Rendering equation

# Rendering equation

Discussion

Encyclopedia
In 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 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 and integral equations, and some problems may be formulated either way...

Radiance and spectral radiance are radiometric measures that describe the amount of radiation such as light or radiant heat that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. They are used to characterize both emission from...

leaving a point is given as the sum of emitted plus reflected radiance under a geometric optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

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 model , by means of computer programs. A scene file contains objects in a strictly defined language or data structure; it would contain geometry, viewpoint, texture, lighting, and shading information as a description of the virtual scene...

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 and spectral radiance are radiometric measures that describe the amount of radiation such as light or radiant heat that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. They are used to characterize both emission from...

, 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.

## Equation form

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 relative ability of its surface to emit energy by radiation. It is the ratio of energy radiated 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 four-dimensional function that defines how light is reflected at an opaque surface...

, 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 prism; it has since been applied by...

and time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as 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 visible to the human eye. Electromagnetic radiation in this range of wavelengths is called visible light or simply light. A typical human eye will respond to wavelengths from about 390 to 750 nm. In terms of...

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 still image or a sequence of images such as a movie or animation. It results when the image being recorded changes during the recording of a single frame, either due to rapid movement or long exposure.- Photography :When a camera...

can be produced by integrating over .

## Limitations

Although the equation is very general, it does not capture every aspect of light reflection. Some missing aspects include the following:
• phosphorescence
Phosphorescence
Phosphorescence is a specific type of photoluminescence related to fluorescence. Unlike fluorescence, a phosphorescent material does not immediately re-emit the radiation it absorbs. The slower time scales of the re-emission are associated with "forbidden" energy state transitions in quantum...

, which occurs when light is absorbed at one moment in time and emitted at a different time,
• fluorescence
Fluorescence
Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation of a different wavelength. It is a form of luminescence. In most cases, emitted light has a longer wavelength, and therefore lower energy, than the absorbed radiation...

, where the absorbed and emitted light have different wavelengths,
• interference, where the wave properties of light are exhibited, and
• subsurface scattering
Subsurface scattering
Subsurface scattering is a mechanism of light transport in which light penetrates the surface of a translucent object, is scattered by interacting with the material, and exits the surface at a different point...

, where the spatial locations for incoming and departing light are different. Surfaces rendered without accounting for subsurface scattering may appear unnaturally opaque.

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 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 in computer simulations of physical and mathematical systems...

s has led to many different algorithms including path tracing
Path Tracing
Path tracing is a computer graphics rendering technique that attempts to simulate the physical behaviour of light as closely as possible. It is a generalisation of conventional ray tracing, tracing rays from the virtual camera through several bounces on or through objects...

, 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. Rays from the light source and rays from the camera are traced independently until some termination criterion is met, then they are connected in a...

, and Metropolis light transport
Metropolis light transport
The Metropolis light transport is a SIGGRAPH 1997 paper by Eric Veach and Leonidas J. Guibas, describing 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...

, among others.

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
Volume rendering
In scientific visualization and computer graphics, volume rendering is a set of techniques used to display a 2D projection of a 3D discretely sampled data set.A typical 3D data set is a group of 2D slice images acquired by aCT, MRI, or MicroCT scanner....

and a transient rendering equation for use with data from a time-of-flight camera
Time-of-flight camera
A time-of-flight camera is a range imaging camera system that resolves distance based on the known speed of light, measuring the time-of-flight of a light signal between the camera and the subject...

.