Abstract

Estimating the illumination and the reflectance properties of an object surface from a few images is an important but challenging problem. The problem becomes even more challenging if we wish to deal with real-world objects that naturally have spatially inhomogeneous reflectance. In this paper, we derive a novel method for estimating the spatially varying specular reflectance properties of a surface of known geometry as well as the illumination distribution of a scene from a specular-only image, for instance, recovered from two images captured with a polarizer to separate reflection components. Unlike previous work, we do not assume the illumination to be a single point light source. We model specular reflection with a spherical statistical distribution and encode its spatial variation with a radial basis function (RBF) network of their parameter values, which allows us to formulate the simultaneous estimation of spatially varying specular reflectance and illumination as a constrained optimization based on the I-divergence measure. To solve it, we derive a variational algorithm based on the expectation maximization principle. At the same time, we estimate optimal encoding of the specular reflectance properties by learning the number, centers, and widths of the RBF hidden units. We demonstrate the effectiveness of the method on images of synthetic and real-world objects.

© 2011 Optical Society of America

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  1. G. Kay and T. Caelli, “Inverting an illumination model from range and intensity maps,” CVGIP, Image Underst. 59, 183–201(1994).
    [CrossRef]
  2. Y. Sato, M. Wheeler, and K. Ikeuchi, “Object shape and reflectance modeling from observation,” in Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 1997), pp. 379–387.
  3. P. Debevec, T. Hawkins, C. Tchou, H.-P. Duiker, W. Sarokin, and M. Sagar, “Acquiring the reflectance field of a human face,” in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2000), pp. 145–156.
  4. S. Marschner and D. Greenberg, “Inverse lighting for photography,” In Proceedings of IS&T/SID Fifth Color Imaging Conference (The Society for Imaging Science and Technology, 1997), pp. 262–265.
  5. K. Hara, K. Nishino, and K. Ikeuchi, “Mixture of spherical distributions for single-view relighting,” IEEE Trans. Patt. Anal. Mach. Intell. 30, 25–35 (2008).
    [CrossRef]
  6. K. Nishino, K. Ikeuchi, and Z. Zhang, “Re-rendering from a sparse set of images,” Tech. Rep. DU-CS-05-12 (Drexel University, 2005).
  7. R. Ramamoorthi and P. Hanrahan, “A signal processing framework for inverse rendering,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2001), pp. 117–128.
  8. H. Lensch, J. Kautz, M. Goesele, W. Heidrich, and H. Seidel, “Image-based reconstruction of spatially varying materials,” in Proceedings of the 12th Eurographics Workshop on Rendering Techniques, S.J.Gortler and K.Myszkowski, eds. (Springer, 2001), pp. 104–115.
  9. T. Zickler, R. Ramamoorthi, S. Enrique, and P. Belhumeur, “Reflectance sharing: predicting appearance from a sparse set of images of a known shape,” IEEE Trans. Patt. Anal. Mach. Intell. 28, 1287–1302 (2006).
    [CrossRef]
  10. D. Goldman, B. Curless, A. Hertzmann, and S. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Patt. Anal. Mach. Intell. 32, 1060–1071 (2010).
    [CrossRef]
  11. I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inverse problems,” Ann. Stat. 19, 2032–2066(1991).
    [CrossRef]
  12. A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–38 (1977).
    [CrossRef]
  13. S. Nayar, K. Ikeuchi, and T. Kanade, “Determining shape and reflectance of Lambertian, specular, and hybrid surfaces using extended sources,” in Proceedings of the International Workshop on Industrial Applications of Machine Intelligence and Vision (IEEE, 1989), pp. 169–175.
    [CrossRef]
  14. Y. Sato and K. Ikeuchi, “Temporal-color space analysis of reflection,” J. Opt. Soc. Am. A 11, 2990–3002 (1994).
    [CrossRef]
