applications==
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| The BRDF is a fundamental [[radiometric]] concept, and accordingly is used in [[computer graphics]] for [[Rendering (computer graphics)|photorealistic rendering]] of synthetic scenes (see the [[Rendering equation]]), as well as in [[computer vision]] for many [[inverse problem]]s such as [[object recognition]].
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| ==Models==
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| BRDFs can be measured directly from real objects using calibrated cameras and lightsources;<ref>{{cite web |author=Rusinkiewicz, S. |title=A Survey of BRDF Representation for Computer Graphics |accessdate=2007-09-05 |url=http://www.cs.princeton.edu/~smr/cs348c-97/surveypaper.html}}</ref> 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:
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| * accommodating [[anisotropic]] reflection
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| * editable using a small number of intuitive parameters
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| * accounting for [[Fresnel equations|Fresnel effects]] at grazing angles
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| * being well-suited to [[Monte Carlo method]]s.
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| Wojciech et al. found that interpolating between measured samples produced realistic results and was easy to understand.<ref>Wojciech Matusik, Hanspeter Pfister, Matt Brand, and Leonard McMillan. [http://people.csail.mit.edu/wojciech/DDRM/index.html A Data-Driven Reflectance Model]. ACM Transactions on Graphics. 22(3) 2002.</ref>
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| ===Some examples===
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| * [[Lambertian|Lambertian model]], representing perfectly diffuse (matte) surfaces by a constant BRDF.
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| * [[Seeliger effect|Lommel–Seeliger]], lunar and Martian reflection.
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| * [[Phong reflection model|Phong reflectance model]], a phenomenological model akin to plastic-like specularity.<ref>B. T. Phong, Illumination for computer generated pictures, Communications of ACM 18 (1975), no. 6, 311–317.</ref>
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| * [[Blinn–Phong shading model|Blinn–Phong model]], resembling Phong, but allowing for certain quantities to be interpolated, reducing computational overhead.<ref>{{cite journal | journal = Proc. 4th annual conference on computer graphics and interactive techniques | title = Models of light reflection for computer synthesized pictures | author = James F. Blinn | year = 1977 | url = http://portal.acm.org/citation.cfm?doid=563858.563893 | doi = 10.1145/563858.563893 | pages = 192}}</ref>
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| * Torrance–Sparrow model, a general model representing surfaces as distributions of perfectly specular microfacets.<ref name='torrance_1976'>K. Torrance and E. Sparrow. [http://www.graphics.cornell.edu/~westin/pubs/TorranceSparrowJOSA1967.pdf Theory for Off-Specular Reflection from Roughened Surfaces]. J. Optical Soc. America, vol. 57. 1976. pp. 1105–1114.</ref>
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| * [[Cook–Torrance|Cook–Torrance model]], a specular-microfacet model (Torrance–Sparrow) accounting for wavelength and thus color shifting.<ref>R. Cook and K. Torrance. "A reflectance model for computer graphics". Computer Graphics (SIGGRAPH '81 Proceedings), Vol. 15, No. 3, July 1981, pp. 301–316.</ref>
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| * [[Ward's anisotropic model|Ward model]], a specular-microfacet model with an elliptical-Gaussian distribution function dependent on surface tangent orientation (in addition to surface normal).<ref name='ward_1992'>
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| {{cite conference
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| | first = Gregory J.
