The light field is a vector function that describes the amount of light flowing in every direction through every point in space. The space of all possible '' light rays'' is given by the five-dimensional 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 radiati ...
. Michael Faraday 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, ultra ...
—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 radiati ...
, 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 W ...
s (W) per steradian (sr) per meter squared (m2). 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 po ...
, 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 human ...
and computer graphics 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 ne ...
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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
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 ...
. 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

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, t ...
, 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 In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously calle ...
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 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, 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 freq ...
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 L_(s,t,u,v) represents a 4-D light field that records light rays traveling from position (u,v) on the first plane to position (s,t) on the second plane, where F is the distance between two planes, a 2-D photograph at any depth \alpha F can be obtained from the following integral transform: :\mathcal_\left _\rights,t)=\iint L_F(u(1-1/\alpha)+s/\alpha,v(1-1/\alpha)+t/\alpha,u,v)~dudv, or more concisely, :\mathcal_\left _\right\boldsymbol)=\frac \int L_\left(\boldsymbol\left(1-\frac\right)+\frac, \boldsymbol\right) d \boldsymbol, where \boldsymbol=(s,t), \boldsymbol=(u,v), and \mathcal_\left cdot\right/math> is the photography operator. In practice, this formula cannot be directly used because a plenoptic camera usually captures discrete samples of the lightfield L_(s,t,u,v), and hence resampling (or interpolation) is needed to compute L_\left(\boldsymbol\left(1-\frac\right)+\frac, \boldsymbol\right). Another problem is high computation complexity. To compute an N\times N 2-D photograph from an N\times N\times N\times N 4-D light field, the complexity of the formula is O(N^4).

Fourier slice photography

One way to reduce the complexity of computation is to adopt the concept of
Fourier slice theorem Fourier may refer to: People named Fourier *Joseph Fourier (1768–1830), French mathematician and physicist *Charles Fourier (1772–1837), French utopian socialist thinker * Peter Fourier (1565–1640), French saint in the Roman Catholic Church ...
: The photography operator \mathcal_\left cdot\right/math> can be viewed as a shear followed by projection. The result should be proportional to a dilated 2-D slice of the 4-D Fourier transform of a light field. More precisely, a refocused image can be generated from the 4-D Fourier spectrum of a light field by extracting a 2-D slice, applying an inverse 2-D transform, and scaling. The asymptotic complexity of the algorithm is O(N^2 \operatornameN).

Discrete focal stack transform

Another way to efficiently compute 2-D photographs is to adopt discrete focal stack transform (DFST). DFST is designed to generate a collection of refocused 2-D photographs, or so-called Focal Stack. This method can be implemeted by fast fractional fourier transform (FrFT). The discrete photography operator \mathcal_\left cdot\right/math> is defined as follows for a lightfield L_(\boldsymbol ,\boldsymbol ) sampled in a 4-D grid \boldsymbol = \Delta s \tilde, \tilde=-\boldsymbol _,...,\boldsymbol _, \boldsymbol = \Delta u \tilde, \tilde=-\boldsymbol _,...,\boldsymbol _: :\mathcal_ \boldsymbol)= \sum_^ L(\boldsymbol q+\boldsymbol, \boldsymbol) \Delta \boldsymbol, \quad \Delta \boldsymbol=\Delta u\Delta v, \quad q=\left(1-\frac\right) Because (\boldsymbol q+\boldsymbol, \boldsymbol) is usually not on the 4-D grid, DFST adopts
trigonometric interpolation In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes through some given data points. For trigonometric interpolation, this function has to be a ...
to compute the non-grid values. The algorithm consists of these steps: * Sample the light field L_(\boldsymbol ,\boldsymbol ) with the sampling period \Delta s and \Delta u and get the discretized light field L^d_(\boldsymbol ,\boldsymbol ). * Pad L^d_(\boldsymbol ,\boldsymbol ) with zeros such that the signal length is enough for FrFT without aliasing. * For every \boldsymbol , compute the Discrete Fourier transform of L^d_(\boldsymbol ,\boldsymbol ), and get the result R1. * For every focal length \alpha F, compute the fractional fourier transform of R1, where the order of the transform depends on \alpha, and get the result R2. * Compute the inverse Discrete Fourier transform of R2. * Remove the marginal pixels of R2 so that each 2-D photograph has the size (2_+1) \times (2_+1)

Methods to create light fields

Light fields are a fundamental representation for light with many methods for defining them. In computer graphics, light fields are typically produced either by rendering a
3D model In 3D computer graphics, 3D modeling is the process of developing a mathematical coordinate-based representation of any surface of an object (inanimate or living) in three dimensions via specialized software by manipulating edges, vertices, a ...
or by photographing a real scene. In either case, to produce a light field, views must be obtained for a large collection of viewpoints. Depending on the parameterization, this collection typically spans some portion of a line, circle, plane, sphere, or other shape, although unstructured collections are possible. Devices for capturing light fields photographically may include a moving handheld camera or a robotically controlled camera, an arc of cameras (as in the bullet time effect used in '' The Matrix''), a dense array of cameras, handheld cameras, Ng 2005 microscopes, or other optical system. How many images should be in a light field? The largest known light field (of Michelangelo's statue of Night) contains 24,000 1.3-megapixel images. At a deeper level, the answer depends on the application. For light field rendering to completely capture an opaque object, images must be taken of at least the front and back. Less obviously, for an object that lies astride the ''st'' plane, finely spaced images must be taken on the ''uv'' plane (in the two-plane parameterization shown above). The number and arrangement of images in a light field, and the resolution of each image, are together called the "sampling" of the 4D light field. Also of interest are the effects of occlusion, lighting and reflection.