  15. A. L. Yuille, D. Snow, R. Epstein, and P. N. Belhumeur, “Determining generative models of objects under varying illumination: shape and albedo from multiple images using SVD and integrability,” Int. J. Comput. Vis. 35, 203–222 (1999).
    [CrossRef]
  16. W. Y. Zhao and R. Chellappa, “Symmetric shape-from-shading using self-ratio image,” Int. J. Comput. Vis. 45, 55–75(2001).
    [CrossRef]
  17. Q. Zheng and R. Chellappa, “Estimation of illuminant direction, albedo and shape from shading,” IEEE Trans. Patt. Anal. Mach. Intell. 13, 680–702 (1991).
    [CrossRef]
  18. W. A. P. Smith and E. R. Hancock, “Recovering facial shape using a statistical model of surface normal direction,” IEEE Trans. Patt. Anal. Mach. Intell. 28, 1914–1930 (2006).
    [CrossRef]
  19. N. Birkbeck, D. Cobzas, P. F. Sturm, and M. Jagersand, “Variational shape and reflectance estimation under changing light and viewpoints,” in Proceedings of the 9th European Conference on Computer Vision (Springer, 2006), pp. 536–549.
  20. N. Alldrin, T. Zickler, and D. Kriegman, “Photometric stereo with non-parametric and spatially-varying reflectance,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.
  21. S. Biswas, G. Aggarwal, and R. Chellappa, “Robust estimation of albedo for illumination-invariant matching and shape recovery,” IEEE Trans. Patt. Anal. Mach. Intell. 31, 884–899(2009).
    [CrossRef]
  22. K. Ikeuchi and K. Sato, “Determining reflectance properties of an object using range and brightness images,” IEEE Trans. Patt. Anal. Mach. Intell. 13, 1139–1153 (1991).
    [CrossRef]
  23. I. Sato, Y. Sato, and K. Ikeuchi, “Illumination from shadows,” IEEE Trans. Patt. Anal. Mach. Intell. 25, 290–300 (2003).
    [CrossRef]
  24. S. Tominaga and N. Tanaka, “Estimating reflection parameters from a single color image,” IEEE Comput. Graph. Appl. 20, 58–66 (2000).
    [CrossRef]
  25. T. Yu, H. Wang, N.Ahuja, and W.-C. Chen, “Sparse lumigraph relighting by illumination and reflectance estimation from multi-view images,” in Rendering Techniques 2006: Eurographics Symposium on Rendering, T.Akenine-Mo¨ller and W.Heidrich, eds. (Eurographics Association, 2006), pp. 41–50.
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    [CrossRef]
  27. R. Fisher, “Dispersion on a sphere,” Proc. R. Soc. Lond., Ser. A. 217, 295–305 (1953).
    [CrossRef]
  28. K. Mardia and P. Jupp, Directional Statistics (Wiley, 2000).
  29. F. Solomon and K. Ikeuchi, “Extracting the shape and roughness of specular lobe objects using four light photometric stereo,” IEEE Trans. Patt. Anal. Mach. Intell. 18, 449–454 (1996).
    [CrossRef]
  30. H. Kameoka, T. Nishimoto, and S. Sagayama, “A multipitch analyzer based on harmonic temporal structured clustering,” IEEE Trans. Audio Speech Lang. Process. 15, 982–994(2007).
    [CrossRef]
  31. N. Ueda and R. Nakano, “Deterministic annealing EM algorithm,” Neural Netw. 11, 271–282 (1998).
    [CrossRef]
  32. M. Lavielle and E. Moulines, “A simulated annealing version of the EM algorithm for non-Gaussian deconvolution,” Stat. Comput. 7, 229–236 (1997).
    [CrossRef]
  33. L. Yingwei, N. Sundararajan, and P. Saratchandran, “A sequential learning scheme for function approximation using minimal radial basis function neural networks,” Neural Comput. 9,461–478 (1997).
    [CrossRef]
  34. S. Nayar, X. Fang, and T. Boult, “Removal of specularities using color and polarization,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1993), pp. 583–590.
    [CrossRef]
  35. L. Wolff and T. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Patt. Anal. Mach. Intell. 13, 635–657 (1991).
    [CrossRef]

2010 (1)

D. Goldman, B. Curless, A. Hertzmann, and S. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Patt. Anal. Mach. Intell. 32, 1060–1071 (2010).
[CrossRef]

2009 (1)

S. Biswas, G. Aggarwal, and R. Chellappa, “Robust estimation of albedo for illumination-invariant matching and shape recovery,” IEEE Trans. Patt. Anal. Mach. Intell. 31, 884–899(2009).
[CrossRef]

2008 (1)

K. Hara, K. Nishino, and K. Ikeuchi, “Mixture of spherical distributions for single-view relighting,” IEEE Trans. Patt. Anal. Mach. Intell. 30, 25–35 (2008).
[CrossRef]

