Abstract

We present a theoretical analysis of the relationship between incoming radiance and irradiance. Specifically, we address the question of whether it is possible to compute the incident radiance from knowledge of the irradiance at all surface orientations. This is a fundamental question in computer vision and inverse radiative transfer. We show that the irradiance can be viewed as a simple convolution of the incident illumination, i.e., radiance and a clamped cosine transfer function. Estimating the radiance can then be seen as a deconvolution operation. We derive a simple closed-form formula for the irradiance in terms of spherical harmonic coefficients of the incident illumination and demonstrate that the odd-order modes of the lighting with order greater than 1 are completely annihilated. Therefore these components cannot be estimated from the irradiance, contradicting a theorem that is due to Preisendorfer. A practical realization of the radiance-from-irradiance problem is the estimation of the lighting from images of a homogeneous convex curved Lambertian surface of known geometry under distant illumination, since a Lambertian object reflects light equally in all directions proportional to the irradiance. We briefly discuss practical and physical considerations and describe a simple experimental test to verify our theoretical results.

© 2001 Optical Society of America

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References

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  1. R. W. Preisendorfer, Hydrologic Optics (U.S. Department of Commerce, Washington, D.C., 1976).
  2. S. R. Marschner, D. P. Greenberg, “Inverse lighting for photography,” in Fifth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 262–265.
  3. I. Sato, Y. Sato, K. Ikeuchi, “Illumination distribution from brightness in shadows: adaptive estimation of illumination distribution with unknown reflectance properties in shadow regions,” in Seventh IEEE International Conference on Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1999), pp. 875–882.
  4. N. McCormick, “Inverse radiative transfer problems: a review,” Nucl. Sci. Eng. 112, 185–198 (1992).
  5. G. Miller, C. Hoffman, “Illumination and reflection maps: simulated objects in simulated and real environments,” in SIGGRAPH 84 Advanced Computer Graphics Animation Seminar Notes (Association for Computing Machinery, New York, 1984).
  6. B. Cabral, M. Olano, P. Nemec, “Reflection space image based rendering,” in SIGGRAPH 99, Computer Graphics Proceedings, Annual Conference Series, A. Rockwood, ed. (Association for Computing Machinery, New York, 1999), pp. 165–170.
  7. J. Haddon, D. Forsyth, “Shape representations from shading primitives,” in Fifth European Conference on Computer Vision, H. Burkhardt, B. Neumann, eds. (Springer-Verlag, Berlin, 1998), pp. 415–431.
  8. D. Jacobs, P. Belhumeur, R. Basri, “Comparing images under variable illumination,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1998), pp. 610–617.
  9. P. Belhumeur, D. Kriegman, “What is the set of images of an object under all possible illumination conditions?” Int. J. Comput. Vision 28, 245–260 (1998).
    [CrossRef]
  10. R. Epstein, P. W. Hallinan, A. Yuille, “5 plus or minus 2 eigenimages suffice: an empirical investigation of low-dimensional lighting models,” in IEEE Workshop on Physics-Based Modeling in Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1995), pp. 108–116.
  11. R. Basri, D. Jacobs, “Lambertian reflectance and linear subspaces,” (NEC Research Institute, Princeton, N.J., 2000).
  12. A. Gershun, “The light field, ” J. Math. Phys. 18, 51–151 (1939) (translated by P. Moon and G. Timoshenko).
  13. R. McCluney, Introduction to Radiometry and Photometry (Artech House, Norwood, Mass., 1994).
  14. S. Gortler, R. Grzeszczuk, R. Szeliski, M. Cohen, “The lumigraph,” in SIGGRAPH 96, Computer Graphics Proceedings, Annual Conference Series, H. Rushmeier, ed. (Association for Computing Machinery, New York, 1996), pp. 43–54.
  15. M. Levoy, P. Hanrahan, “Light field rendering,” in SIGGRAPH 96, Computer Graphics Proceedings, Annual Conference Series, H. Rushmeier, ed. (Association for Computing Machinery, New York, 1996), pp. 31–42.
  16. D. Wood, D. Azuma, K. Aldinger, B. Curless, T. Duchamp, D. Salesin, W. Stuetzle, “Surface light fields for 3D photography,” in SIGGRAPH 2000, Computer Graphics Proceedings, Annual Conference Series, K. Akeley, ed. (Association for Computing Machinery, New York, 2000), pp. 287–296.
  17. R. Ramamoorthi, P. Hanrahan, “Analysis of planar light fields from homogeneous convex curved surfaces under distant illumination,” http://graphics.stanford.edu/papers/planarlf .
  18. T. Inui, Y. Tanabe, Y. Onodera, Group Theory and its Applications in Physics (Springer Verlag, New York, 1990).
  19. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  20. T. M. MacRobert, Spherical Harmonics; An Elementary Treatise on Harmonic Functions, with Applications (Dover, New York, 1948).
  21. M. D’Zmura, “Shading ambiguity: reflectance and illumination,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass.1991), pp. 187–207.
  22. B. Cabral, N. Max, R. Springmeyer, “Bidirectional reflection functions from surface bump maps,” Comput. Graph. 21(4), 273–281 (1987).
    [CrossRef]
  23. S. Westin, J. Arvo, K. Torrance, “Predicting reflectance functions from complex surfaces,” Comput. Graph. 26(2), 255–264 (1992).
    [CrossRef]
  24. F. Sillion, J. Arvo, S. Westin, D. Greenberg, “A global illumination solution for general reflectance distributions,” Comput. Graph. 25(4), 187–196 (1991).
    [CrossRef]
  25. J. Koenderink, A. van Doorn, “Phenomenological description of bidirectional surface reflection,” J. Opt. Soc. Am. A 15, 2903–2912 (1998).
    [CrossRef]

