Abstract

The structure of light fields of natural scenes is highly complex due to high frequencies in the radiance distribution function. However it is the low-order properties of light that determine the appearance of common matte materials. We describe the local light field in terms of spherical harmonics and analyze the qualitative properties and physical meaning of the low-order components. We take a first step in the further development of Gershun's classical work on the light field by extending his description beyond the 3D vector field, toward a more complete description of the illumination using tensors. We show that the three first components, namely, the monopole (density of light), the dipole (light vector), and the quadrupole (squash tensor) suffice to describe a wide range of qualitatively different light fields.

In this paper we address a related issue, namely, the spatial properties of light fields within natural scenes. We want to find out to what extent local light fields change from point to point and how different orders behave. We found experimentally that the low-order components of the light field are rather constant over the scenes whereas high-order components are not. Using very simple models, we found a strong relationship between the low-order components and the geometrical layouts of the scenes.

© 2007 Optical Society of America

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References

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  1. A. Adams, The Negative (Little, Brown and Co., 1981).
  2. T. S. Jacobs, Drawing with an Open Mind (Watson-Guptill, 1991).
  3. L. Michel, Light: the Space of Shape (Wiley, 1995).
  4. M. Baxandall, Shadows and Enlightenment (Yale U. Press, 1995).
  5. R. W. Fleming, R. O. Dror, and E. H. Adelson, "Real-world illumination and the perception of surface reflectance properties," J. Vision 3, 347-368 (2003).
    [CrossRef]
  6. R. O. Dror, T. K. Leung, E. H. Adelson, and A. S. Willsky, "Statistics of real-world illumination," in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 164-171.
  7. R. O. Dror, A. S. Willsky, and E. H. Adelson, "Statistical characterization of real-world illumination," J. Vision 4, 821-837 (2004).
    [CrossRef]
  8. A. Gershun, "The light field," J. Math. Phys. 18, 51151 (1939) (translated by P. Moon and G. Timoshenko).
  9. P. Moon and D. E. Spencer, The Photic Field (MIT Press, 1981).
  10. E. H. Adelson and J. Bergen, "The plenoptic function and the elements of early vision," in Computational Models of Visual Processing, M. Landy and J. Movshon, eds. (MIT Press, 1991), pp. 3-20.
  11. S. J. Gortler, R. Grzeszczuk, R. Szeliski, and M. F. Cohen, "The lumigraph," in Computer Graphics, Proc. SIGGRAPH96 (1996), pp. 43-54.
  12. M. Levoy and P. Hanrahan, "Light field rendering," in Proc. SIGGRAPH 96 (1996), pp. 31-42.
    [CrossRef]
  13. J. Huang and D. Mumford, "Statistics of natural images and models," in Computer Vision and Pattern Recognition (IEEE, 1999), pp. 541-547.
  14. S. Teller, M. Antone, M. Bosse, S. Coorg, M. Jethwa, and N. Master, "Calibrated, registered images of an extended urban area," in Proc. IEEE Computer Vision and Pattern Recognition (CVPR, 2001), pp. 93-107.
  15. T. MacRobert, Spherical Harmonics: An Elementary Treatise on Harmonic Functions, with Applications (Dover, 1947).
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    [CrossRef]
  18. P. N. Belhumeur and D. J. Kriegman, "What is the set of images of an object under all possible illumination conditions?"Int. J. Comput. Vision 28, 245-260 (1998).
    [CrossRef]
  19. M. Kazhdan, T. Funkhouser, and S. Rusinkiewicz, "Rotation invariant spherical harmonic representation of 3D shape descriptors," in Eurographics Symposium on Geometry Processing, L. Kobbelt, P. Schrder, and H. Hoppe, eds. (EG Digital Library, 2003).
  20. R. D. Stock and M. W. Siegel, "Orientation invariant light source parameters," J. Opt. Eng. 35, 2651-2660 (1996).
    [CrossRef]
  21. P. E. Debevec and J. Malik, "Recovering high dynamic range radiance maps from photographs," in Proceedings of ACM SIGGRAPH (1997), Annual Conference Series pp. 369-378, 1997.
  22. http://www.debevec.org/Probes/.
  23. J. A. Endler, "The color of light in forests and its implications," in Ecological Monographs 63 (Ecological Society of America, 1993), pp. 1-27.
    [CrossRef]

