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Light field
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The light field is a function that describes the amount of light traveling in every direction through every point in space. Michael Faraday was the first to propose (in an 1846 lecture entitled "Thoughts on Ray Vibrations") 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 Alexander Gershun in a classic paper on the radiometric properties of light in three-dimensional space (1936).
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The light field is a function that describes the amount of light traveling in every direction through every point in space. Michael Faraday was the first to propose (in an 1846 lecture entitled "Thoughts on Ray Vibrations") 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 Alexander Gershun in a classic paper on the radiometric properties of light in three-dimensional space (1936). The phrase has been redefined by researchers in computer graphics to mean something slightly different. To understand this difference, we'll need a bit of terminology.
The 5D plenoptic function
If we restrict ourselves 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 (W) per steradian (sr) per meter squared (m2). 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 (Adelson 1991). 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, and is more 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 and , as shown at left, it is a five-dimensional function. (One can consider time, wavelength, and polarization angle as additional variables, yielding higher-dimensional functions.)
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. Equivalently, one can imagine an infinite collection of infinitesimal surfaces placed at that point, one per direction, with different values of irradiance assigned to each surface.
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 (Arvo, 1994). 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.
The 4D light field
In a plenoptic function, if the region of interest contains a concave object (think of a cupped hand), 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 (think shrink-wrap) of the object, then we can measure the plenoptic function easily 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 (1981), while researchers in computer graphics call it the 4D light field (Levoy 1996) or Lumigraph (Gortler 1996). 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 (below). 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 (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.
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 (Buehler 2001).
Devices for capturing light fields photographically may include a moving handheld camera, a robotically controlled camera (Levoy, 2002) an arc of
cameras (as in the bullet time effect used in The Matrix), a dense array of cameras (Kanade 1998; Yang 2002; Wilburn 2005), or a handheld camera (Ng 2005; Georgiev 2006), microscope (Levoy 2006), or other optical system in which an array of microlenses has been inserted in the optical path: see plenoptic camera. Some public domain archives of light field datasets are listed below.
How many images should be in a light field? The largest known light field (of ) contains 24,000 1.3-megapixel images. At a deeper level, the answer depends on the application. For light field
rendering (see the Application section below), 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 (in the two-plane parameterization shown above), 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 (2000). Also of interest is Durand (2005) for the effects of occlusion, Ramamoorthi (2006) for the effects of lighting and reflection, and Ng (2005) and Zwicker (2006) for applications to plenoptic cameras and 3D displays, respectively.
Applications of light fields
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:
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. An example is shown at right. A more modern study is (Ashdown 1993).
Light field rendering. By extracting appropriate 2D slices from the 4D light field of a scene, one can produce novel views of the scene (Levoy 1996; Gortler 1996). Depending on the parameterization of the light field and slices, these views might be perspective, orthographic, crossed-slit (Zomet 2003), multi-perspective (Rademacher 1998), or another type of projection. Light field rendering is one form of image-based rendering.
Synthetic aperture photography. By integrating an appropriate 4D subset of the samples in a light field, one 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. By shearing or warping the light field before performing this integration, one can focus on different fronto-parallel (Isaksen 2000) or oblique (Vaish 2005) planes in the scene. If the light field is captured using a handheld camera (Ng 2005), this essentially constitutes a digital camera whose photographs can be refocused after they are taken.
3D display. By presenting a light field using technology that maps each sample to the appropriate ray in physical space, one obtains 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. If the latter is combined with an array of video cameras, one can capture and display a time-varying light field. This essentially constitutes a 3D television system (Javidi 2002; Matusik 2004).
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 (Halle 1991, 1994).
Glare reduction. Glare arises due to multiple scattering of light inside the camera’s body and lens optics and 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). By statistically analyzing the ray-space inside a camera, one can classify and remove 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 could be used to reduce glare without significantly compromising image resolution (Raskar 2008).
Theory
Adelson, E.H., Bergen, J.R. (1991).
,
In Computation Models of Visual Processing,
M. Landy and J.A. Movshon, eds., MIT Press, Cambridge, 1991, pp. 3-20.
Arvo, J. (1994).
,
Proc. ACM SIGGRAPH,
ACM Press, pp. 335-342.
Faraday, M.,
,
Philosophical Magazine,
S.3, Vol XXVIII, N188, May 1846.
Gershun, A. (1936).
"The Light Field",
Moscow, 1936. Translated by P. Moon and G. Timoshenko in
Journal of Mathematics and Physics,
Vol. XVIII, MIT, 1939, pp. 51-151.
Gortler, S.J., Grzeszczuk, R., Szeliski, R., Cohen, M. (1996).