1998

P. Belhumeur, D. 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. Koenderink, A. van Doorn, “Phenomenological description of bidirectional surface reflection,” J. Opt. Soc. Am. A 15, 2903–2912 (1998).
[CrossRef]

1992

S. Westin, J. Arvo, K. Torrance, “Predicting reflectance functions from complex surfaces,” Comput. Graph. 26(2), 255–264 (1992).
[CrossRef]

N. McCormick, “Inverse radiative transfer problems: a review,” Nucl. Sci. Eng. 112, 185–198 (1992).

1991

F. Sillion, J. Arvo, S. Westin, D. Greenberg, “A global illumination solution for general reflectance distributions,” Comput. Graph. 25(4), 187–196 (1991).
[CrossRef]

1987

B. Cabral, N. Max, R. Springmeyer, “Bidirectional reflection functions from surface bump maps,” Comput. Graph. 21(4), 273–281 (1987).
[CrossRef]

1939

A. Gershun, “The light field, ” J. Math. Phys. 18, 51–151 (1939) (translated by P. Moon and G. Timoshenko).

Aldinger, K.

D. Wood, D. Azuma, K. Aldinger, B. Curless, T. Duchamp, D. Salesin, W. Stuetzle, “Surface light fields for 3D photography,” in SIGGRAPH 2000, Computer Graphics Proceedings, Annual Conference Series, K. Akeley, ed. (Association for Computing Machinery, New York, 2000), pp. 287–296.

Arvo, J.

S. Westin, J. Arvo, K. Torrance, “Predicting reflectance functions from complex surfaces,” Comput. Graph. 26(2), 255–264 (1992).
[CrossRef]

F. Sillion, J. Arvo, S. Westin, D. Greenberg, “A global illumination solution for general reflectance distributions,” Comput. Graph. 25(4), 187–196 (1991).
[CrossRef]

Azuma, D.

D. Wood, D. Azuma, K. Aldinger, B. Curless, T. Duchamp, D. Salesin, W. Stuetzle, “Surface light fields for 3D photography,” in SIGGRAPH 2000, Computer Graphics Proceedings, Annual Conference Series, K. Akeley, ed. (Association for Computing Machinery, New York, 2000), pp. 287–296.

Basri, R.

R. Basri, D. Jacobs, “Lambertian reflectance and linear subspaces,” (NEC Research Institute, Princeton, N.J., 2000).

D. Jacobs, P. Belhumeur, R. Basri, “Comparing images under variable illumination,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1998), pp. 610–617.

Belhumeur, P.

