Light field

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. 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.[1] 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.[2]

The plenoptic function

Radiance L along a ray can be thought of as the amount of light traveling along all possible straight lines through a tube whose size is determined by its solid angle and cross-sectional area.

For geometric optics—i.e., to incoherent light and to objects larger than the wavelength of light—then the fundamental carrier of light is a ray. The measure for the amount of light traveling along a ray is radiance, denoted by L and measured in watts (W) per steradian (sr) per meter squared (m2). The steradian is a measure of solid angle, and meters squared are used as a measure of cross-sectional area, as shown at right.

Parameterizing a ray in 3D space by position (x, y, z) and direction (θ, ϕ).

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.[3] The plenoptic illumination function is an idealized function used in computer vision 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.[4] 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 equivalent to the product of 3D Euclidean space and the 2-sphere.

Summing the irradiance vectors D1 and D2 arising from two light sources I1 and I2 produces a resultant vector D having the magnitude and direction shown [5]

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 is called the vector irradiance field.[6] 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, and polarization angle can be treated as additional dimensions, yielding higher-dimensional functions, accordingly.

The 4D light field

Radiance along a ray remains constant if there are no blockers.

In a plenoptic function, if the region of interest contains a concave 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 (e.g., shrink-wrap), then 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[7] or Lumigraph.[8] 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.

Some alternative parameterizations of the 4D light field, which represents the flow of light through an empty region of three-dimensional space. Left: points on a plane or curved surface and directions leaving each point. Center: pairs of points on the surface of a sphere. Right: pairs of points on two planes in general (meaning any) position.

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, a sound wave front 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 principal of refocusing is to obtaining 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 determined plane.

Assuming ${\displaystyle L_{F}(s,t,u,v)}$ represents a 4-D light field that records light rays traveling from position ${\displaystyle (u,v)}$ on the first plane to position ${\displaystyle (s,t)}$ on the second plane, where ${\displaystyle F}$ is the distance between two planes, a 2-D photograph at any depth ${\displaystyle \alpha F}$ can be obtained from the following integral transform:[9]

${\displaystyle {\mathcal {P}}_{\alpha }\left[L_{F}\right](s,t)={1 \over \alpha ^{2}F^{2}}\iint L_{F}(u(1-1/\alpha )+s/\alpha ,v(1-1/\alpha )+t/\alpha ,u,v)~dudv}$,

or more concisely,

${\displaystyle {\mathcal {P}}_{\alpha }\left[L_{F}\right]({\boldsymbol {s}})={\frac {1}{\alpha ^{2}F^{2}}}\int L_{F}\left({\boldsymbol {u}}\left(1-{\frac {1}{\alpha }}\right)+{\frac {\boldsymbol {s}}{\alpha }},{\boldsymbol {u}}\right)d{\boldsymbol {u}}}$,

where ${\displaystyle {\boldsymbol {s}}=(s,t)}$, ${\displaystyle {\boldsymbol {u}}=(u,v)}$, and ${\displaystyle {\mathcal {P}}_{\alpha }\left[\cdot \right]}$ is the Photography operator.

In practice, this formula cannot be directly used because a plenoptic camera usually captures discrete samples of the lightfield ${\displaystyle L_{F}(s,t,u,v)}$, and hence resampling (or interpolation) is needed to compute ${\displaystyle L_{F}\left({\boldsymbol {u}}\left(1-{\frac {1}{\alpha }}\right)+{\frac {\boldsymbol {s}}{\alpha }},{\boldsymbol {u}}\right)}$. Another problem is high computation complexity. To compute an ${\displaystyle N\times N}$ 2-D photograph from an ${\displaystyle N\times N\times N\times N}$ 4-D light field, the complexity of the formula is ${\displaystyle O(N^{4})}$.[9]

Fourier slice photography

One way to reduce the complexity of computation is to adopt the concept of Fourier slice theorem:[9] The photography operator ${\displaystyle {\mathcal {P}}_{\alpha }\left[\cdot \right]}$ 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 ${\displaystyle O(N^{2}\operatorname {log} N)}$.

