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The Oren–Nayar reflectance model, developed by Michael Oren and Shree K. Nayar, is a
reflectivity The reflectance of the surface of a material is its effectiveness in Reflection (physics), reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the respon ...
model for
diffuse reflection Diffuse reflection is the reflection of light or other waves or particles from a surface such that a ray incident on the surface is scattered at many angles rather than at just one angle as in the case of specular reflection. An ''ideal'' dif ...
from rough surfaces. It has been shown to accurately predict the appearance of a wide range of natural surfaces, such as concrete, plaster, sand, etc.


Introduction

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 ...
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:
diffuse Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
and
specular Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface. The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surf ...
. In
computer vision Computer vision tasks include methods for image sensor, acquiring, Image processing, processing, Image analysis, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical ...
and
computer graphics Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...
, the diffuse component is often assumed to be Lambertian. A surface that obeys Lambert's Law appears equally bright from all viewing directions. This model for diffuse reflection was proposed by
Johann Heinrich Lambert Johann Heinrich Lambert (; ; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, at that time allied to the Switzerland, Swiss Confederacy, who made important contributions to the subjects of mathematics, phys ...
in 1760 and has been perhaps the most widely used reflectance model in
computer vision Computer vision tasks include methods for image sensor, acquiring, Image processing, processing, Image analysis, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical ...
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. Rough surfaces can be modelled as a set of facets with different slopes, where each facet is a small planar patch. Since photo receptors of the
retina The retina (; or retinas) is the innermost, photosensitivity, light-sensitive layer of tissue (biology), tissue of the eye of most vertebrates and some Mollusca, molluscs. The optics of the eye create a focus (optics), focused two-dimensional ...
and
pixels In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest addressable element in a dot matrix display device. In most digital display devices, pixels are the sma ...
in a camera are both finite-area detectors, substantial
macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic. Overview When applied to physical phenome ...
(much larger than the wavelength of incident light) surface roughness is often projected onto a single detection element, which in turn produces an aggregate 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 viewing directions, and thus the surface appearance will be view-dependent. 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
moon The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...
. The Oren–Nayar reflectance model, developed by Michael Oren and Shree K. Nayar 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 masking, shadowing and interreflections 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
human vision Visual perception is the ability to detect light and use it to form an image of the surrounding Biophysical environment, environment. Photodetection without image formation is classified as ''light sensing''. In most vertebrates, visual percept ...
and
computer vision Computer vision tasks include methods for image sensor, acquiring, Image processing, processing, Image analysis, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical ...
problems, such as
shape from shading Photometric stereo is a technique in computer vision for estimating the surface normals of objects by observing that object under different lighting conditions (photometry). It is based on the fact that the amount of light reflected by a surface ...
,
photometric stereo Photometric stereo is a technique in computer vision for estimating the surface normals of objects by observing that object under different lighting conditions (photometry). It is based on the fact that the amount of light reflected by a surface ...
, etc.


Formulation

The surface roughness model used in the derivation of the Oren-Nayar model is the microfacet model, proposed by Torrance and Sparrow, which assumes the surface to be composed of long symmetric V-cavities. Each cavity consists of two planar facets. The roughness of the surface is specified using a probability function for the distribution of facet slopes. In particular, the
Gaussian distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is f(x ...
is often used, and thus the
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
of the Gaussian distribution, \sigma^2, is a measure of the roughness of the surfaces. The standard deviation of the facet slopes (gradient of the surface elevation), \sigma ranges in \cos(\phi_i-\phi_r), )\tan\frac2\right), :L_2 = 0.17\frac\pi E_0 \cos\theta_i\frac\left[1-\cos(\phi_i-\phi_r)\left(\frac\pi\right)^2\right where :C_1 = 1-0.5\frac, :C_2 = \begin 0.45\frac\sin\alpha & \text\cos(\phi_i-\phi_r)\ge0,\\ 0.45\frac\left(\sin\alpha-\left(\frac\pi\right)^3\right) & \text \end :C_3 = 0.125\frac\left(\frac\right)^2, :\alpha = \max(\theta_i, \theta_r), :\beta = \min(\theta_i, \theta_r), and \rho is the albedo of the surface, and \sigma is the roughness of the surface. In the case of \sigma=0 (i.e., all facets in the same plane), we have C_1=1, and C_2=C_3=L_2=0, and thus the Oren-Nayar model simplifies to the Lambertian model: :L_r = \frac E_0 \cos \theta_i.


Results

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. Here are rendered images of a sphere using the Oren-Nayar model, corresponding to different surface roughnesses (i.e. different \sigma values):


Connection with other microfacet reflectance models


See also

*
List of common shading algorithms {{Short description, none This article lists common shading algorithms used in computer graphics. Interpolation techniques These techniques can be combined with any illumination model: * Flat shading * Gouraud shading * Phong shading Illuminat ...
*
Phong reflection model The Phong reflection model (also called Phong illumination or Phong lighting) is an empirical model of the local illumination of points on a surface designed by the computer graphics researcher Bui Tuong Phong. In 3D computer graphics, it is ...
*
Gamma correction Gamma correction or gamma is a Nonlinearity, nonlinear operation used to encode and decode Relative luminance, luminance or CIE 1931 color space#Tristimulus values, tristimulus values in video or still image systems. Gamma correction is, in the s ...


References


External links


The official project page for the Oren-Nayar model
at Shree Nayar'
CAVE research group webpage
{{DEFAULTSORT:Oren-Nayar reflectance model Scattering, absorption and radiative transfer (optics) Shading