2004

R. O. Dror, A. S. Willsky, and E. H. Adelson, "Statistical characterization of real-world illumination," J. Vision 4, 821-837 (2004).
[CrossRef]

2003

R. W. Fleming, R. O. Dror, and E. H. Adelson, "Real-world illumination and the perception of surface reflectance properties," J. Vision 3, 347-368 (2003).
[CrossRef]

2001

1998

P. N. Belhumeur and D. J. Kriegman, "What is the set of images of an object under all possible illumination conditions?"Int. J. Comput. Vision 28, 245-260 (1998).
[CrossRef]

1996

R. D. Stock and M. W. Siegel, "Orientation invariant light source parameters," J. Opt. Eng. 35, 2651-2660 (1996).
[CrossRef]

1939

A. Gershun, "The light field," J. Math. Phys. 18, 51151 (1939) (translated by P. Moon and G. Timoshenko).

Int. J. Comput. Vision

P. N. Belhumeur and D. J. Kriegman, "What is the set of images of an object under all possible illumination conditions?"Int. J. Comput. Vision 28, 245-260 (1998).
[CrossRef]

J. Math. Phys.

A. Gershun, "The light field," J. Math. Phys. 18, 51151 (1939) (translated by P. Moon and G. Timoshenko).

J. Opt. Eng.

R. D. Stock and M. W. Siegel, "Orientation invariant light source parameters," J. Opt. Eng. 35, 2651-2660 (1996).
[CrossRef]

J. Opt. Soc. Am. A

J. Vision

R. O. Dror, A. S. Willsky, and E. H. Adelson, "Statistical characterization of real-world illumination," J. Vision 4, 821-837 (2004).
[CrossRef]

R. W. Fleming, R. O. Dror, and E. H. Adelson, "Real-world illumination and the perception of surface reflectance properties," J. Vision 3, 347-368 (2003).
[CrossRef]

Other

R. O. Dror, T. K. Leung, E. H. Adelson, and A. S. Willsky, "Statistics of real-world illumination," in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 164-171.

A. Adams, The Negative (Little, Brown and Co., 1981).

T. S. Jacobs, Drawing with an Open Mind (Watson-Guptill, 1991).

L. Michel, Light: the Space of Shape (Wiley, 1995).

M. Baxandall, Shadows and Enlightenment (Yale U. Press, 1995).

P. Moon and D. E. Spencer, The Photic Field (MIT Press, 1981).

E. H. Adelson and J. Bergen, "The plenoptic function and the elements of early vision," in Computational Models of Visual Processing, M. Landy and J. Movshon, eds. (MIT Press, 1991), pp. 3-20.

S. J. Gortler, R. Grzeszczuk, R. Szeliski, and M. F. Cohen, "The lumigraph," in Computer Graphics, Proc. SIGGRAPH96 (1996), pp. 43-54.

M. Levoy and P. Hanrahan, "Light field rendering," in Proc. SIGGRAPH 96 (1996), pp. 31-42.
[CrossRef]

J. Huang and D. Mumford, "Statistics of natural images and models," in Computer Vision and Pattern Recognition (IEEE, 1999), pp. 541-547.

S. Teller, M. Antone, M. Bosse, S. Coorg, M. Jethwa, and N. Master, "Calibrated, registered images of an extended urban area," in Proc. IEEE Computer Vision and Pattern Recognition (CVPR, 2001), pp. 93-107.

T. MacRobert, Spherical Harmonics: An Elementary Treatise on Harmonic Functions, with Applications (Dover, 1947).

J. Jacson, Classical Electrodynamics (Wiley, 1975).

P. E. Debevec and J. Malik, "Recovering high dynamic range radiance maps from photographs," in Proceedings of ACM SIGGRAPH (1997), Annual Conference Series pp. 369-378, 1997.

http://www.debevec.org/Probes/.