2007 (1)

H. Kameoka, T. Nishimoto, and S. Sagayama, “A multipitch analyzer based on harmonic temporal structured clustering,” IEEE Trans. Audio Speech Lang. Process. 15, 982–994(2007).
[CrossRef]

2006 (2)

T. Zickler, R. Ramamoorthi, S. Enrique, and P. Belhumeur, “Reflectance sharing: predicting appearance from a sparse set of images of a known shape,” IEEE Trans. Patt. Anal. Mach. Intell. 28, 1287–1302 (2006).
[CrossRef]

W. A. P. Smith and E. R. Hancock, “Recovering facial shape using a statistical model of surface normal direction,” IEEE Trans. Patt. Anal. Mach. Intell. 28, 1914–1930 (2006).
[CrossRef]

2003 (1)

I. Sato, Y. Sato, and K. Ikeuchi, “Illumination from shadows,” IEEE Trans. Patt. Anal. Mach. Intell. 25, 290–300 (2003).
[CrossRef]

2001 (1)

W. Y. Zhao and R. Chellappa, “Symmetric shape-from-shading using self-ratio image,” Int. J. Comput. Vis. 45, 55–75(2001).
[CrossRef]

2000 (1)

S. Tominaga and N. Tanaka, “Estimating reflection parameters from a single color image,” IEEE Comput. Graph. Appl. 20, 58–66 (2000).
[CrossRef]

1999 (1)

A. L. Yuille, D. Snow, R. Epstein, and P. N. Belhumeur, “Determining generative models of objects under varying illumination: shape and albedo from multiple images using SVD and integrability,” Int. J. Comput. Vis. 35, 203–222 (1999).
[CrossRef]

1998 (1)

N. Ueda and R. Nakano, “Deterministic annealing EM algorithm,” Neural Netw. 11, 271–282 (1998).
[CrossRef]

1997 (2)

M. Lavielle and E. Moulines, “A simulated annealing version of the EM algorithm for non-Gaussian deconvolution,” Stat. Comput. 7, 229–236 (1997).
[CrossRef]

L. Yingwei, N. Sundararajan, and P. Saratchandran, “A sequential learning scheme for function approximation using minimal radial basis function neural networks,” Neural Comput. 9,461–478 (1997).
[CrossRef]

1996 (1)

F. Solomon and K. Ikeuchi, “Extracting the shape and roughness of specular lobe objects using four light photometric stereo,” IEEE Trans. Patt. Anal. Mach. Intell. 18, 449–454 (1996).
[CrossRef]

1994 (2)

G. Kay and T. Caelli, “Inverting an illumination model from range and intensity maps,” CVGIP, Image Underst. 59, 183–201(1994).
[CrossRef]

Y. Sato and K. Ikeuchi, “Temporal-color space analysis of reflection,” J. Opt. Soc. Am. A 11, 2990–3002 (1994).
[CrossRef]

1991 (4)

Q. Zheng and R. Chellappa, “Estimation of illuminant direction, albedo and shape from shading,” IEEE Trans. Patt. Anal. Mach. Intell. 13, 680–702 (1991).
[CrossRef]

I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inverse problems,” Ann. Stat. 19, 2032–2066(1991).
[CrossRef]

K. Ikeuchi and K. Sato, “Determining reflectance properties of an object using range and brightness images,” IEEE Trans. Patt. Anal. Mach. Intell. 13, 1139–1153 (1991).
[CrossRef]

L. Wolff and T. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Patt. Anal. Mach. Intell. 13, 635–657 (1991).
[CrossRef]

1977 (1)

A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–38 (1977).
[CrossRef]

1967 (1)

1953 (1)

R. Fisher, “Dispersion on a sphere,” Proc. R. Soc. Lond., Ser. A. 217, 295–305 (1953).
[CrossRef]

Aggarwal, G.

S. Biswas, G. Aggarwal, and R. Chellappa, “Robust estimation of albedo for illumination-invariant matching and shape recovery,” IEEE Trans. Patt. Anal. Mach. Intell. 31, 884–899(2009).
[CrossRef]

Ahuja, N.

