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Analysis of this phenomenon has a long history and can be traced back almost a century. Past work has resulted in empirical models designed to fit experimental data as well as theoretical results derived from first principles. Much of this work was motivated by the non-Lambertian reflectance of the
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brightness value over many facets. Whereas
Lambert’s law may hold well when observing a single planar facet, a collection of such facets with different orientations is guaranteed to violate Lambert’s law. The primary reason for this is that the foreshortened facet areas will change for different
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and graphics. For a large number of real-world surfaces, such as concrete, plaster, sand, etc., however, the
Lambertian model is an inadequate approximation of the diffuse component. This is primarily because the Lambertian model does not take the roughness of the surface into account.
1261:{\displaystyle C_{2}={\begin{cases}0.45{\frac {\sigma ^{2}}{\sigma ^{2}+0.09}}\sin \alpha &{\text{if }}\cos(\phi _{i}-\phi _{r})\geq 0,\\0.45{\frac {\sigma ^{2}}{\sigma ^{2}+0.09}}\left(\sin \alpha -\left({\frac {2\beta }{\pi }}\right)^{3}\right)&{\text{otherwise,}}\end{cases}}}
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Here is a real image of a matte vase illuminated from the viewing direction, along with versions rendered using the
Lambertian and Oren-Nayar models. It shows that the Oren-Nayar model predicts the diffuse reflectance for rough surfaces more accurately than the Lambertian model.
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is a physical property of a material that describes how it reflects incident light. The appearance of various materials are determined to a large extent by their reflectance properties. Most reflectance models can be broadly classified into two categories:
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between points on the surface facets. It can be viewed as a generalization of
Lambert’s law. Today, it is widely used in computer graphics and animation for rendering rough surfaces. It also has important implications for
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747:{\displaystyle L_{1}={\frac {\rho }{\pi }}E_{0}\cos \theta _{i}\left(C_{1}+C_{2}\cos(\phi _{i}-\phi _{r})\tan \beta +C_{3}(1-|\cos(\phi _{i}-\phi _{r})|)\tan {\frac {\alpha +\beta }{2}}\right),}
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in 1993, predicts reflectance from rough diffuse surfaces for the entire hemisphere of source and sensor directions. The model takes into account complex physical phenomena such as
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166:(much larger than the wavelength of incident light) surface roughness is often projected onto a single detection element, which in turn produces an
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can be modelled as a set of facets with different slopes, where each facet is a small planar patch. Since photo receptors of the
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Here are rendered images of a sphere using the Oren-Nayar model, corresponding to different surface roughnesses (i.e. different
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Comparison of a matte vase with the rendering based on the
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942:{\displaystyle L_{2}=0.17{\frac {\rho ^{2}}{\pi }}E_{0}\cos \theta _{i}{\frac {\sigma ^{2}}{\sigma ^{2}+0.13}}\left,}
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of the surface is specified using a probability function for the distribution of facet slopes. In particular, the
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Plot of the brightness of the rendered images, compared with the measurements on a cross section of the real vase
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The surface roughness model used in the derivation of the Oren-Nayar model is the microfacet model, proposed by
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Proceedings of the 21st annual conference on
Computer graphics and interactive techniques - SIGGRAPH '94
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Torrance, K. E.; Sparrow, E. M. (1967). "Theory for off-specular reflection from roughened surfaces".
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appears equally bright from all viewing directions. This model for diffuse reflection was proposed by
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In the Oren–Nayar reflectance model, each facet is assumed to be
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of the light reflected by the faceted surface, according to the Oren-Nayar model, is
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Oren, M.; Nayar, S. K. (1994). "Generalization of
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that describes bounces of light between the facets are defined as follows.
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viewing directions, and thus the surface appearance will be view-dependent.
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in 1760 and has been perhaps the most widely used reflectance model in
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is the irradiance when the facet is illuminated head-on, the radiance
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1019:{\displaystyle C_{1}=1-0.5{\frac {\sigma ^{2}}{\sigma ^{2}+0.33}}}
1727:{\displaystyle L_{r}={\frac {\rho }{\pi }}E_{0}\cos \theta _{i}.}
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The Oren–Nayar reflectance model, developed by
Michael Oren and
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1981:"Microfacet Models for Refraction through Rough Surfaces"
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in a camera are both finite-area detectors, substantial
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Connection with other microfacet reflectance models
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1905:. pp. 239–246.
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484:and the term
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136:Lambert's Law
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209:human vision
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95:Introduction
86:
80:
77:reflectivity
68:
66:
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46:Please help
41:
227:Formulation
164:macroscopic
108:Reflectance
2015:Categories
1922:0897916670
1877:References
1813:Oren-Nayar
1248:otherwise,
306:ranges in
130:Lambertian
79:model for
1985:Egsr 2007
1793:values):
1781:σ
1713:θ
1709:
1691:π
1688:ρ
1552:σ
1532:σ
1508:ρ
1475:θ
1462:θ
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1298:σ
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1207:−
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1164:σ
1145:≥
1133:ϕ
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1120:ϕ
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1065:σ
999:σ
988:σ
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537:ρ
323:∞
294:σ
268:σ
249:roughness
200:shadowing
168:aggregate
56:July 2019
1855:See also
1106:if
257:variance
241:Torrance
117:specular
2026:Shading
1954:Bibcode
1738:Results
1520:is the
245:Sparrow
223:, etc.
196:masking
113:diffuse
75:, is a
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1522:albedo
952:where
160:pixels
156:retina
1927:S2CID
1292:0.125
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185:moon
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