Selected applications: * Illumination engineering–Gershun's reason for studying the light field was to derive (in closed form) illumination patterns that would be observed on surfaces due to light sources of various shapes positioned above these surface. The branch of optics devoted to illumination engineering is nonimaging optics. It extensively uses the concept of flow lines (Gershun's flux lines) and vector flux (Gershun's light vector). However, the light field (in this case the positions and directions defining the light rays) is commonly described in terms of
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
Hamiltonian optics Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrica ...
. * Light field rendering–Extracting appropriate 2D slices from the 4D light field of a scene, enables novel views of the scene. Depending on the parameterization of the light field and slices, these views might be perspective, orthographic, crossed-slit, general linear cameras, multi-perspective, or another type of projection. Light field rendering is one form of image-based rendering. * Synthetic aperture photography–Integrating an appropriate 4D subset of the samples in a light field can approximate the view that would be captured by a camera having a finite (i.e., non-
pinhole A hole is an opening in or through a particular medium, usually a solid body. Holes occur through natural and artificial processes, and may be useful for various purposes, or may represent a problem needing to be addressed in many fields of en ...
) aperture. Such a view has a finite
depth of field The depth of field (DOF) is the distance between the nearest and the furthest objects that are in acceptably sharp focus in an image captured with a camera. Factors affecting depth of field For cameras that can only focus on one object dist ...
. Shearing or warping the light field before performing this integration can focus on different fronto-parallel or oblique planes. Images captured by digital cameras that capture the light field can be refocused. * 3D display–Presenting a light field using technology that maps each sample to the appropriate ray in physical space produces an
autostereoscopic Autostereoscopy is any method of displaying stereoscopic images (adding binocular perception of 3D depth) without the use of special headgear, glasses, something that affects vision, or anything for eyes on the part of the viewer. Because head ...
visual effect akin to viewing the original scene. Non-digital technologies for doing this include integral photography, parallax panoramagrams, and
holography Holography is a technique that enables a wavefront to be recorded and later re-constructed. Holography is best known as a method of generating real three-dimensional images, but it also has a wide range of other applications. In principle, it ...
; digital technologies include placing an array of lenslets over a high-resolution display screen, or projecting the imagery onto an array of lenslets using an array of video projectors. An array of video cameras can capture and display a time-varying light field. This essentially constitutes a
3D television 3D television (3DTV) is television that conveys depth perception to the viewer by employing techniques such as stereoscopic display, multi-view display, 2D-plus-depth, or any other form of 3D display. Most modern 3D television sets use an ...
system. * Brain imaging–Neural activity can be recorded optically by genetically encoding neurons with reversible fluorescent markers such as GCaMP that indicate the presence of
calcium ions Calcium ions (Ca2+) contribute to the physiology and biochemistry of organisms' cells. They play an important role in signal transduction pathways, where they act as a second messenger, in neurotransmitter release from neurons, in contraction ...
in real time. Since light field microscopy captures full volume information in a single frame, it is possible to monitor neural activity in individual neurons randomly distributed in a large volume at video framerate. Quantitative measurement of neural activity can be done despite optical aberrations in brain tissue and without reconstructing a volume image, and be used to monitor activity in thousands of neurons. * Generalized Scene Reconstruction (GSR)–This is a method of 3D reconstruction from multiple images that creates a scene model representing a generalized light field and a relightable matter field.Leffingwell, 2018 The light field represents light flowing in every direction through every point in the scene. The matter field represents the light interaction properties of matter occupying every point in the scene. GSR can be performed using Neural Radiance Fields (NeRFs), Plenoxels and Inverse Light Transport. * Holographic stereograms–Image generation and predistortion of synthetic imagery for holographic stereograms is one of the earliest examples of computed light fields. * Glare reduction– Glare arises due to multiple scattering of light inside the camera body and lens optics that reduces image contrast. While glare has been analyzed in 2D image space,Talvala 2007 it is useful to identify it as a 4D ray-space phenomenon.Raskar 2008 Statistically analyzing the ray-space inside a camera allows the classification and removal of glare artifacts. In ray-space, glare behaves as high frequency noise and can be reduced by outlier rejection. Such analysis can be performed by capturing the light field inside the camera, but it results in the loss of spatial resolution. Uniform and non-uniform ray sampling can be used to reduce glare without significantly compromising image resolution.

See also

* Light-field camera * Angle–sensitive pixel * Lytro * Reflectance paper * Raytrix * Dual photography




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Light field cameras

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Stanford Spherical Gantry
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Light field displays

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Light field archives

"The Stanford Light Field Archive"

"UCSD/MERL Light Field Repository"

"The HCI Light Field Benchmark"

"Synthetic Light Field Archive"


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