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 the 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 for several years. The phrase light field was coined by Andrey Gershun in a classic paper on the radiometric properties of light in three-dimensional space.

The 5D plenoptic function

If the concept is restricted to 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 ' per steradian ' per meter squared . The steradian is a measure of solid angle, and meters squared are used here 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 never actually 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.
Like Adelson, Gershun defined the light field at each point in space as a 5D function. However, he treated it 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 at right, reproduced from Gershun's paper, 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. The vector direction at each point in the field can be interpreted as the orientation one would face a flat surface placed at that point to most brightly illuminate it.

Higher dimensionality

One can consider time, wavelength, and polarization angle as additional variables, yielding higher-dimensional functions.

The 4D light field

In a plenoptic function, if the region of interest contains a concave object, then light leaving one point on the object may travel only a short distance before being blocked by another point on the object. No practical device could measure the function in such a region.
However, if we restrict ourselves to locations outside the convex hull of the object, i.e. in free space, then we can measure the plenoptic function by taking many photos using a digital camera. Moreover, in this case the function contains redundant information, because the radiance along a ray remains constant from point to point along its length, as shown at left. In fact, the redundant information is exactly one dimension, leaving us with a four-dimensional function. Parry Moon dubbed this function the photic field, while researchers in computer graphics call it the 4D light field or Lumigraph. Formally, the 4D light 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, a few of which are shown below. Of these, the most common is the two-plane parameterization shown at right. While this parameterization cannot represent all rays, for example rays parallel to the two planes if the planes are parallel to each other, it has the advantage of relating closely to the analytic geometry of perspective imaging. Indeed, a simple way to think about a two-plane light field is as a collection of perspective images of the st plane, 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, while by Huygens–Fresnel principle, a sound wave front can be modeled as spherical waves : light moves in a single direction, while sound simply 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.

Ways to create light fields

Light fields are a fundamental representation for light. As such, there are as many ways of creating light fields as there are computer programs capable of creating images or instruments capable of capturing 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 employed, this collection will typically span some portion of a line, circle, plane, sphere, or other shape, although unstructured collections of viewpoints are also possible.
Devices for capturing light fields photographically may include a moving handheld camera or a robotically controlled camera, an arc of cameras, a dense array of cameras, handheld cameras, microscopes, or other optical system.
How many images should be in a light field? The largest known light field contains 24,000 1.3-megapixel images. At a deeper level, the answer depends on the application. For light field rendering, if you want to walk completely around an opaque object, then of course you need to photograph its back side. Less obviously, if you want to walk close to the object, and the object lies astride the st plane, then you need images taken at finely spaced positions on the uv plane, which is now behind you, and these images need to have high spatial resolution.
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; a good starting point is Chai. Also of interest is Durand for the effects of occlusion, Ramamoorthi for the effects of lighting and reflection, and Ng and Zwicker for applications to plenoptic cameras and 3D displays, respectively.

Applications

Computational imaging refers to any image formation method that involves a digital computer. Many of these methods operate at visible wavelengths, and many of those produce light fields. As a result, listing all applications of light fields would require surveying all uses of computational imaging in art, science, engineering, and medicine. In computer graphics, some selected applications are:
Image generation and predistortion of synthetic imagery for holographic stereograms is one of the earliest examples of computed light fields, anticipating and later motivating the geometry used in Levoy and Hanrahan's work.
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.
184026/http://adaptiveimagesynthesis.com/ "Adaptive Image Synthesis for Compressive Displays"], ACM Transactions on Graphics