T. Yu, H. Wang, N.Ahuja, and W.-C. Chen, “Sparse lumigraph relighting by illumination and reflectance estimation from multi-view images,” in Rendering Techniques 2006: Eurographics Symposium on Rendering, T.Akenine-Mo¨ller and W.Heidrich, eds. (Eurographics Association, 2006), pp. 41–50.

Alldrin, N.

N. Alldrin, T. Zickler, and D. Kriegman, “Photometric stereo with non-parametric and spatially-varying reflectance,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.

Belhumeur, P.

T. Zickler, R. Ramamoorthi, S. Enrique, and P. Belhumeur, “Reflectance sharing: predicting appearance from a sparse set of images of a known shape,” IEEE Trans. Patt. Anal. Mach. Intell. 28, 1287–1302 (2006).
[CrossRef]

Belhumeur, P. N.

A. L. Yuille, D. Snow, R. Epstein, and P. N. Belhumeur, “Determining generative models of objects under varying illumination: shape and albedo from multiple images using SVD and integrability,” Int. J. Comput. Vis. 35, 203–222 (1999).
[CrossRef]

Birkbeck, N.

N. Birkbeck, D. Cobzas, P. F. Sturm, and M. Jagersand, “Variational shape and reflectance estimation under changing light and viewpoints,” in Proceedings of the 9th European Conference on Computer Vision (Springer, 2006), pp. 536–549.

Biswas, S.

S. Biswas, G. Aggarwal, and R. Chellappa, “Robust estimation of albedo for illumination-invariant matching and shape recovery,” IEEE Trans. Patt. Anal. Mach. Intell. 31, 884–899(2009).
[CrossRef]

Boult, T.

L. Wolff and T. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Patt. Anal. Mach. Intell. 13, 635–657 (1991).
[CrossRef]

S. Nayar, X. Fang, and T. Boult, “Removal of specularities using color and polarization,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1993), pp. 583–590.
[CrossRef]

Caelli, T.

G. Kay and T. Caelli, “Inverting an illumination model from range and intensity maps,” CVGIP, Image Underst. 59, 183–201(1994).
[CrossRef]

Chellappa, R.

S. Biswas, G. Aggarwal, and R. Chellappa, “Robust estimation of albedo for illumination-invariant matching and shape recovery,” IEEE Trans. Patt. Anal. Mach. Intell. 31, 884–899(2009).
[CrossRef]

W. Y. Zhao and R. Chellappa, “Symmetric shape-from-shading using self-ratio image,” Int. J. Comput. Vis. 45, 55–75(2001).
[CrossRef]

Q. Zheng and R. Chellappa, “Estimation of illuminant direction, albedo and shape from shading,” IEEE Trans. Patt. Anal. Mach. Intell. 13, 680–702 (1991).
[CrossRef]

Chen, W.-C.

T. Yu, H. Wang, N.Ahuja, and W.-C. Chen, “Sparse lumigraph relighting by illumination and reflectance estimation from multi-view images,” in Rendering Techniques 2006: Eurographics Symposium on Rendering, T.Akenine-Mo¨ller and W.Heidrich, eds. (Eurographics Association, 2006), pp. 41–50.

Cobzas, D.

N. Birkbeck, D. Cobzas, P. F. Sturm, and M. Jagersand, “Variational shape and reflectance estimation under changing light and viewpoints,” in Proceedings of the 9th European Conference on Computer Vision (Springer, 2006), pp. 536–549.

Csiszár, I.

I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inverse problems,” Ann. Stat. 19, 2032–2066(1991).
[CrossRef]

Curless, B.

D. Goldman, B. Curless, A. Hertzmann, and S. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Patt. Anal. Mach. Intell. 32, 1060–1071 (2010).
[CrossRef]

Debevec, P.

P. Debevec, T. Hawkins, C. Tchou, H.-P. Duiker, W. Sarokin, and M. Sagar, “Acquiring the reflectance field of a human face,” in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2000), pp. 145–156.

Dempster, A.

A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–38 (1977).
[CrossRef]

Duiker, H.-P.

P. Debevec, T. Hawkins, C. Tchou, H.-P. Duiker, W. Sarokin, and M. Sagar, “Acquiring the reflectance field of a human face,” in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2000), pp. 145–156.

Enrique, S.

T. Zickler, R. Ramamoorthi, S. Enrique, and P. Belhumeur, “Reflectance sharing: predicting appearance from a sparse set of images of a known shape,” IEEE Trans. Patt. Anal. Mach. Intell. 28, 1287–1302 (2006).
[CrossRef]

Epstein, R.