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

D. Jacobs, P. Belhumeur, R. Basri, “Comparing images under variable illumination,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1998), pp. 610–617.

Cabral, B.

B. Cabral, N. Max, R. Springmeyer, “Bidirectional reflection functions from surface bump maps,” Comput. Graph. 21(4), 273–281 (1987).
[CrossRef]

B. Cabral, M. Olano, P. Nemec, “Reflection space image based rendering,” in SIGGRAPH 99, Computer Graphics Proceedings, Annual Conference Series, A. Rockwood, ed. (Association for Computing Machinery, New York, 1999), pp. 165–170.

Cohen, M.

S. Gortler, R. Grzeszczuk, R. Szeliski, M. Cohen, “The lumigraph,” in SIGGRAPH 96, Computer Graphics Proceedings, Annual Conference Series, H. Rushmeier, ed. (Association for Computing Machinery, New York, 1996), pp. 43–54.

Curless, B.

D. Wood, D. Azuma, K. Aldinger, B. Curless, T. Duchamp, D. Salesin, W. Stuetzle, “Surface light fields for 3D photography,” in SIGGRAPH 2000, Computer Graphics Proceedings, Annual Conference Series, K. Akeley, ed. (Association for Computing Machinery, New York, 2000), pp. 287–296.

D’Zmura, M.

M. D’Zmura, “Shading ambiguity: reflectance and illumination,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass.1991), pp. 187–207.

Duchamp, T.

D. Wood, D. Azuma, K. Aldinger, B. Curless, T. Duchamp, D. Salesin, W. Stuetzle, “Surface light fields for 3D photography,” in SIGGRAPH 2000, Computer Graphics Proceedings, Annual Conference Series, K. Akeley, ed. (Association for Computing Machinery, New York, 2000), pp. 287–296.

Epstein, R.

R. Epstein, P. W. Hallinan, A. Yuille, “5 plus or minus 2 eigenimages suffice: an empirical investigation of low-dimensional lighting models,” in IEEE Workshop on Physics-Based Modeling in Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1995), pp. 108–116.

Forsyth, D.

J. Haddon, D. Forsyth, “Shape representations from shading primitives,” in Fifth European Conference on Computer Vision, H. Burkhardt, B. Neumann, eds. (Springer-Verlag, Berlin, 1998), pp. 415–431.

Gershun, A.

A. Gershun, “The light field, ” J. Math. Phys. 18, 51–151 (1939) (translated by P. Moon and G. Timoshenko).

Gortler, S.

S. Gortler, R. Grzeszczuk, R. Szeliski, M. Cohen, “The lumigraph,” in SIGGRAPH 96, Computer Graphics Proceedings, Annual Conference Series, H. Rushmeier, ed. (Association for Computing Machinery, New York, 1996), pp. 43–54.

Greenberg, D.

F. Sillion, J. Arvo, S. Westin, D. Greenberg, “A global illumination solution for general reflectance distributions,” Comput. Graph. 25(4), 187–196 (1991).
[CrossRef]

Greenberg, D. P.

S. R. Marschner, D. P. Greenberg, “Inverse lighting for photography,” in Fifth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 262–265.

Grzeszczuk, R.

S. Gortler, R. Grzeszczuk, R. Szeliski, M. Cohen, “The lumigraph,” in SIGGRAPH 96, Computer Graphics Proceedings, Annual Conference Series, H. Rushmeier, ed. (Association for Computing Machinery, New York, 1996), pp. 43–54.

Haddon, J.

J. Haddon, D. Forsyth, “Shape representations from shading primitives,” in Fifth European Conference on Computer Vision, H. Burkhardt, B. Neumann, eds. (Springer-Verlag, Berlin, 1998), pp. 415–431.

Hallinan, P. W.

R. Epstein, P. W. Hallinan, A. Yuille, “5 plus or minus 2 eigenimages suffice: an empirical investigation of low-dimensional lighting models,” in IEEE Workshop on Physics-Based Modeling in Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1995), pp. 108–116.

Hanrahan, P.

