Abstract

Light fields [J. Math. Phys. 18, 51 (1936);The Photic Field (MIT, 1981)] of natural scenes are highly complex and vary within a scene from point to point. However, in many applications complex lighting can be successfully replaced by its low-order approximation [J. Opt. Soc. Am. A 18, 2448 (2001); Appl. Opt. 46, 7308 (2007)]. The purpose of this research is to investigate the structure of light fields in natural scenes. We describe the structure of light fields in terms of spherical harmonics and analyze their spatial variation and qualitative properties over scenes. We consider several types of natural scene geometries. Empirically and via modeling, we study the typical behavior of the first- and second-order approximation of the local light field in those scenes. The first-order term is generally known as the “light vector” and has an immediate physical meaning. The quadrupole component, which we named “squash tensor,” is a useful addition as we show in this paper. The measurements were done with a custom-made device of novel design, called a “Plenopter,” which was constructed to measure the light field in terms of spherical harmonics up to the second order. In different scenes of similar geometries, we found structurally similar light fields, which suggests that in some way the light field can be thought of as a property of the geometry. Furthermore, the smooth variation of the light field’s low-order components suggests that, instead of specifying the complete light field of the scene, it is often sufficient to measure the light field only in a few points and rely on interpolation to recover the light field at arbitrary points of the scene.

© 2009 Optical Society of America

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References

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  1. C. Cuttle, Lighting by Design (Architectural Press, 2003).
  2. C. Cuttle, “Lighting patterns and the flow of light,” Light. Res. Technol. 3, 171-189 (1971).
  3. L. Michel, Light: the Shape of Space (Wiley, 1995).
  4. T. S. Jacobs, Drawing with an Open Mind (Watson-Guptill, 1991).
  5. M. Baxandall, Shadows and Enlightenment (Yale University, 1995).
  6. A. Adams, The Negative (Little, Brown and Company, 1981).
  7. 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.
  8. 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]
  9. R. O. Dror, A. S. Willsky, and E. H. Adelson, “Statistical characterization of real-world illumination,” J. Vision 4, 821-837(2004).
    [CrossRef]
  10. J. Huang and D. Mumford, “Statistics of natural images and models,” in Computer Vision and Pattern Recognition (IEEE, 1999), pp. 541-547.
  11. S. Teller, M. Antone, M. Bosse, S. Coorg, M. Jethwa, and N. Master, “Calibrated, registered images of an extended urban area,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 93-107.
  12. A. A. Mury, S. C. Pont, and J. J. Koenderink, “Light field constancy within natural scenes,” Appl. Opt. 46, 7308-7316(2007).
    [CrossRef] [PubMed]
  13. A. Gershun, “The light field,” J. Math. Phys. 18, 51-151 (1936) (translated by P. Moon and G. Timoshenko).
  14. E. H. Adelson and J. R. Bergen, “The plenoptic function and the elements of early vision,” in Computational Models of Visual Processing, M. Landy and J. A. Movshon, eds. (MIT, 1991), pp. 3-20.
  15. R. Ramamoorthi and P. Hanrahan, “On the relationship between radiance and irradiance: determining the illumination from images of a convex Lambertian object,” J. Opt. Soc. Am. A 18, 2448-2459 (2001).
    [CrossRef]
  16. J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes, Computer Graphics, Principles and Practice (Addison Wesley, 1990).
  17. C. J. Copi, D. Huterer, D. J. Schwarz, and G. D. Starkman, “On the large-angle anomalies of the microwave sky,” Mon. Not. R. Astron. Soc. 1-27 (2005).
  18. M. R. Dennis and K. Land, “Probability density of the multipole vectors for a Gaussian cosmic microwave background,” Mon. Not. R. Astron. Soc. 383(2), 424-434 (2007).
    [CrossRef]
  19. J. J. Koenderink and A. J. van Doorn, “Geometrical modes as a general method to treat diffuse interreflections in radiometry,” J. Opt. Soc. Am. 73, 843-850 (1983).
    [CrossRef]
  20. P. Moon and D. E. Spencer, The Photic Field (MIT, 1981).
  21. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University, 1988).
  22. S. Darula and R. Kittler, “CIE General Sky standard defining luminance distributions,” in Proceedings ESim Conference (IBPSA, 2002), pp. 1113.
  23. H. Akbari, L. S. Rose, and H. Taha, “Characterizing the fabric of the urban environment: a case study of Sacramento, California,” LBNL-44688, Lawrence Berkeley National Laboratory, Berkeley, California, 1999.
  24. S. C. Pont and J. J. Koenderink, “Reflectance from locally glossy thoroughly pitted surfaces,” Comput. Vision Image Understand. 98 (2), 211-222 (2005).
    [CrossRef]
  25. S. J. Gortler, R. Grzeszczuk, R. Szeliski, and M. F. Cohen, “The lumigraph,” in Computer Graphics (SIGGRAPH96) Proceedings (ACM, 1996), pp. 43-54.
  26. H. M. Fox and G. Vevers, The Nature of Animal Colours (Sidqwick and Jackson, 1960).
  27. D. Gomez and M. Thery, “Simultaneous crypsis and conspicuousness in color patterns: comparative analysis of a neotropical rainforest bird community,” Am. Nat. 169, S42-S61 (2007).
    [CrossRef]
  28. G. H. Thayer, Concealing Coloration in the Animal Kingdom (Macmillan, 1909/1918).
  29. 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]