A. L. Yuille, D. Snow, R. Epstein, and P. N. Belhumeur, “Determining generative models of objects under varying illumination: shape and albedo from multiple images using SVD and integrability,” Int. J. Comput. Vis. 35, 203–222 (1999).
[CrossRef]

Fang, X.

S. Nayar, X. Fang, and T. Boult, “Removal of specularities using color and polarization,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1993), pp. 583–590.
[CrossRef]

Fisher, R.

R. Fisher, “Dispersion on a sphere,” Proc. R. Soc. Lond., Ser. A. 217, 295–305 (1953).
[CrossRef]

Goesele, M.

H. Lensch, J. Kautz, M. Goesele, W. Heidrich, and H. Seidel, “Image-based reconstruction of spatially varying materials,” in Proceedings of the 12th Eurographics Workshop on Rendering Techniques, S.J.Gortler and K.Myszkowski, eds. (Springer, 2001), pp. 104–115.

Goldman, D.

D. Goldman, B. Curless, A. Hertzmann, and S. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Patt. Anal. Mach. Intell. 32, 1060–1071 (2010).
[CrossRef]

Greenberg, D.

S. Marschner and D. Greenberg, “Inverse lighting for photography,” In Proceedings of IS&T/SID Fifth Color Imaging Conference (The Society for Imaging Science and Technology, 1997), pp. 262–265.

Hancock, E. R.

W. A. P. Smith and E. R. Hancock, “Recovering facial shape using a statistical model of surface normal direction,” IEEE Trans. Patt. Anal. Mach. Intell. 28, 1914–1930 (2006).
[CrossRef]

Hanrahan, P.

R. Ramamoorthi and P. Hanrahan, “A signal processing framework for inverse rendering,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2001), pp. 117–128.

Hara, K.

K. Hara, K. Nishino, and K. Ikeuchi, “Mixture of spherical distributions for single-view relighting,” IEEE Trans. Patt. Anal. Mach. Intell. 30, 25–35 (2008).
[CrossRef]

Hawkins, T.

P. Debevec, T. Hawkins, C. Tchou, H.-P. Duiker, W. Sarokin, and M. Sagar, “Acquiring the reflectance field of a human face,” in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2000), pp. 145–156.

Heidrich, W.

H. Lensch, J. Kautz, M. Goesele, W. Heidrich, and H. Seidel, “Image-based reconstruction of spatially varying materials,” in Proceedings of the 12th Eurographics Workshop on Rendering Techniques, S.J.Gortler and K.Myszkowski, eds. (Springer, 2001), pp. 104–115.

Hertzmann, A.

D. Goldman, B. Curless, A. Hertzmann, and S. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Patt. Anal. Mach. Intell. 32, 1060–1071 (2010).
[CrossRef]

Ikeuchi, K.

K. Hara, K. Nishino, and K. Ikeuchi, “Mixture of spherical distributions for single-view relighting,” IEEE Trans. Patt. Anal. Mach. Intell. 30, 25–35 (2008).
[CrossRef]

I. Sato, Y. Sato, and K. Ikeuchi, “Illumination from shadows,” IEEE Trans. Patt. Anal. Mach. Intell. 25, 290–300 (2003).
[CrossRef]

F. Solomon and K. Ikeuchi, “Extracting the shape and roughness of specular lobe objects using four light photometric stereo,” IEEE Trans. Patt. Anal. Mach. Intell. 18, 449–454 (1996).
[CrossRef]

Y. Sato and K. Ikeuchi, “Temporal-color space analysis of reflection,” J. Opt. Soc. Am. A 11, 2990–3002 (1994).
[CrossRef]

K. Ikeuchi and K. Sato, “Determining reflectance properties of an object using range and brightness images,” IEEE Trans. Patt. Anal. Mach. Intell. 13, 1139–1153 (1991).
[CrossRef]

S. Nayar, K. Ikeuchi, and T. Kanade, “Determining shape and reflectance of Lambertian, specular, and hybrid surfaces using extended sources,” in Proceedings of the International Workshop on Industrial Applications of Machine Intelligence and Vision (IEEE, 1989), pp. 169–175.
[CrossRef]

K. Nishino, K. Ikeuchi, and Z. Zhang, “Re-rendering from a sparse set of images,” Tech. Rep. DU-CS-05-12 (Drexel University, 2005).

