Oren–Nayar Reflectance Model
Encyclopedia
The Oren-Nayar reflectance model, developed by Michael Oren and Shree K. Nayar
, is a reflectance model for diffuse reflection
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.
and specular
. In computer vision
and computer graphics
, 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
in 1760 and has been perhaps the most widely used reflectance model in computer vision
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
and pixels in a camera are both finite-area detectors, substantial macroscopic
(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 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
and computer vision
problems, such as shape from shading, photometric stereo
, etc.
of the Gaussian distribution,
, is a measure of the roughness of the surfaces (ranging from 0 to 1).
In the Oren-Nayar reflectance model, each facet is assumed to be Lambertian in reflectance. As shown in the image at right, given the radiance of the incoming light
, the radiance of the reflected light 
, according to the Oren-Nayar model, is
where
,
,
,
,
and
is the albedo
of the surface, and
is the roughness of the surface (ranging from 0 to 1). In the case of 
(i.e., all facets in the same plane), we have 
, and 
, and thus the Oren-Nayar model simplifies to the Lambertian model:
Here are rendered images of a sphere using the Oren-Nayar model, corresponding to different surface roughnesses (i.e. different
values):
Shree K. Nayar
Shree K. Nayar is an engineer and computer scientist known for his work in the fields of computer vision, computer graphics and computational cameras. He is the T. C. Chang Professor of the at Columbia University. Nayar co-directs the and is the head of the , which develops advanced computer...
, is a reflectance model for diffuse reflection
Diffuse reflection
Diffuse reflection is the reflection of light from a surface such that an incident ray is reflected at many angles rather than at just one angle as in the case of specular reflection...
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 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: diffuseDiffuse reflection
Diffuse reflection is the reflection of light from a surface such that an incident ray is reflected at many angles rather than at just one angle as in the case of specular reflection...
and specular
Specular reflection
Specular reflection is the mirror-like reflection of light from a surface, in which light from a single incoming direction is reflected into a single outgoing direction...
. In computer vision
Computer vision
Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...
and computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
, 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
Johann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...
in 1760 and has been perhaps the most widely used reflectance model in computer vision
Computer vision
Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...
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
Retina
The vertebrate retina is a light-sensitive tissue lining the inner surface of the eye. The optics of the eye create an image of the visual world on the retina, which serves much the same function as the film in a camera. Light striking the retina initiates a cascade of chemical and electrical...
and pixels in a camera are both finite-area detectors, substantial macroscopic
Macroscopic
The macroscopic scale is the length scale on which objects or processes are of a size which is measurable and observable by the naked eye.When applied to phenomena and abstract objects, the macroscopic scale describes existence in the world as we perceive it, often in contrast to experiences or...
(much larger than the wavelength of incident light) surface roughness is often projected onto a single detection element, which in turn produces an aggregate
Aggregate data
In statistics, aggregate data describes data combined from several measurements.In economics, aggregate data or data aggregates describes high-level data that is composed of a multitude or combination of other more individual data....
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
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
.
The Oren-Nayar reflectance model, developed by Michael Oren and Shree K. Nayar
Shree K. Nayar
Shree K. Nayar is an engineer and computer scientist known for his work in the fields of computer vision, computer graphics and computational cameras. He is the T. C. Chang Professor of the at Columbia University. Nayar co-directs the and is the head of the , which develops advanced computer...
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
Visual perception is the ability to interpret information and surroundings from the effects of visible light reaching the eye. The resulting perception is also known as eyesight, sight, or vision...
and computer vision
Computer vision
Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...
problems, such as shape from shading, photometric stereo
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....
, 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 is often used, and thus the varianceVariance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
of the Gaussian distribution,
, is a measure of the roughness of the surfaces (ranging from 0 to 1).In the Oren-Nayar reflectance model, each facet is assumed to be Lambertian in reflectance. As shown in the image at right, given the radiance of the incoming light
, the radiance of the reflected light , according to the Oren-Nayar model, iswhere
,,,,and
is the albedoAlbedo
Albedo , or reflection coefficient, is the diffuse reflectivity or reflecting power of a surface. It is defined as the ratio of reflected radiation from the surface to incident radiation upon it...
of the surface, and
is the roughness of the surface (ranging from 0 to 1). In the case of (i.e., all facets in the same plane), we have , and , and thus the Oren-Nayar model simplifies to the Lambertian model: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
values):Connection with other microfacet reflectance models
| Oren-Nayar model | Torrance-Sparrow 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.-Microfacets:The... model |
Microfacet model for refraction |
|---|---|---|
| Rough opaque diffuse surfaces | Rough opaque specular surfaces (glossy surfaces) | Rough transparent surfaces |
| Each facet is Lambertian (diffuse) | Each facet is a mirror (specular) | Each facet is made of glass (transparent) |
See also
- List of common shading algorithms
- Lambertian ModelLambertian reflectanceIf 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. More technically, the surface luminance is isotropic...
- Phong Reflection ModelPhong reflection modelThe Phong reflection model is an empirical model of the local illumination of points on a surface...