M. Levoy, P. Hanrahan, “Light field rendering,” in SIGGRAPH 96, Computer Graphics Proceedings, Annual Conference Series, H. Rushmeier, ed. (Association for Computing Machinery, New York, 1996), pp. 31–42.

Hoffman, C.

G. Miller, C. Hoffman, “Illumination and reflection maps: simulated objects in simulated and real environments,” in SIGGRAPH 84 Advanced Computer Graphics Animation Seminar Notes (Association for Computing Machinery, New York, 1984).

Ikeuchi, K.

I. Sato, Y. Sato, K. Ikeuchi, “Illumination distribution from brightness in shadows: adaptive estimation of illumination distribution with unknown reflectance properties in shadow regions,” in Seventh IEEE International Conference on Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1999), pp. 875–882.

Inui, T.

T. Inui, Y. Tanabe, Y. Onodera, Group Theory and its Applications in Physics (Springer Verlag, New York, 1990).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Jacobs, D.

D. Jacobs, P. Belhumeur, R. Basri, “Comparing images under variable illumination,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1998), pp. 610–617.

R. Basri, D. Jacobs, “Lambertian reflectance and linear subspaces,” (NEC Research Institute, Princeton, N.J., 2000).

Koenderink, J.

Kriegman, D.

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

Levoy, M.

M. Levoy, P. Hanrahan, “Light field rendering,” in SIGGRAPH 96, Computer Graphics Proceedings, Annual Conference Series, H. Rushmeier, ed. (Association for Computing Machinery, New York, 1996), pp. 31–42.

MacRobert, T. M.

T. M. MacRobert, Spherical Harmonics; An Elementary Treatise on Harmonic Functions, with Applications (Dover, New York, 1948).

Marschner, S. R.

S. R. Marschner, D. P. Greenberg, “Inverse lighting for photography,” in Fifth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 262–265.

Max, N.

B. Cabral, N. Max, R. Springmeyer, “Bidirectional reflection functions from surface bump maps,” Comput. Graph. 21(4), 273–281 (1987).
[CrossRef]

McCluney, R.

R. McCluney, Introduction to Radiometry and Photometry (Artech House, Norwood, Mass., 1994).

McCormick, N.

N. McCormick, “Inverse radiative transfer problems: a review,” Nucl. Sci. Eng. 112, 185–198 (1992).

Miller, G.

G. Miller, C. Hoffman, “Illumination and reflection maps: simulated objects in simulated and real environments,” in SIGGRAPH 84 Advanced Computer Graphics Animation Seminar Notes (Association for Computing Machinery, New York, 1984).

Nemec, P.

B. Cabral, M. Olano, P. Nemec, “Reflection space image based rendering,” in SIGGRAPH 99, Computer Graphics Proceedings, Annual Conference Series, A. Rockwood, ed. (Association for Computing Machinery, New York, 1999), pp. 165–170.

Olano, M.

B. Cabral, M. Olano, P. Nemec, “Reflection space image based rendering,” in SIGGRAPH 99, Computer Graphics Proceedings, Annual Conference Series, A. Rockwood, ed. (Association for Computing Machinery, New York, 1999), pp. 165–170.

Onodera, Y.

T. Inui, Y. Tanabe, Y. Onodera, Group Theory and its Applications in Physics (Springer Verlag, New York, 1990).

Preisendorfer, R. W.

R. W. Preisendorfer, Hydrologic Optics (U.S. Department of Commerce, Washington, D.C., 1976).

Salesin, D.

D. Wood, D. Azuma, K. Aldinger, B. Curless, T. Duchamp, D. Salesin, W. Stuetzle, “Surface light fields for 3D photography,” in SIGGRAPH 2000, Computer Graphics Proceedings, Annual Conference Series, K. Akeley, ed. (Association for Computing Machinery, New York, 2000), pp. 287–296.

Sato, I.

I. Sato, Y. Sato, K. Ikeuchi, “Illumination distribution from brightness in shadows: adaptive estimation of illumination distribution with unknown reflectance properties in shadow regions,” in Seventh IEEE International Conference on Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1999), pp. 875–882.

Sato, Y.

I. Sato, Y. Sato, K. Ikeuchi, “Illumination distribution from brightness in shadows: adaptive estimation of illumination distribution with unknown reflectance properties in shadow regions,” in Seventh IEEE International Conference on Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1999), pp. 875–882.