Y. Sato, M. Wheeler, and K. Ikeuchi, “Object shape and reflectance modeling from observation,” in Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 1997), pp. 379–387.

Jagersand, M.

N. Birkbeck, D. Cobzas, P. F. Sturm, and M. Jagersand, “Variational shape and reflectance estimation under changing light and viewpoints,” in Proceedings of the 9th European Conference on Computer Vision (Springer, 2006), pp. 536–549.

Jupp, P.

K. Mardia and P. Jupp, Directional Statistics (Wiley, 2000).

Kameoka, H.

H. Kameoka, T. Nishimoto, and S. Sagayama, “A multipitch analyzer based on harmonic temporal structured clustering,” IEEE Trans. Audio Speech Lang. Process. 15, 982–994(2007).
[CrossRef]

Kanade, T.

S. Nayar, K. Ikeuchi, and T. Kanade, “Determining shape and reflectance of Lambertian, specular, and hybrid surfaces using extended sources,” in Proceedings of the International Workshop on Industrial Applications of Machine Intelligence and Vision (IEEE, 1989), pp. 169–175.
[CrossRef]

Kautz, J.

H. Lensch, J. Kautz, M. Goesele, W. Heidrich, and H. Seidel, “Image-based reconstruction of spatially varying materials,” in Proceedings of the 12th Eurographics Workshop on Rendering Techniques, S.J.Gortler and K.Myszkowski, eds. (Springer, 2001), pp. 104–115.

Kay, G.

G. Kay and T. Caelli, “Inverting an illumination model from range and intensity maps,” CVGIP, Image Underst. 59, 183–201(1994).
[CrossRef]

Kriegman, D.

N. Alldrin, T. Zickler, and D. Kriegman, “Photometric stereo with non-parametric and spatially-varying reflectance,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.

Laird, N.

A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–38 (1977).
[CrossRef]

Lavielle, M.

M. Lavielle and E. Moulines, “A simulated annealing version of the EM algorithm for non-Gaussian deconvolution,” Stat. Comput. 7, 229–236 (1997).
[CrossRef]

Lensch, H.

H. Lensch, J. Kautz, M. Goesele, W. Heidrich, and H. Seidel, “Image-based reconstruction of spatially varying materials,” in Proceedings of the 12th Eurographics Workshop on Rendering Techniques, S.J.Gortler and K.Myszkowski, eds. (Springer, 2001), pp. 104–115.

Mardia, K.

K. Mardia and P. Jupp, Directional Statistics (Wiley, 2000).

Marschner, S.

S. Marschner and D. Greenberg, “Inverse lighting for photography,” In Proceedings of IS&T/SID Fifth Color Imaging Conference (The Society for Imaging Science and Technology, 1997), pp. 262–265.

Moulines, E.

M. Lavielle and E. Moulines, “A simulated annealing version of the EM algorithm for non-Gaussian deconvolution,” Stat. Comput. 7, 229–236 (1997).
[CrossRef]

Nakano, R.

N. Ueda and R. Nakano, “Deterministic annealing EM algorithm,” Neural Netw. 11, 271–282 (1998).
[CrossRef]

Nayar, S.

S. Nayar, X. Fang, and T. Boult, “Removal of specularities using color and polarization,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1993), pp. 583–590.
[CrossRef]

S. Nayar, K. Ikeuchi, and T. Kanade, “Determining shape and reflectance of Lambertian, specular, and hybrid surfaces using extended sources,” in Proceedings of the International Workshop on Industrial Applications of Machine Intelligence and Vision (IEEE, 1989), pp. 169–175.
[CrossRef]

Nishimoto, T.

H. Kameoka, T. Nishimoto, and S. Sagayama, “A multipitch analyzer based on harmonic temporal structured clustering,” IEEE Trans. Audio Speech Lang. Process. 15, 982–994(2007).
[CrossRef]

Nishino, K.

K. Hara, K. Nishino, and K. Ikeuchi, “Mixture of spherical distributions for single-view relighting,” IEEE Trans. Patt. Anal. Mach. Intell. 30, 25–35 (2008).
[CrossRef]

K. Nishino, K. Ikeuchi, and Z. Zhang, “Re-rendering from a sparse set of images,” Tech. Rep. DU-CS-05-12 (Drexel University, 2005).