,
Proc. ACM SIGGRAPH,
ACM Press, pp. 43-54.
Levoy, M., Hanrahan, P. (1996).
,
Proc. ACM SIGGRAPH,
ACM Press, pp. 31-42.
Moon, P., Spencer, D.E. (1981).
The Photic Field,
MIT Press.
Analysis
Ramamoorthi, R., Mahajan, D., Belhumeur, P. (2006).
,
ACM TOG.
Zwicker, M., Matusik, W., Durand, F., Pfister, H. (2006).
,
Eurographics Symposium on Rendering, 2006.
Ng, R. (2005).
,
Proc. ACM SIGGRAPH, ACM Press, pp. 735-744.
Durand, F., Holzschuch, N., Soler, C., Chan, E., Sillion, F. X. (2005).
,
Proc. ACM SIGGRAPH, ACM Press, pp. 1115-1126.
Chai, J.-X., Tong, X., Chan, S.-C., Shum, H. (2000).
,
Proc. ACM SIGGRAPH, ACM Press, pp. 307-318.
Halle, M. (1994)
,
in SPIE Proc. Vol. #2176: Practical Holography VIII, S.A. Benton, ed., pp. 73-84.
Devices
Liang, C.K., Lin, T.H., Wong, B.Y., Liu, C., Chen, H. H. (2008).
, Proc. ACM SIGGRAPH.
Veeraraghavan, A., Raskar, R., Agrawal, A., Mohan, A., Tumblin, J. (2007).
, Proc. ACM SIGGRAPH.
Georgiev, T., Zheng, C., Nayar, S., Curless, B., Salesin, D., Intwala, C. (2006).
, Proc. EGSR 2006.
Kanade, T., Saito, H., Vedula, S. (1998).
, Tech report CMU-RI-TR-98-34, December 1998.
Levoy, M. (2002).
.
Levoy, M., Ng, R., Adams, A., Footer, M., Horowitz, M. (2006).
,
ACM Transactions on Graphics (Proc. SIGGRAPH),
Vol. 25, No. 3.
Ng, R., Levoy, M., Brédif, M., Duval, G., Horowitz, M., Hanrahan, P. (2005).
, Stanford Tech Report CTSR 2005-02, April, 2005.
Wilburn, B., Joshi, N., Vaish, V., Talvala, E., Antunez, E., Barth, A., Adams, A., Levoy, M., Horowitz, M. (2005).
,
ACM Transactions on Graphics (Proc. SIGGRAPH),
Vol. 24, No. 3, pp. 765-776.
Yang, J.C., Everett, M., Buehler, C., McMillan, L. (2002).
,
Proc. Eurographics Rendering Workshop 2002.
Archives of light fields
Applications
Ashdown, I. (1993).
,
Journal of the Illuminating Engineering Society,
Vol. 22, No. 1, Winter, 1993, pp. 163-180.
Buehler, C., Bosse, M., McMillan, L., Gortler, S., Cohen, M. (2001).
,
Proc. ACM SIGGRAPH,
ACM Press.
Isaksen, A., McMillan, L., Gortler, S.J. (2000).
, Proc. ACM SIGGRAPH, ACM Press, pp. 297-306.
Javidi, B., Okano, F., eds. (2002).
Three-Dimensional Television, Video and Display Technologies,
Springer-Verlag.
Matusik, W., Pfister, H. (2004).
,
Proc. ACM SIGGRAPH, ACM Press.
Rademacher, P., Bishop, G. (1998).
,
Proc. ACM SIGGRAPH,
ACM Press.
Vaish, V., Garg, G., Talvala, E., Antunez, E., Wilburn, B.,
Horowitz, M., Levoy, M. (2005).
,
Proc. Workshop on Advanced 3D Imaging for Safety and Security,
in conjunction with CVPR 2005.
Zomet, A., Feldman, D., Peleg, S., Weinshall, D. (2003).
,
IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI),
Vol. 25, No. 6, June 2003, pp. 741-754.
Halle, M., Benton, S., Klug, M., Underkoffler, J. (1991).
,
SPIE Vol. 1461, Practical Holography V, S.A. Benton, ed., pp. 142-155.
Talvala, E-V., Adams, A., Horowitz, M., Levoy, M. (2007).
,
Proc. ACM SIGGRAPH.
Raskar, R., Agrawal, A., Wilson, C., Veeraraghavan, A. (2008).
,
Proc. ACM SIGGRAPH.
Pérez, F., Marichal, J. G., Rodriguez, J.M. (2008).
,
Proc. EUSIPCO
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