2007 (3)

M. R. Dennis and K. Land, “Probability density of the multipole vectors for a Gaussian cosmic microwave background,” Mon. Not. R. Astron. Soc. 383(2), 424-434 (2007).
[CrossRef]

D. Gomez and M. Thery, “Simultaneous crypsis and conspicuousness in color patterns: comparative analysis of a neotropical rainforest bird community,” Am. Nat. 169, S42-S61 (2007).
[CrossRef]

A. A. Mury, S. C. Pont, and J. J. Koenderink, “Light field constancy within natural scenes,” Appl. Opt. 46, 7308-7316(2007).
[CrossRef] [PubMed]

2005 (2)

C. J. Copi, D. Huterer, D. J. Schwarz, and G. D. Starkman, “On the large-angle anomalies of the microwave sky,” Mon. Not. R. Astron. Soc. 1-27 (2005).

S. C. Pont and J. J. Koenderink, “Reflectance from locally glossy thoroughly pitted surfaces,” Comput. Vision Image Understand. 98 (2), 211-222 (2005).
[CrossRef]

2004 (1)

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

2003 (1)

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 (1)

1983 (1)

1971 (1)

C. Cuttle, “Lighting patterns and the flow of light,” Light. Res. Technol. 3, 171-189 (1971).

1936 (1)

A. Gershun, “The light field,” J. Math. Phys. 18, 51-151 (1936) (translated by P. Moon and G. Timoshenko).

Adams, A.

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

Adelson, E. H.

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]

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

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.

Akbari, H.

H. Akbari, L. S. Rose, and H. Taha, “Characterizing the fabric of the urban environment: a case study of Sacramento, California,” LBNL-44688, Lawrence Berkeley National Laboratory, Berkeley, California, 1999.

Antone, M.

S. Teller, M. Antone, M. Bosse, S. Coorg, M. Jethwa, and N. Master, “Calibrated, registered images of an extended urban area,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 93-107.

Baxandall, M.

M. Baxandall, Shadows and Enlightenment (Yale University, 1995).

Bergen, J. R.

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

Bosse, M.

S. Teller, M. Antone, M. Bosse, S. Coorg, M. Jethwa, and N. Master, “Calibrated, registered images of an extended urban area,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 93-107.

Cohen, M. F.

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

Coorg, S.

S. Teller, M. Antone, M. Bosse, S. Coorg, M. Jethwa, and N. Master, “Calibrated, registered images of an extended urban area,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 93-107.

Copi, C. J.

C. J. Copi, D. Huterer, D. J. Schwarz, and G. D. Starkman, “On the large-angle anomalies of the microwave sky,” Mon. Not. R. Astron. Soc. 1-27 (2005).

Cuttle, C.

C. Cuttle, “Lighting patterns and the flow of light,” Light. Res. Technol. 3, 171-189 (1971).

C. Cuttle, Lighting by Design (Architectural Press, 2003).

Darula, S.

S. Darula and R. Kittler, “CIE General Sky standard defining luminance distributions,” in Proceedings ESim Conference (IBPSA, 2002), pp. 1113.

Dennis, M. R.

M. R. Dennis and K. Land, “Probability density of the multipole vectors for a Gaussian cosmic microwave background,” Mon. Not. R. Astron. Soc. 383(2), 424-434 (2007).
[CrossRef]

Dror, R. O.

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]

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.

Endler, J. A.

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]

Feiner, S. K.

J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes, Computer Graphics, Principles and Practice (Addison Wesley, 1990).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University, 1988).

Fleming, R. W.

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]

Foley, J. D.

J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes, Computer Graphics, Principles and Practice (Addison Wesley, 1990).

Fox, H. M.

H. M. Fox and G. Vevers, The Nature of Animal Colours (Sidqwick and Jackson, 1960).

Gershun, A.

A. Gershun, “The light field,” J. Math. Phys. 18, 51-151 (1936) (translated by P. Moon and G. Timoshenko).

Gomez, D.

D. Gomez and M. Thery, “Simultaneous crypsis and conspicuousness in color patterns: comparative analysis of a neotropical rainforest bird community,” Am. Nat. 169, S42-S61 (2007).
[CrossRef]

Gortler, S. J.