Ramamoorthi, R.

T. Zickler, R. Ramamoorthi, S. Enrique, and P. Belhumeur, “Reflectance sharing: predicting appearance from a sparse set of images of a known shape,” IEEE Trans. Patt. Anal. Mach. Intell. 28, 1287–1302 (2006).
[CrossRef]

R. Ramamoorthi and P. Hanrahan, “A signal processing framework for inverse rendering,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2001), pp. 117–128.

Rubin, D.

A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–38 (1977).
[CrossRef]

Sagar, M.

P. Debevec, T. Hawkins, C. Tchou, H.-P. Duiker, W. Sarokin, and M. Sagar, “Acquiring the reflectance field of a human face,” in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2000), pp. 145–156.

Sagayama, S.

H. Kameoka, T. Nishimoto, and S. Sagayama, “A multipitch analyzer based on harmonic temporal structured clustering,” IEEE Trans. Audio Speech Lang. Process. 15, 982–994(2007).
[CrossRef]

Saratchandran, P.

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[CrossRef]

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N. Alldrin, T. Zickler, and D. Kriegman, “Photometric stereo with non-parametric and spatially-varying reflectance,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.

K. Nishino, K. Ikeuchi, and Z. Zhang, “Re-rendering from a sparse set of images,” Tech. Rep. DU-CS-05-12 (Drexel University, 2005).

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Y. Sato, M. Wheeler, and K. Ikeuchi, “Object shape and reflectance modeling from observation,” in Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 1997), pp. 379–387.

P. Debevec, T. Hawkins, C. Tchou, H.-P. Duiker, W. Sarokin, and M. Sagar, “Acquiring the reflectance field of a human face,” in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (Association for Computing Machinery, 2000), pp. 145–156.

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Figures (9)

Fig. 1
Fig. 1

Spherical Torrance–Sparrow reflection model [5] represents the microfacet orientation distribution of the Torrance–Sparrow reflection model [26] with a directional statistics distribution, namely, the von Mises–Fisher distribution [27].

Fig. 2
Fig. 2

We formulate the joint estimation of spatially varying, inhomogeneous reflectance and illumination as learning the hidden units of a pseudo-RBF network.

Fig. 3
Fig. 3

(a) Surface geometry used to render the synthetic specular image. (b) Synthetic specular image. (c) Initial values of c j and r j . (d) Final values of c j and r j .

Fig. 4
Fig. 4

(a) K s ( x ) (ground truth values). (b) K s ( x ) (estimated values). (c) κ ( x ) (ground truth values). (d) κ ( x ) (estimated values).

Fig. 5
Fig. 5

Synthesized image under novel lighting conditions.

Fig. 6
Fig. 6

(a), (e) One out the input images. (b), (f) Other input image taken with orthogonal polarization filters to observe diffuse reflection only. (c), (g) Specular image computed by subtracting (b) from (a). (d), (h) Synthesized specular image using the estimated illumination and specular reflectance properties.

Fig. 7
Fig. 7

Estimated spatially varying specular reflectance properties: (a), (c) K s ( x ) , (b), (d) κ ( x ) .

Fig. 8
Fig. 8

Synthesized images of the scene under novel lighting conditions.

Fig. 9
Fig. 9

(a) Image captured under a particular lighting condition. (b) Specular image separated from (a). (c) K s ( x ) estimated (b). (d) κ ( x ) estimated (b). (e) Image captured under a lighting condition different from (a). (f) Specular image separated from (e). (g) K s ( x ) estimated (f). (h) κ ( x ) estimated (f).

Tables (5)

Tables Icon

Table 1 Experimental Conditions

Tables Icon

Table 2 Estimated Light Intensities

Tables Icon

Table 3 Estimated Light Intensities

Equations (32)