Sillion, F.

F. Sillion, J. Arvo, S. Westin, D. Greenberg, “A global illumination solution for general reflectance distributions,” Comput. Graph. 25(4), 187–196 (1991).
[CrossRef]

Springmeyer, R.

B. Cabral, N. Max, R. Springmeyer, “Bidirectional reflection functions from surface bump maps,” Comput. Graph. 21(4), 273–281 (1987).
[CrossRef]

Stuetzle, W.

D. Wood, D. Azuma, K. Aldinger, B. Curless, T. Duchamp, D. Salesin, W. Stuetzle, “Surface light fields for 3D photography,” in SIGGRAPH 2000, Computer Graphics Proceedings, Annual Conference Series, K. Akeley, ed. (Association for Computing Machinery, New York, 2000), pp. 287–296.

Szeliski, R.

S. Gortler, R. Grzeszczuk, R. Szeliski, M. Cohen, “The lumigraph,” in SIGGRAPH 96, Computer Graphics Proceedings, Annual Conference Series, H. Rushmeier, ed. (Association for Computing Machinery, New York, 1996), pp. 43–54.

Tanabe, Y.

T. Inui, Y. Tanabe, Y. Onodera, Group Theory and its Applications in Physics (Springer Verlag, New York, 1990).

Torrance, K.

S. Westin, J. Arvo, K. Torrance, “Predicting reflectance functions from complex surfaces,” Comput. Graph. 26(2), 255–264 (1992).
[CrossRef]

van Doorn, A.

Westin, S.

S. Westin, J. Arvo, K. Torrance, “Predicting reflectance functions from complex surfaces,” Comput. Graph. 26(2), 255–264 (1992).
[CrossRef]

F. Sillion, J. Arvo, S. Westin, D. Greenberg, “A global illumination solution for general reflectance distributions,” Comput. Graph. 25(4), 187–196 (1991).
[CrossRef]

Wood, D.

D. Wood, D. Azuma, K. Aldinger, B. Curless, T. Duchamp, D. Salesin, W. Stuetzle, “Surface light fields for 3D photography,” in SIGGRAPH 2000, Computer Graphics Proceedings, Annual Conference Series, K. Akeley, ed. (Association for Computing Machinery, New York, 2000), pp. 287–296.

Yuille, A.

R. Epstein, P. W. Hallinan, A. Yuille, “5 plus or minus 2 eigenimages suffice: an empirical investigation of low-dimensional lighting models,” in IEEE Workshop on Physics-Based Modeling in Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1995), pp. 108–116.

Comput. Graph.

B. Cabral, N. Max, R. Springmeyer, “Bidirectional reflection functions from surface bump maps,” Comput. Graph. 21(4), 273–281 (1987).
[CrossRef]

S. Westin, J. Arvo, K. Torrance, “Predicting reflectance functions from complex surfaces,” Comput. Graph. 26(2), 255–264 (1992).
[CrossRef]

F. Sillion, J. Arvo, S. Westin, D. Greenberg, “A global illumination solution for general reflectance distributions,” Comput. Graph. 25(4), 187–196 (1991).
[CrossRef]

Int. J. Comput. Vision

P. Belhumeur, D. 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, 51–151 (1939) (translated by P. Moon and G. Timoshenko).

J. Opt. Soc. Am. A

Nucl. Sci. Eng.

N. McCormick, “Inverse radiative transfer problems: a review,” Nucl. Sci. Eng. 112, 185–198 (1992).

Other

G. Miller, C. Hoffman, “Illumination and reflection maps: simulated objects in simulated and real environments,” in SIGGRAPH 84 Advanced Computer Graphics Animation Seminar Notes (Association for Computing Machinery, New York, 1984).

B. Cabral, M. Olano, P. Nemec, “Reflection space image based rendering,” in SIGGRAPH 99, Computer Graphics Proceedings, Annual Conference Series, A. Rockwood, ed. (Association for Computing Machinery, New York, 1999), pp. 165–170.