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

Grzeszczuk, R.

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

Hanrahan, P.

Huang, J.

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

Hughes, J. F.

J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes, Computer Graphics, Principles and Practice (Addison Wesley, 1990).

Huterer, D.

C. J. Copi, D. Huterer, D. J. Schwarz, and G. D. Starkman, “On the large-angle anomalies of the microwave sky,” Mon. Not. R. Astron. Soc. 1-27 (2005).

Jacobs, T. S.

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

Jethwa, M.

S. Teller, M. Antone, M. Bosse, S. Coorg, M. Jethwa, and N. Master, “Calibrated, registered images of an extended urban area,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 93-107.

Kittler, R.

S. Darula and R. Kittler, “CIE General Sky standard defining luminance distributions,” in Proceedings ESim Conference (IBPSA, 2002), pp. 1113.

Koenderink, J. J.

Land, K.

M. R. Dennis and K. Land, “Probability density of the multipole vectors for a Gaussian cosmic microwave background,” Mon. Not. R. Astron. Soc. 383(2), 424-434 (2007).
[CrossRef]

Leung, T. K.

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.

Master, N.

S. Teller, M. Antone, M. Bosse, S. Coorg, M. Jethwa, and N. Master, “Calibrated, registered images of an extended urban area,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 93-107.

Michel, L.

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

Moon, P.

A. Gershun, “The light field,” J. Math. Phys. 18, 51-151 (1936) (translated by P. Moon and G. Timoshenko).

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

Mumford, D.

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

Mury, A. A.

Pont, S. C.

A. A. Mury, S. C. Pont, and J. J. Koenderink, “Light field constancy within natural scenes,” Appl. Opt. 46, 7308-7316(2007).
[CrossRef] [PubMed]

S. C. Pont and J. J. Koenderink, “Reflectance from locally glossy thoroughly pitted surfaces,” Comput. Vision Image Understand. 98 (2), 211-222 (2005).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University, 1988).

Ramamoorthi, R.

Rose, L. S.

H. Akbari, L. S. Rose, and H. Taha, “Characterizing the fabric of the urban environment: a case study of Sacramento, California,” LBNL-44688, Lawrence Berkeley National Laboratory, Berkeley, California, 1999.

Schwarz, D. J.

C. J. Copi, D. Huterer, D. J. Schwarz, and G. D. Starkman, “On the large-angle anomalies of the microwave sky,” Mon. Not. R. Astron. Soc. 1-27 (2005).

Spencer, D. E.

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

Starkman, G. D.

C. J. Copi, D. Huterer, D. J. Schwarz, and G. D. Starkman, “On the large-angle anomalies of the microwave sky,” Mon. Not. R. Astron. Soc. 1-27 (2005).

Szeliski, R.

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

Taha, H.

H. Akbari, L. S. Rose, and H. Taha, “Characterizing the fabric of the urban environment: a case study of Sacramento, California,” LBNL-44688, Lawrence Berkeley National Laboratory, Berkeley, California, 1999.

Teller, S.

S. Teller, M. Antone, M. Bosse, S. Coorg, M. Jethwa, and N. Master, “Calibrated, registered images of an extended urban area,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 93-107.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University, 1988).

Thayer, G. H.

G. H. Thayer, Concealing Coloration in the Animal Kingdom (Macmillan, 1909/1918).

Thery, M.

D. Gomez and M. Thery, “Simultaneous crypsis and conspicuousness in color patterns: comparative analysis of a neotropical rainforest bird community,” Am. Nat. 169, S42-S61 (2007).
[CrossRef]

Timoshenko, G.

A. Gershun, “The light field,” J. Math. Phys. 18, 51-151 (1936) (translated by P. Moon and G. Timoshenko).

van Dam, A.

J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes, Computer Graphics, Principles and Practice (Addison Wesley, 1990).

van Doorn, A. J.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University, 1988).

Vevers, G.

H. M. Fox and G. Vevers, The Nature of Animal Colours (Sidqwick and Jackson, 1960).

Willsky, A. S.

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

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.

Am. Nat. (1)

D. Gomez and M. Thery, “Simultaneous crypsis and conspicuousness in color patterns: comparative analysis of a neotropical rainforest bird community,” Am. Nat. 169, S42-S61 (2007).
[CrossRef]

Appl. Opt. (1)

Comput. Vision Image Understand. (1)

S. C. Pont and J. J. Koenderink, “Reflectance from locally glossy thoroughly pitted surfaces,” Comput. Vision Image Understand. 98 (2), 211-222 (2005).
[CrossRef]

J. Math. Phys. (1)

A. Gershun, “The light field,” J. Math. Phys. 18, 51-151 (1936) (translated by P. Moon and G. Timoshenko).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Vision (2)

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]

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

Light. Res. Technol. (1)

C. Cuttle, “Lighting patterns and the flow of light,” Light. Res. Technol. 3, 171-189 (1971).

Mon. Not. R. Astron. Soc. (2)

C. J. Copi, D. Huterer, D. J. Schwarz, and G. D. Starkman, “On the large-angle anomalies of the microwave sky,” Mon. Not. R. Astron. Soc. 1-27 (2005).

