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Rendering equation
 
 
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In computer graphics
Computer graphics

Computer graphics are graphics created by computers and, more generally, the representation and manipulation of pictorial data by a computer....
, the rendering equation is an integral equation
Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential equation and integral equations, and some problems may be formulated either way....
 in which the equilibrium radiance
Radiance

Radiance and spectral radiance are radiometry measures that describe the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction....
 leaving a point is given as the sum of emitted plus reflected radiance under a geometric optics
Optics

Optics is the study of the behavior and properties of light including its optical phenomena with matter and its imaging by optical instruments....
 approximation. It was simultaneously introduced into computer graphics by David Immel et al. and James Kajiya in 1986. The various realistic rendering
Rendering (computer graphics)

Rendering is the process of generating an image from a 3D model, by means of computer programs. The model is a description of three-dimensional objects in a strictly defined language or data structure....
 techniques in computer graphics attempt to solve this equation.

The physical basis for the rendering equation is the law of conservation of energy.






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In computer graphics
Computer graphics

Computer graphics are graphics created by computers and, more generally, the representation and manipulation of pictorial data by a computer....
, the rendering equation is an integral equation
Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential equation and integral equations, and some problems may be formulated either way....
 in which the equilibrium radiance
Radiance

Radiance and spectral radiance are radiometry measures that describe the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction....
 leaving a point is given as the sum of emitted plus reflected radiance under a geometric optics
Optics

Optics is the study of the behavior and properties of light including its optical phenomena with matter and its imaging by optical instruments....
 approximation. It was simultaneously introduced into computer graphics by David Immel et al. and James Kajiya in 1986. The various realistic rendering
Rendering (computer graphics)

Rendering is the process of generating an image from a 3D model, by means of computer programs. The model is a description of three-dimensional objects in a strictly defined language or data structure....
 techniques in computer graphics attempt to solve this equation.

The physical basis for the rendering equation is the law of conservation of energy. Assuming that L denotes radiance
Radiance

Radiance and spectral radiance are radiometry measures that describe the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction....
, we have that at each particular position and direction, the outgoing light (Lo) is the sum of the emitted light (Le) and the reflected light. The reflected light itself is the sum of the incoming light (Li) from all directions, multiplied by the surface reflection and cosine of the incident angle.

The rendering equation may be written in the form



where
  • is a particular wavelength of light
  • is time
  • is the total amount of light of wavelength directed outward along direction at time , from a particular position
  • is emitted
    Emissivity

    The emissivity of a material is the ratio of energy Radiation by a particular material to energy radiated by a black body at the same temperature....
     light
  • is an integral over a hemisphere of inward directions
  • is the bidirectional reflectance distribution function
    Bidirectional reflectance distribution function

    The bidirectional reflectance distribution function is a 4-dimensional function that defines how light is reflected at an opaque surface. The function takes an incoming light direction, , and outgoing direction, , both defined with respect to the surface normal , and returns the ratio of reflected radiance exiting along to the irradiance...
    , the proportion of light reflected from to at position , time , and at wavelength
  • is light of wavelength coming inward toward from direction at time
  • is the attenuation of inward light due to incident angle


Two noteworthy features are: its linearity—it is composed only of multiplications and additions, and its spatial homogeneity—it is the same in all positions and orientations. These mean a wide range of factorings and rearrangements of the equation are possible.

Note this equation's spectral
Spectrum

A spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a Continuum . The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a triangular prism ; it has since been applied by analogy to many fields other than op...
 and time
Time

Time is a component of the measurement used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects....
 dependence— may be sampled at or integrated over sections of the visible spectrum
Visible spectrum

The visible spectrum is the portion of the electromagnetic spectrum that is visual perception to the human eye. Electromagnetic radiation in this range of wavelengths is called visible light or simply light....
 to obtain, for example, a trichromatic color sample. A pixel value for a single frame in an animation may be obtained by fixing ; motion blur
Motion blur

Motion blur is the apparent streaking of rapidly moving objects in a Photography or a sequence of images such as a film or animation....
 can be produced by integrating over .

Solving the rendering equation for any given scene is the primary challenge in realistic rendering. One approach to solving the equation is based on finite element methods, leading to the radiosity
Radiosity

Radiosity is a global illumination algorithm used in 3D computer graphics rendering . Radiosity is an application of the finite element method to solving the rendering equation for scenes with purely diffuse surfaces....
 algorithm. Another approach using Monte Carlo method
Monte Carlo method

Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used when computer simulation physics and mathematics systems....
s has led to many different algorithms including path tracing
Path Tracing

Path tracing is a photorealistic computer graphics rendering technique by James Kajiya when he presented his paper on the rendering equation in the 1980s....
, photon mapping
Photon mapping

In computer graphics, photon mapping is a two-pass global illumination algorithm developed by Henrik Wann Jensen that solves the rendering equation....
, and Metropolis light transport
Metropolis light transport

This SIGGRAPH 1997 paper by Eric Veach and Leonidas J. Guibas describes an application of a variant of the Monte Carlo method called the Metropolis-Hastings algorithm to the rendering equation for generating images from detailed physical descriptions of three dimensional scenes....
, among others.

External links

  • from Stanford University course CS 348B, Computer Graphics: Image Synthesis Techniques