J. A. Endler, "The color of light in forests and its implications," in Ecological Monographs 63 (Ecological Society of America, 1993), pp. 1-27.
[CrossRef]

M. Kazhdan, T. Funkhouser, and S. Rusinkiewicz, "Rotation invariant spherical harmonic representation of 3D shape descriptors," in Eurographics Symposium on Geometry Processing, L. Kobbelt, P. Schrder, and H. Hoppe, eds. (EG Digital Library, 2003).

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Figures (9)

Fig. 1
Fig. 1

(Color online) Spherical harmonics basis functions. The first row (the sphere) represents the zero order, the second row shows the basis functions of the dipole, the third row shows the basis functions of the quadrupole.

Fig. 2
Fig. 2

(Color online) Second-order representation of a light field in the (a) original arbitrary orientation and (b) orientation with regard to the quadrupole. On the left side the first nine coefficients are presented (note the change after rotation); on the right side the quadrupoles are presented graphically. Note that the shape of the quadrupole does not change after rotation whereas the mathematical description is simplified and depends only on two coefficients q + = 0.093 and q = 0.020 . The coefficients that make up the quadrupole are framed.

Fig. 3
Fig. 3

Schematics of the second-order light field in (a) original orientation and (b) orientation aligned according to the light vector (c) orientation aligned according to the quadrupole. The SH coefficients are presented on the left side. The mutual orientation of the components D , q + , and q is shown on the right side. The length of the light gray arrow corresponds to the value d 1 (strength of the light vector), the lengths of the dark gray and black arrows correspond to values q + and q .

Fig. 4
Fig. 4

(Color online) Qualitative properties of the quadrupole. Extreme cases of light fields due to the quadrupole: (a) light clamp, q + = 1 , q = 0 , (b) light ring, q + = 0.5 , q = 3 / 2 . The light vector is assumed to be zero, the monopole d 0 is chosen as small as possible such that the resulting function is nonnegative everywhere.

Fig. 5
Fig. 5

Geometrical layouts of the measured scenes (a) open field, (b) wall, (c) street, and (d) forest.

Fig. 6
Fig. 6

Simplest models of the light field in the following geometries (a) street scene, (b) wall scene, and (c) forest scene.

Fig. 7
Fig. 7

(Color online) Light probes of (a) street scene under a clear sky, (b) street scene under an overcast sky, (c) forest under clear sky, (d) forest under an overcast sky, (e) “wall” scene measured along the wall, and (f) wall measured across. To the right of the panoramic images the cross sections through the S H 6 approximations of the corresponding local light fields are depicted; the directions of the cross sections are indicated by black circles in the panoramic images.

Fig. 8
Fig. 8

Schematic of the second-order approximations of the local light field measurements. The letters represent the same scenes as in Fig. 7. For each scene figures that run from left to right correspond to the figures that run from top to bottom in Fig. 7.

Fig. 9
Fig. 9

Strengths d 1 of the light field components up to tenth order. The letters represent the same scenes as in Fig. 7; the bars of different gray levels represent three samples within the scene.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

f ( ϑ , φ ) = l = 0 m = l l f l m Y l m ( ϑ , φ ) ,
Y l m ( ϑ , φ ) = K l m e i m φ P l m ( cos ϑ ) , l N , l m l ,
K l m = ( 2 l + 1 ) 4 π ( l m ) ! ( l + m ) ! ,
Y l m ( ϑ , φ ) = { 2 K l m cos ( m φ ) P l m ( cos ϑ ) , m > 0 2 K l | m | sin ( | m | φ ) P l | m | ( cos ϑ ) , m < 0 K l 0 P l 0 ( cos ϑ ) , m = 0 .
f l m = φ = 0 2 π ϑ = 0 π f ( ϑ , φ ) Y l m ( ϑ , φ ) sin ( ϑ ) d ϑ d φ .
d l = m = l l f l m 2 .
S H 2 Q ( L F ) = { M Q , D Q , Q Q } = { { d 0 } , { f 1 1 Q , f 10 Q , f 11 Q } , { 0 , 0 , q + , 0 , q } } .

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