J. Haddon, D. Forsyth, “Shape representations from shading primitives,” in Fifth European Conference on Computer Vision, H. Burkhardt, B. Neumann, eds. (Springer-Verlag, Berlin, 1998), pp. 415–431.

D. Jacobs, P. Belhumeur, R. Basri, “Comparing images under variable illumination,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1998), pp. 610–617.

R. Epstein, P. W. Hallinan, A. Yuille, “5 plus or minus 2 eigenimages suffice: an empirical investigation of low-dimensional lighting models,” in IEEE Workshop on Physics-Based Modeling in Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1995), pp. 108–116.

R. Basri, D. Jacobs, “Lambertian reflectance and linear subspaces,” (NEC Research Institute, Princeton, N.J., 2000).

R. W. Preisendorfer, Hydrologic Optics (U.S. Department of Commerce, Washington, D.C., 1976).

S. R. Marschner, D. P. Greenberg, “Inverse lighting for photography,” in Fifth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 262–265.

I. Sato, Y. Sato, K. Ikeuchi, “Illumination distribution from brightness in shadows: adaptive estimation of illumination distribution with unknown reflectance properties in shadow regions,” in Seventh IEEE International Conference on Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1999), pp. 875–882.

R. McCluney, Introduction to Radiometry and Photometry (Artech House, Norwood, Mass., 1994).

S. Gortler, R. Grzeszczuk, R. Szeliski, M. Cohen, “The lumigraph,” in SIGGRAPH 96, Computer Graphics Proceedings, Annual Conference Series, H. Rushmeier, ed. (Association for Computing Machinery, New York, 1996), pp. 43–54.

M. Levoy, P. Hanrahan, “Light field rendering,” in SIGGRAPH 96, Computer Graphics Proceedings, Annual Conference Series, H. Rushmeier, ed. (Association for Computing Machinery, New York, 1996), pp. 31–42.

D. Wood, D. Azuma, K. Aldinger, B. Curless, T. Duchamp, D. Salesin, W. Stuetzle, “Surface light fields for 3D photography,” in SIGGRAPH 2000, Computer Graphics Proceedings, Annual Conference Series, K. Akeley, ed. (Association for Computing Machinery, New York, 2000), pp. 287–296.

R. Ramamoorthi, P. Hanrahan, “Analysis of planar light fields from homogeneous convex curved surfaces under distant illumination,” http://graphics.stanford.edu/papers/planarlf .

T. Inui, Y. Tanabe, Y. Onodera, Group Theory and its Applications in Physics (Springer Verlag, New York, 1990).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

T. M. MacRobert, Spherical Harmonics; An Elementary Treatise on Harmonic Functions, with Applications (Dover, New York, 1948).

M. D’Zmura, “Shading ambiguity: reflectance and illumination,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass.1991), pp. 187–207.

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Figures (5)

Fig. 1
Fig. 1

Local geometry. Quantities are primed because they are all in local coordinates.

Fig. 2
Fig. 2

Diagram showing how the rotation corresponding to (α, β, γ) transforms between local (primed) and global (unprimed) coordinates.

Fig. 3
Fig. 3

The solid curve is a plot of An versus n. It can be seen that odd terms with n>1 have An=0. Also, as n increases, the coefficients rapidly decay.

Fig. 4
Fig. 4

Successive approximations to the clamped-cosine function by adding more spherical harmonic terms. For n=2, we already get a very good approximation.

Fig. 5
Fig. 5

(A) One of the photographs of the mirror sphere (left) and the Teflon sphere (right), (B) the real lighting as recovered by using the mirrored sphere, (C) the lighting obtained by considering only the first nine coefficients of (B), i.e., up to order 2, (D) the recovered lighting obtained by calculating the first nine coefficients of the light from the radiant exitance of the Teflon sphere, (E) the real lighting up to order 4, (F) an attempt to recover the lighting up to order 4 by also calculating the nine order-4 modes. Images (B)–(F) are visualizations obtained by unwrapping spherical coordinates of the lighting. θi ranges over [0, π] uniformly from top to bottom, and ϕi ranges over [0, 2π] uniformly from left to right. The zero of the lighting is the gray color used for the background of image (B).

