The light field is a
vector function
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could b ...
that describes the amount of
light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
flowing in every direction through every point in space. The space of all possible ''
light rays
In optics a ray is an idealized geometrical model of light, obtained by choosing a curve that is perpendicular to the ''wavefronts'' of the actual light, and that points in the direction of energy transfer, energy flow. Rays are used to model t ...
'' is given by the
five-dimensional
A five-dimensional space is a space with five dimensions. In mathematics, a sequence of ''N'' numbers can represent a location in an ''N''-dimensional space. If interpreted physically, that is one more than the usual three spatial dimensions a ...
plenoptic function, and the magnitude of each ray is given by its
radiance
In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiatio ...
.
Michael Faraday
Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic inducti ...
was the first to propose that light should be interpreted as a field, much like the magnetic fields on which he had been working. The phrase ''light field'' was coined by
Andrey Gershun in a classic 1936 paper on the radiometric properties of light in three-dimensional space.
Modern approaches to light field display explore co-designs of optical elements and compressive computation to achieve higher resolutions, increased contrast, wider fields of view, and other benefits.
The term “radiance field” may also be used to refer to similar concepts. The term is used in modern research such as
neural radiance fields.
The plenoptic function
For 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, ultraviole ...
—i.e., to
incoherent light and to objects larger than the wavelength of light—the fundamental carrier of light is a
ray. The measure for the amount of light traveling along a ray is
radiance
In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiatio ...
, denoted by ''L'' and measured in , i.e.,
watt
The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Wa ...
s (W) per
steradian
The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radian ...
(sr) per meter squared (m
2). The steradian is a measure of
solid angle
In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point.
The poi ...
, and meters squared are used as a measure of cross-sectional area, as shown at right.
The radiance along all such rays in a region of three-dimensional space illuminated by an unchanging arrangement of lights is called the plenoptic function. The plenoptic illumination function is an idealized function used in
computer vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the hum ...
and
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 de ...
to express the image of a scene from any possible viewing position at any viewing angle at any point in time. It is not used in practice computationally, but is conceptually useful in understanding other concepts in vision and graphics. Since rays in space can be parameterized by three coordinates, ''x'', ''y'', and ''z'' and two angles ''θ'' and ''ϕ'', as shown at left, it is a five-dimensional function, that is, a function over a five-dimensional
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
equivalent to the product of 3D
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
and the
2-sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
.
The light field at each point in space can be treated as an infinite collection of vectors, one per direction impinging on the point, with lengths proportional to their radiances.
Integrating these vectors over any collection of lights, or over the entire sphere of directions, produces a single scalar value—the total irradiance at that point, and a resultant direction. The figure shows this calculation for the case of two light sources. In computer graphics, this vector-valued function of
3D space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
is called the vector irradiance field. The vector direction at each point in the field can be interpreted as the orientation of a flat surface placed at that point to most brightly illuminate it.
Higher dimensionality
Time,
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, tro ...
, and
polarization angle can be treated as additional dimensions, yielding higher-dimensional functions, accordingly.
The 4D light field
In a plenoptic function, if the region of interest contains a
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
* The concavity
In ca ...
object (e.g., a cupped hand), then light leaving one point on the object may travel only a short distance before another point on the object blocks it. No practical device could measure the function in such a region.
However, for locations outside the object's
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
(e.g., shrink-wrap), the plenoptic function can be measured by capturing multiple images. In this case the function contains redundant information, because the radiance along a ray remains constant throughout its length. The redundant information is exactly one dimension, leaving a four-dimensional function variously termed the photic field, the 4D light field or lumigraph. Formally, the field is defined as radiance along rays in empty space.
The set of rays in a light field can be parameterized in a variety of ways. The most common is the two-plane parameterization. While this parameterization cannot represent all rays, for example rays parallel to the two planes if the planes are parallel to each other, it relates closely to the
analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...
of perspective imaging. A simple way to think about a two-plane light field is as a collection of perspective images of the ''st'' plane (and any objects that may lie astride or beyond it), each taken from an observer position on the ''uv'' plane. A light field parameterized this way is sometimes called a light slab.
Sound analog
The analog of the 4D light field for sound is the sound field or wave field'','' as in
wave field synthesis
Wave field synthesis (WFS) is a spatial audio rendering technique, characterized by creation of virtual acoustic space, acoustic environments. It produces ''artificial'' wavefronts synthesized by a large number of individually driven loudspeake ...
, and the corresponding parametrization is the
Kirchhoff-Helmholtz integral, which states that, in the absence of obstacles, a sound field over time is given by the pressure on a plane. Thus this is two dimensions of information at any point in time, and over time, a 3D field.
This two-dimensionality, compared with the apparent four-dimensionality of light, is because light travels in rays (0D at a point in time, 1D over time), while by the
Huygens–Fresnel principle
The Huygens–Fresnel principle (named after Dutch physicist Christiaan Huygens and French physicist Augustin-Jean Fresnel) states that every point on a wavefront is itself the source of spherical wavelets, and the secondary wavelets emanating ...
, a sound
wave front
In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
can be modeled as spherical waves (2D at a point in time, 3D over time): light moves in a single direction (2D of information), while sound expands in every direction. However, light travelling in non-vacuous media may scatter in a similar fashion, and the irreversibility or information lost in the scattering is discernible in the apparent loss of a system dimension.
Image refocusing
Because light field provides spatial and angular information, we can alter the position of focal planes after exposure, which is often termed ''refocusing''. The principle of refocusing is to obtain conventional 2-D photographs from a light field through the integral transform. The transform takes a lightfield as its input and generates a photograph focused on a specific plane.
Assuming
represents a 4-D light field that records light rays traveling from position
on the first plane to position
on the second plane, where
is the distance between two planes, a 2-D photograph at any depth
can be obtained from the following integral transform:
:
,
or more concisely,
:
,
where
,
, and