Discrete focal stack transform

Another way to efficiently compute 2-D photographs is to adopt discrete focal stack transform (DFST).[10] More precisely, 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 ${\displaystyle {\mathcal {P}}_{\alpha }\left[\cdot \right]}$ is defined as follows for a lightfield ${\displaystyle L_{F}({\boldsymbol {s}},{\boldsymbol {u}})}$ sampled in a 4-D grid ${\displaystyle {\boldsymbol {s}}=\Delta s{\tilde {\boldsymbol {s}}},{\tilde {\boldsymbol {s}}}=-{\boldsymbol {n}}_{\boldsymbol {s}},...,{\boldsymbol {n}}_{\boldsymbol {s}}}$, ${\displaystyle {\boldsymbol {u}}=\Delta u{\tilde {\boldsymbol {u}}},{\tilde {\boldsymbol {u}}}=-{\boldsymbol {n}}_{\boldsymbol {u}},...,{\boldsymbol {n}}_{\boldsymbol {u}}}$:

${\displaystyle {\mathcal {P}}_{q}[L]({\boldsymbol {s}})=\sum _{{\tilde {\boldsymbol {u}}}=-{\boldsymbol {n}}_{\boldsymbol {u}}}^{{\boldsymbol {n}}_{\boldsymbol {u}}}L({\boldsymbol {u}}q+{\boldsymbol {s}},{\boldsymbol {u}})\Delta {\boldsymbol {u}},\quad \Delta {\boldsymbol {u}}=\Delta u\Delta v,\quad q=\left(1-{\frac {1}{\alpha }}\right)}$

Because ${\displaystyle ({\boldsymbol {u}}q+{\boldsymbol {s}},{\boldsymbol {u}})}$ is usually not on the 4-D grid, DFST adopts trigonometric interpolation to compute the non-grid values.

The whole algorithm consists of these steps:

• Sample the light field ${\displaystyle L_{F}({\boldsymbol {s}},{\boldsymbol {u}})}$ with the sampling period ${\displaystyle \Delta s}$ and ${\displaystyle \Delta u}$ and get the discretized light field ${\displaystyle L_{F}^{d}({\boldsymbol {s}},{\boldsymbol {u}})}$.
• Padding ${\displaystyle L_{F}^{d}({\boldsymbol {s}},{\boldsymbol {u}})}$ with zeros such that the signal length is enough for FrFT without aliasing.
• For every ${\displaystyle {\boldsymbol {u}}}$, compute the Discrete Fourier transform of ${\displaystyle L_{F}^{d}({\boldsymbol {s}},{\boldsymbol {u}})}$, and get the result ${\displaystyle R1}$.
• For every focal length ${\displaystyle \alpha F}$, compute the fractional fourier transform of ${\displaystyle R1}$, where the order of the transform depends on ${\displaystyle \alpha }$, and get the result ${\displaystyle R2}$.
• Compute the inverse Discrete Fourier transform of ${\displaystyle R2}$.
• Remove the marginal pixels of ${\displaystyle R2}$ so that each 2-D photograph has the size ${\displaystyle (2{n}_{\boldsymbol {s}}+1)\times (2{n}_{\boldsymbol {s}}+1)}$

Methods to create light fields

There are many 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 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 of viewpoints are also possible.[11]

Devices for capturing light fields photographically may include a moving handheld camera or a robotically controlled camera,[12] an arc of cameras (as in the bullet time effect used in The Matrix), a dense array of cameras,[13] handheld cameras,[14][15] microscopes,[16] or other optical system.[17]

How many images should be in a light field? The largest known light field (of Michelangelo's statue of Night)[18] 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 take 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. Analyses of light field sampling have been undertaken by many researchers.[19] Also of interest are the effects of occlusion,[20] the effects of lighting and reflection,[21] and applications to plenoptic cameras[22] and 3D displays.[23]

Applications

Computational imaging refers to any image formation method that involves a computer. Many of these methods operate at visible wavelengths, and many of those produce light fields.Some selected applications are:

A downward-facing light source (F-F') induces a light field whose irradiance vectors curve outwards. Using calculus, Gershun could compute the irradiance falling on points (P1, P2) on a surface. [24])
• Illumination engineering–Gershun's reason for studying the light field was to derive (in closed form if possible) the illumination patterns that would be observed on surfaces due to light sources of various shapes positioned above these surface.[25]
The branch of optics devoted to illumination engineering is nonimaging optics.[26] 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 and Hamiltonian optics.
• Light field rendering–By extracting appropriate 2D slices from the 4D light field of a scene, one can produce novel views of the scene .[27] Depending on the parameterization of the light field and slices, these views might be perspective, orthographic, crossed-slit,[28] general linear cameras,[29] multi-perspective,[30] 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) aperture. Such a view has a finite depth of field. Shearing or warping the light field before performing this integration can focus on different fronto-parallel[31] or oblique[32] planes in the scene. Images captured by digital cameras able to capture the light field,[14] 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 visual effect akin to viewing the original scene. Non-digital technologies for doing this include integral photography, parallax panoramagrams, and holography; 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 system.[33]
• Brain imaging–Neural activity can be recorded optically by genetically encoding neurons with reversible fluorescent markers e.g. GCaMP, that indicate the presence of calcium ions in real time. Since light field microscopy captures full volume information in a single frame, it is possible to monitor neural activity in many individual neurons randomly distributed in a large volume at video framerate.[34] Quantitative measurement of neural activity can be done despite optical aberrations in brain tissue and without reconstructing a volume image,[35] and be used to monitor activity in thousands of mammalian neurons.[36]
• 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.[37] 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),[38] Plenoxels[39] and Inverse Light Transport.[40]
• Holographic stereograms–Image generation and predistortion of synthetic imagery for holographic stereograms is one of the earliest examples of computed light fields.[41]
• 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,[42] it is useful to identify it as a 4D ray-space phenomenon.[43] 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.[43]

Notes

1. ^ Faraday, Michael (30 April 2009). "LIV. Thoughts on ray-vibrations". Philosophical Magazine. Series 3. 28 (188): 345–350. doi:10.1080/14786444608645431. Archived from the original on 2013-02-18.
2. ^ Wetzstein 2012, 2011; Lanman 2011, 2010
4. ^ Wong 2002
5. ^ Gershun, fig 17
6. ^ Arvo, 1994
7. ^ Levoy 1996
8. ^ Gortler 1996
9. ^ a b c Ng, Ren (2005). "Fourier slice photography". ACM SIGGRAPH 2005 Papers on - SIGGRAPH '05. New York, New York, USA: ACM Press. doi:10.1145/1186822.1073256.
10. ^ Nava, F. Pérez; Marichal-Hernández, J.G.; Rodríguez-Ramos, J.M. (August 2008). "The Discrete Focal Stack Transform". 2008 16th European Signal Processing Conference: 1–5.
11. ^ Buehler 2001
12. ^ Levoy 2002
13. ^ Kanade 1998; Yang 2002; Wilburn 2005
14. ^ a b Ng 2005
15. ^ Georgiev 2006; Marwah 2013
16. ^ Levoy 2006
17. ^ Bolles 1987
18. ^ "A light field of Michelangelo's statue of Night". accademia.stanford.edu. Retrieved 2022-02-08.
19. ^ Chai (2000)
20. ^ Durand (2005)
21. ^ Ramamoorthi (2006)
22. ^ Ng (2005)
23. ^ Zwicker (2006)
24. ^ Gershun, fig 24
25. ^ Ashdown 1993
26. ^ Chaves 2015; Winston 2005
27. ^ Levoy 1996; Gortler 1996
28. ^ Zomet 2003
29. ^ Yu and McMillan 2004
31. ^ Isaksen 2000
32. ^ Vaish 2005
33. ^ Javidi 2002; Matusik 2004
34. ^ Grosenick, 2009, 2017; Perez, 2015
35. ^ Pegard, 2016
36. ^ Grosenick, 2017
37. ^ Leffingwell, 2018
38. ^ Mildenhall, 2020
39. ^ Yu & Fridovich-Keil, 2021
40. ^ Leffingwell, 2018
41. ^ Halle 1991, 1994
42. ^ Talvala 2007
43. ^ a b Raskar 2008