Light field

A light field, or lightfield, is a physical field that describes the amount of light flowing in every direction through every point in a three-dimensional space. The mathematical space of all possible light rays is given by the five-dimensional plenoptic function (with three position coordinates and two direction angles as arguments), 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. The term light field was coined by Andrey Gershun in a classic 1936 paper on the radiometric properties of light in three-dimensional space. The term "radiance field" may also be used to refer to similar, or identical concepts. The term is used in modern research such as neural radiance fields.

Formulation

For geometric optics—i.e., in the regime of incoherent light and 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, denoted by L and expressed in units of W·sr−1·m−2; i.e., watts (W) per steradian (sr) per square meter (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.

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 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 equivalent to the product of 3D Euclidean space and the 2-sphere.

Irradiance field

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 vector-valued function of 3D space 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. The vector magnitude is a scalar-valued function of 3D space, called the irradiance.

Higher dimensionality Time, wavelength, 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 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 (conceptually, the shape a shrink-wrap foil around it would become), 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, 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.

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