M. R. Dennis and K. Land, “Probability density of the multipole vectors for a Gaussian cosmic microwave background,” Mon. Not. R. Astron. Soc. 383(2), 424-434 (2007).
[CrossRef]

Other (18)

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

H. M. Fox and G. Vevers, The Nature of Animal Colours (Sidqwick and Jackson, 1960).

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

C. Cuttle, Lighting by Design (Architectural Press, 2003).

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

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

M. Baxandall, Shadows and Enlightenment (Yale University, 1995).

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

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.

G. H. Thayer, Concealing Coloration in the Animal Kingdom (Macmillan, 1909/1918).

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]

J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes, Computer Graphics, Principles and Practice (Addison Wesley, 1990).

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 Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 93-107.

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

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University, 1988).

S. Darula and R. Kittler, “CIE General Sky standard defining luminance distributions,” in Proceedings ESim Conference (IBPSA, 2002), pp. 1113.

H. Akbari, L. S. Rose, and H. Taha, “Characterizing the fabric of the urban environment: a case study of Sacramento, California,” LBNL-44688, Lawrence Berkeley National Laboratory, Berkeley, California, 1999.

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Figures (9)

Fig. 1
Fig. 1

From left to right, a matte convex object under a collimated source from above on a black, absorbing ground (vertically oriented dipole) and on a white ground causing a secondary source from below (combination of vertically oriented dipole and quadrupole). Next the object was illuminated by collimated sunlight from the left plus ambient light (monopole plus almost horizontally oriented dipole) and with a white screen at the right causing a secondary source from the right (monopole plus almost horizontally oriented dipole and quadrupole).

Fig. 2
Fig. 2

Schematic graphical representation of the second-order light field. 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 . Note that these dark gray and black arrows are perpendicular to each other and that there are always two dark gray and two black arrows opposite each other, together representing a quadrupole (two positive and two negative poles perpendicular to each other).

Fig. 3
Fig. 3

Special cases of light fields due to the squash tensor: (a) a light clamp and (b) a light ring. The light vector is assumed to be zero.

Fig. 4
Fig. 4

Our custom-made light measuring device which we named “Plenopter.”

Fig. 5
Fig. 5

Schematic descriptions of the scenes: (a) wall, (b) street, and (c) room.

Fig. 6
Fig. 6

Comparison of models (left) and measurements (right) for street scene configurations, for three streets (a), (b) and (c) in (1) clear and (2) overcast sky conditions. The vectors represent the light field up to the second order (see Figure 2). We considered from seven to nine points per scene (depending on the scene dimensions).

Fig. 7
Fig. 7

Measurements of second-order light fields for the wall scene in the case of (a) a clear sky and (b) an overcast sky. The Sun was not visible in either case.

Fig. 8
Fig. 8

Measurements for the room scene: (a) and (b) white wall; (c) and (d) black wall; (a) and (c) view from above; (b) and (d) view from a side.

Fig. 9
Fig. 9

Room scene: At the left we show the vector representations for the points near the wall for the white and the black cases. At the right we show the ratios of the magnitudes of the mono-, di-, and quadruples with the monopole.

Equations (12)

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 ) = { 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 .
S H 2 rot d ( L F ) = { M d , D d , Q d } = { { d 0 } , { 0 , 0 , v } , { f 2 , 2 d , f 2 , 1 d , f 2 , 0 d , f 2 , 1 d , f 2 , 2 d } } .
S H 2 rot q ( L F ) = { M q , D q , Q q } = { { d 0 } , { f 1 , 1 q , f 1 , 0 q , f 1 , 1 q } , { 0 , 0 , q + , 0 , q } } .
P j = S j ( θ , ϕ ) · L F ( θ , ϕ ) d Ω , j = 1 , , 12 ,
S j ( θ , ϕ ) = l m s l , m j ϒ l , m ( θ , ϕ ) + ε j .
L F ( θ , ϕ ) = l m c l , m ϒ l , m ( θ , ϕ ) + ε .
P j = [ lm s l , m j ϒ l , m ( θ , ϕ ) ] [ l m c l , m ϒ l , m ( θ , ϕ ) ] d Ω = l l , m m s l , m j c l , m ϒ l , m ( θ , ϕ ) ϒ l , m ( θ , ϕ ) d Ω
ϒ l , m ( θ , ϕ ) ϒ l , m ( θ , ϕ ) = δ l , l δ m , m ,
P j = l m s l , m j v l , m = ( s j , c ) .
P j = k = 1 9 s k j c k .

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