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

I S = π π 0 π 2 K S F G cos θ r L i ( θ i , ϕ i ) exp [ 2 κ sin 2 α ] sin θ i d θ i d ϕ i ,
I S ( x ) = π π 0 π 2 K S ( x ) cos θ r ( x ) L i ( θ i , ϕ i ) exp [ 2 κ ( x ) sin 2 α ( x ) ] sin θ i d θ i d ϕ i .
I S ( x ) I S ( x ) L ,
I S ( x ) = 2 π N L K S ( x ) cos θ r ( x ) l = 1 L L l exp [ 2 κ ( x ) sin 2 α l ( x ) ] ,
I ˜ ( x ) = l = 1 L I l ( x ) ,
I l ( x ) = L ˜ l K s ( x ) exp [ κ ( x ) ω l ( x ) ] ,
l = 1 L L ˜ l = 1 ,
K s ( x ) K s ( x ; Θ ) = exp [ i = 1 I K i Φ i ( x ) ] ,
κ ( x ) κ ( x ; θ ) = j = 1 J κ j ϕ j ( x ) ,
Φ i ( x ) = exp [ x C i 2 2 R i 2 ] ( i = 1 , 2 , , I ) ,
ϕ j ( x ) = exp [ x c j 2 2 r j 2 ] ( j = 1 , 2 , , J ) ,
I ˜ ( x ) I ˜ ( x ; Ω ) = l = 1 L I l ( x ; Ω l ) ,
I l ( x ; Ω l ) = L ˜ l exp [ i = 1 I K i Φ i ( x ) ω l ( x ) j = 1 J κ j ϕ j ( x ) ] = exp [ ln L ˜ l + i = 1 I K i Φ i ( x ) ω l ( x ) j = 1 J κ j ϕ j ( x ) ] ,
E ( Ω ) = S { D ( x ) log D ( x ) I ˜ ( x ; Ω ) [ D ( x ) I ˜ ( x ; Ω ) ] } d x ,
E ( Ω ) E + ( Ω , m ) = S { l = 1 L m l ( x ) D ( x ) log m l ( x ) D ( x ) I l ( x ; Ω l ) l = 1 L [ m l ( x ) D ( x ) I l ( x ; Ω l ) ] } d x = l = 1 L S { D l ( x ) log D l ( x ) I l ( x ; Ω l ) [ D l ( x ) I l ( x ; Ω l ) ] } d x ,
m l ( x ) = I l ( x ; Ω l ) l = 1 L I l ( x ; Ω l ) ( l = 1 , 2 , , L ) .
E + K = l = 1 L S [ D l ( x ) I l ( x ; Ω l ) ] Φ ( x ) d x ,
E + κ = l = 1 L S [ D l ( x ) I l ( x ; Ω l ) ] ω l ( x ) ϕ ( x ) d x ,
J ( Ω , m , λ ) = E + ( Ω , m ) λ ( l = 1 L L ˜ l 1 ) ,
J L ˜ l = 1 L ˜ l S [ D l ( x ) + I l ( x ; Ω l ) ] d x λ ( l = 1 , , L ) ,
J λ = l = 1 L L ˜ l + 1 .
L ˜ l = q l ( t + 1 ) λ p l ( t + 1 ) ( l = 1 , , L ) ,
l = 1 L L ˜ l + 1 = 0 ,
p l ( t ) = S K s ( t ) ( x ) exp [ κ ( t ) ( x ) ω l ( x ) ] d x ( l = 1 , , L ) ,
q l ( t ) = S D l ( t ) ( x ) d x ( l = 1 , , L ) ,
g ( t + 1 ) ( λ ) = l = 1 L q l ( t + 1 ) λ p l ( t + 1 ) + 1 = 0 .
λ ( τ λ + 1 ) = λ ( τ λ ) g ( t + 1 ) ( λ ( τ λ ) ) g ( t + 1 ) ( λ ( τ λ ) ) .
L ˜ l ( t + 1 ) = q l ( t + 1 ) λ ( τ λ ) p l ( t + 1 ) ( l = 1 , , L ) ,
E + C = l = 1 L S [ D l ( x ) I l ( x ; Ω l , Γ ) ] [ Ψ 1 ( x ) x T Ψ 2 ( x ) ] d x ,
E + c = l = 1 L S [ D l ( x ) I l ( x ; Ω l , Γ ) ] ω l ( x ) [ ψ 1 ( x ) x T ψ 2 ( x ) ] d x ,
R i = 1 M i M i C i C i , r j = 1 N j N j c j c j ,
K i ( t ) Φ i ( x ) i = 1 I K i ( t ) Φ ( x ) δ , ω l ( x ) κ j ( t ) ϕ j ( x ) j = 1 J ω l ( x ) κ j ( t ) ϕ j ( x ) δ .

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