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| | last = Ward
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| | title = Measuring and modeling anisotropic reflection
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| | booktitle = Proceedings of SIGGRAPH
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| | pages = 265–272
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| | year = 1992
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| | doi = 10.1145/133994.134078
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| | accessdate = 2008-02-03
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| }}</ref>
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| * [[Oren–Nayar diffuse model|Oren–Nayar model]], a "directed-diffuse" microfacet model, with perfectly diffuse (rather than specular) microfacets.<ref>S.K. Nayar and M. Oren, "[http://www1.cs.columbia.edu/CAVE/publications/pdfs/Nayar_IJCV95.pdf Generalization of the Lambertian Model and Implications for Machine Vision]". International Journal on Computer Vision, Vol. 14, No. 3, pp. 227–251, Apr, 1995</ref>
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| * Ashikhmin-Shirley model, allowing for anisotropic reflectance, along with a diffuse substrate under a specular surface.<ref>Michael Ashikhmin, Peter Shirley, An Anisotropic Phong BRDF Model, Journal of Graphics Tools 2000</ref>
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| * HTSG (He,Torrance,Sillion,Greenberg), a comprehensive physically based model.<ref>X. He, K. Torrance, F. Sillon, and D. Greenberg, A comprehensive physical model for light reflection, Computer Graphics 25 (1991), no. Annual Conference Series, 175–186.</ref>
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| * Fitted Lafortune model, a generalization of Phong with multiple specular lobes, and intended for parametric fits of measured data.<ref>E. Lafortune, S. Foo, K. Torrance, and D. Greenberg, Non-linear approximation of reflectance functions. In Turner Whitted, editor, SIGGRAPH 97 Conference Proceedings, Annual Conference Series, pp. 117–126. ACM SIGGRAPH, Addison Wesley, August 1997.</ref>
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| * Lebedev model for analytical-grid BRDF approximation.<ref>Ilyin A., Lebedev A., Sinyavsky V., Ignatenko, A., [http://data.lebedev.as/LebedevGraphicon2009.pdf Image-based modelling of material reflective properties of flat objects (In Russian)]. In: GraphiCon'2009.; 2009. p. 198-201.</ref>
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| ==Acquisition==
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| Traditionally, BRDF measurements were taken for a specific lighting and viewing direction at a time using [[gonioreflectometer]]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.
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| There is an alternative way to measure BRDF based on [[High Dynamic Range Imaging|HDR images]]. The standard algorithm is to measure the BRDF point cloud from images and optimize it by one of the BRDF models.<ref>[http://lebedev.as/index.php?p=1_7_BRDFRecon BRDFRecon project]</ref>
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| ==See also==
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| {{portal|Computer graphics}}
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| *[[Albedo]]
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| *[[Bidirectional scattering distribution function|BSDF]]
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| *[[Gonioreflectometer]]
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| *[[Opposition spike]]
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| *[[Photometry (astronomy)]]
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| *[[Radiometry]]
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| *[[Reflectance]]
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| *[[Schlick's approximation]]
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| *[[Specular highlight]]
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| ==References==
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| <references/>
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| ==Further reading==
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| *{{Cite book
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| | edition = 1st
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| | publisher = Springer
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| | isbn = 3-540-43097-0
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| | last = Lubin
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| | first = Dan
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| | coauthors = Robert Massom
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| | title = Polar Remote Sensing
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| | others= Volume I: ''Atmosphere and Oceans''
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| | date = 2006-02-10
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| | page = 756
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| }}
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| *{{Cite book
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| | edition = 1st
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| | publisher = Morgan Kauffmann
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| | isbn = 0-12-553180-X
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| | last = Matt
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| | first = Pharr
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| | coauthors = Greg Humphreys
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| | title = Physically Based Rendering
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| | year = 2004
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| | page = 1019
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| }}
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| *{{Cite journal
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| | volume = 103
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| | issue = 1
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| | pages = 27–42
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| | last = Schaepman-Strub
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| | first = G.
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| | coauthors = M. E. Schaepman, T. H. Painter, S. Dangel, J. V. Martonchik
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| | title = Reflectance quantities in optical remote sensing: definitions and case studies
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| | journal = Remote Sensing of Environment
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| | accessdate = 2007-10-18
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| | date = 2006-07-15
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| | url = http://www.sciencedirect.com/science/article/B6V6V-4K427VX-1/2/d8f9855bc59ae8233e2ee9b111252701
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| | doi = 10.1016/j.rse.2006.03.002
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| }}
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| {{DEFAULTSORT:Bidirectional Reflectance Distribution Function}}
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| [[Category:3D rendering]]
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| [[Category:Astrophysics]]
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| [[Category:Optics]]
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| [[Category:Radiometry]] <!--N.B.: Usage of term "photometry" varies between astronomy and optics. The categories are named following the optics convention. -->
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| [[Category:Remote sensing]]
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