Tables (1)

Tables Icon

Table 1 Comparison of Recovered and Real Lighting Coefficients

Equations (47)

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

E(x)=ΩL(x, θi, ϕi)cos θi dΩ,
B(x)=ρE(x),
E(n)=ΩL(θi, ϕi)cos θi dΩ.
n=(sin α cos β, sin α sin β, cos α);
E(α, β, γ)=ΩL(θi, ϕi)A(θi)dΩ.
Rα,β,γ=Rz(β)Ry(α)Rz(γ),
(θi, ϕi)=Rα,β,γ(θi,ϕi),
L(θi, ϕi)=L(Rα,β,γ(θi, ϕi)).
E(α, β, γ)=ΩL(Rα,β,γ(θi, ϕi))A(θi)dΩ.
Yl,m(θ, ϕ)=Nl,mPl,m(cos θ)exp(Imϕ),
L(θi, ϕi)=l=0 m=-l+lLl,mYl,m(θi, ϕi).
Rz(β){Yl,m(θi, ϕi)}=Yl,m(Rz(β){θi, ϕi}).
Rz(β){Yl,m(θi, ϕi)}=Yl,m(θi, ϕi+β)=exp(Imβ)Yl,m(θi, ϕi).
Ry(α){Yl,m(θi, ϕi)}=m=-llDm,ml(α)Yl,m(θi, ϕi),
Rα,β,γ{Yl,m(θi, ϕi)}=Rz(β)Ry(α)Rz(γ){Yl,m(θi, ϕi)}=m=-llD˜m,ml(α, β, γ)Yl,m(θi, ϕi),
D˜m,ml(α, β, γ)=Dm,ml(α)exp(Imβ)exp(Imγ).
Rα,β,γ{Yl,m(0, ϕi)}=2l+14π1/2D˜m,0l(α, β, γ).
A(θi)=cos θi=n=0AnYn,0(θi).
An=2π0π/2Yn,0(θi)cos θi sin θi dθi.
An=2π2n+14π1/201Pn(u)P1(u)du.
-11Pa(u)Pb(u)=22a+1δa,b.
n=1:An=π/3,
n>1,odd:An=0,
neven:An=2π2n+14π1/2 (-1)n/2-1(n+2)(n-1)×n!2n(n!/2)2.
E(α, β, γ)=n=0 l=0 m=-ll m=-llLl,mAnDm,ml(α)×exp(Imβ)exp(Imγ)Tn,l,m,
Tn,l,m=ϕi=02πθi=0πYl,m(θi, ϕi)Yn,0(θi, ϕi)×sin θi dθidϕi.
ϕi=02πθi=0πYl,m(θi, ϕi)Yn,0(θi, ϕi)sin θi dθidϕi=δl,nδm,0.
E(α, β, γ)=l=0 m=-1lLl,mAlDm,0l(α)exp(Imβ).
Dm,0l(α)exp(Imβ)=4π2l+11/2Yl,m(α, β).
E(α, β, γ)=l=0 m=-ll4π2l+11/2AlLl,mYl,m(α, β).
E(α, β)=l=0 m=-llEl,mYl,m(α, β).
El,m=4π2l+11/2AlLl,m.
Ll,m=2l+14π1/2 El,mAl.
ΔE(α, β)=C(α, β)0π/2Y3,0(θi)cos θi sin θidθi.
ΔE(α, β)=C(α, β)0π/253 cos3 θi-cos θi×cos θi sin θidθi=C(α, β)0153u3-uu du=C(α, β)0153u4-u2du=C(α, β)53 15u501-13u301=C(α, β)13-13=0.
L1(θi, ϕi)=1,
L2(θi, ϕi)=1+53 cos3 θi-cos θi.
ΔL(θi, ϕi)=-ΔL(π-θi, π+ϕi).
-L(θi, ϕi)ΔL(θi, ϕi)L(π-θi, π+ϕi).
 |ΔL(θi, ϕi)|max[L(θi, ϕi),L(π-θi, π+ϕi)].
E0,0=3.142L0,0,
E1,m=2.094L1,m,
E2,m=0.785L2,m,
E3,m=0,
E4,m=-0.131L4,m,
E5,m=0,
E6,m=0.049L6,m.

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