In
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great deal ...
, the rendering equation is an
integral equation in which the equilibrium
radiance leaving a point is given as the sum of emitted plus reflected radiance under a geometric
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultra ...
approximation. It was simultaneously introduced into computer graphics by David Immel et al. and
James Kajiya in 1986. The various realistic
rendering 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, we have that at each particular position and direction, the outgoing light (L
o) is the sum of the emitted light (L
e) and the reflected light. The reflected light itself is the sum from all directions of the incoming light (L
i) multiplied by the surface reflection and cosine of the incident angle.
Equation form
The rendering equation may be written in the form
:
where
*
is the total
spectral radiance of wavelength
directed outward along direction
at time
, from a particular position
*
is the location in space
*
is the direction of the outgoing light
*
is a particular wavelength of light
*
is time
*
is
emitted spectral radiance
*
is an
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
over
*
is the unit
hemisphere centered around
containing all possible values for
where
*
is the
bidirectional reflectance distribution function, the proportion of light reflected from
to
at position
, time
, and at wavelength
*
is the negative direction of the incoming light
*
is spectral radiance of wavelength
coming inward toward
from direction
at time
*
is the
surface normal
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...
at
*
is the weakening factor of outward
irradiance due to
incident angle, as the light flux is smeared across a surface whose area is larger than the projected area perpendicular to the ray. This is often written as
.
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. It is a
Fredholm integral equation In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to ...
of the second kind, similar to those that arise in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
.
Note this equation's
spectral and
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
dependence —
may be sampled at or integrated over sections of the
visible spectrum 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 is the apparent streaking of moving objects in a photograph or a sequence of frames, such as a film or animation. It results when the image being recorded changes during the recording of a single exposure, due to rapid movement or ...
can be produced by
averaging over some given time interval (by integrating over the time interval and dividing by the length of the interval).
Note that a solution to the rendering equation is the function
. The function
is related to
via a ray-tracing operation: The incoming radiance from some direction at one point is the outgoing radiance at some other point in the opposite direction.
Applications
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
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of struct ...
methods, leading to the
radiosity algorithm. Another approach using
Monte Carlo method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deter ...
s has led to many different algorithms including
path tracing,
photon mapping, and
Metropolis light transport, among others.
Limitations
Although the equation is very general, it does not capture every aspect of light reflection. Some missing aspects include the following:
*
Transmission, which occurs when light is transmitted through the surface, such as when it hits a
glass
Glass is a non-Crystallinity, crystalline, often transparency and translucency, transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most ...
object or a
water
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
surface,
*
Subsurface scattering, where the spatial locations for incoming and departing light are different. Surfaces rendered without accounting for subsurface scattering may appear unnaturally opaque — however, it is not necessary to account for this if transmission is included in the equation, since that will effectively include also light scattered under the surface,
*
Polarization
Polarization or polarisation may refer to:
Mathematics
*Polarization of an Abelian variety, in the mathematics of complex manifolds
*Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
, where different light polarizations will sometimes have different reflection distributions, for example when light bounces at a water surface,
*
Phosphorescence, which occurs when light or other
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible ...
is
absorbed at one moment and emitted at a later moment, usually with a longer
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
(unless the absorbed electromagnetic radiation is very intense),
*
Interference, where the wave properties of light are exhibited,
*
Fluorescence
Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation. It is a form of luminescence. In most cases, the emitted light has a longer wavelength, and therefore a lower photon energy, ...
, where the absorbed and emitted light have different
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
s,
*
Non-linear effects, where very intense light can increase the
energy level of an
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary partic ...
with more energy than that of a single
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
(this can occur if the electron is hit by two photons at the same time), and
emission
Emission may refer to:
Chemical products
* Emission of air pollutants, notably:
**Flue gas, gas exiting to the atmosphere via a flue
** Exhaust gas, flue gas generated by fuel combustion
** Emission of greenhouse gases, which absorb and emit radi ...
of light with higher frequency than the frequency of the light that hit the surface suddenly becomes possible, and
*
Doppler effect
The Doppler effect or Doppler shift (or simply Doppler, when in context) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, ...
, where light that bounces off an object moving at a very high speed will get its wavelength changed: if the light bounces off an object that is moving towards it, the light will be
blueshifted and the
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
s will be packed more closely so the photon flux will be increased; if it bounces off an object moving away from it, it will be
redshifted and the photon flux will be decreased. This effect becomes apparent only at speeds comparable to the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
, which is not the case for most rendering applications.
For scenes that are either not composed of simple surfaces in a vacuum or for which the travel time for light is an important factor, researchers have generalized the rendering equation to produce a ''volume rendering equation'' suitable for
volume rendering and a ''transient rendering equation''
[{{Cite techreport, first=Adam M., last=Smith , author2=Skorupski, James , author3=Davis, James, title=Transient Rendering, number=UCSC-SOE-08-26, institution=UC Santa Cruz, year=2008 , url=http://classes.soe.ucsc.edu/cmps290b/Fall07/TransientRendering/ucsc-soe-08-26.pdf] for use with data from a
time-of-flight camera.
References
External links
Lecture notesfrom Stanford University course CS 348B, ''Computer Graphics: Image Synthesis Techniques''
Computer graphics