Abstract

The theory of illuminance flow estimation by structure tensors is generalized for oblique viewing of anisotropic texture. An added benefit is that the theory predicts the behavior of unsupervised illuminant tilt estimators. The previous theory is refined with general matrix formulations and compacted by exploiting general properties of the structure tensor. Theoretical predictions based on the revised theory are presented and compared with experimental results on rendered images. The predicted curves are shown to conform well to expectations when the deviations from normal viewing and surface anisotropy are not large.

© 2009 Optical Society of America

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  1. J. J. Koenderink and S. C. Pont, “Irradiation direction from texture,” J. Opt. Soc. Am. A 20, 1875-1882 (2003).
    [CrossRef]
  2. J. Bigun and G. H. Granlund, “Optimal orientation detection of linear symmetry,” in Proceedings of International Conference on Computer Vision (IEEE, 1987), pp. 433-438.
  3. S. C. Pont and J. J. Koenderink, “Irradiation orientation from obliquely viewed texture,” in Proceedings of the Conference on Deep Structure, Singularities, and Computer Vision (EU,DSSCV05) (Springer, 2005), pp. 205-210.
    [CrossRef]
  4. S. Karlsson, S. C. Pont, and J. J. Koenderink, “Illuminance flow over anisotropic surfaces,” J. Opt. Soc. Am. A 25, 282-291 (2008).
    [CrossRef]
  5. M. Varma and A. Zisserman, “Estimating illumination direction from textured images,” in Proceedings of Computer Vision and Pattern Recognition Workshop (IEEE, 2004) pp. 179-186.
  6. D. Knill, “Estimating illuminant direction and degree of surface relief,” J. Opt. Soc. Am. A 7, 759-775 (1990).
    [CrossRef] [PubMed]
  7. M. Chantler and G. Delguste, “Illuminant-tilt estimation from images of isotropic texture,” IEE Proc. Vision Image Signal Process. 144, 213-219 (1997).
    [CrossRef]
  8. X. Lladó, A. Oliver, M. Petrou, J. Freixenet, and J. Martí, “Simultaneous surface texture classification and illumination tilt angle prediction,” in Proceedings of the British Machine Vision Conference, Norwich, England (2003).
  9. M. Berry and V. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809-1821 (1977).
    [CrossRef]
  10. M. S. Longuet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London, Ser. A 249, 321-64 (1956).
  11. P. Kube and A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704-707 (1988).
    [CrossRef]
  12. A. Pentland, “The visual inference of shape: computation from local features,” Ph.D. thesis (Massachusetts Institute of Technology, 1982).

2008

2003

1997

M. Chantler and G. Delguste, “Illuminant-tilt estimation from images of isotropic texture,” IEE Proc. Vision Image Signal Process. 144, 213-219 (1997).
[CrossRef]

1990

1988

P. Kube and A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704-707 (1988).
[CrossRef]

1977

M. Berry and V. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809-1821 (1977).
[CrossRef]

1956

M. S. Longuet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London, Ser. A 249, 321-64 (1956).

Berry, M.

M. Berry and V. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809-1821 (1977).
[CrossRef]

Bigun, J.

J. Bigun and G. H. Granlund, “Optimal orientation detection of linear symmetry,” in Proceedings of International Conference on Computer Vision (IEEE, 1987), pp. 433-438.

Chantler, M.

M. Chantler and G. Delguste, “Illuminant-tilt estimation from images of isotropic texture,” IEE Proc. Vision Image Signal Process. 144, 213-219 (1997).
[CrossRef]

Delguste, G.

M. Chantler and G. Delguste, “Illuminant-tilt estimation from images of isotropic texture,” IEE Proc. Vision Image Signal Process. 144, 213-219 (1997).
[CrossRef]

Freixenet, J.

X. Lladó, A. Oliver, M. Petrou, J. Freixenet, and J. Martí, “Simultaneous surface texture classification and illumination tilt angle prediction,” in Proceedings of the British Machine Vision Conference, Norwich, England (2003).

Granlund, G. H.

J. Bigun and G. H. Granlund, “Optimal orientation detection of linear symmetry,” in Proceedings of International Conference on Computer Vision (IEEE, 1987), pp. 433-438.

Hannay, V.

M. Berry and V. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809-1821 (1977).
[CrossRef]

Karlsson, S.

Knill, D.

Koenderink, J. J.

S. Karlsson, S. C. Pont, and J. J. Koenderink, “Illuminance flow over anisotropic surfaces,” J. Opt. Soc. Am. A 25, 282-291 (2008).
[CrossRef]

J. J. Koenderink and S. C. Pont, “Irradiation direction from texture,” J. Opt. Soc. Am. A 20, 1875-1882 (2003).
[CrossRef]

S. C. Pont and J. J. Koenderink, “Irradiation orientation from obliquely viewed texture,” in Proceedings of the Conference on Deep Structure, Singularities, and Computer Vision (EU,DSSCV05) (Springer, 2005), pp. 205-210.
[CrossRef]

Kube, P.

P. Kube and A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704-707 (1988).
[CrossRef]

Lladó, X.

X. Lladó, A. Oliver, M. Petrou, J. Freixenet, and J. Martí, “Simultaneous surface texture classification and illumination tilt angle prediction,” in Proceedings of the British Machine Vision Conference, Norwich, England (2003).

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London, Ser. A 249, 321-64 (1956).

Martí, J.

X. Lladó, A. Oliver, M. Petrou, J. Freixenet, and J. Martí, “Simultaneous surface texture classification and illumination tilt angle prediction,” in Proceedings of the British Machine Vision Conference, Norwich, England (2003).

Oliver, A.

X. Lladó, A. Oliver, M. Petrou, J. Freixenet, and J. Martí, “Simultaneous surface texture classification and illumination tilt angle prediction,” in Proceedings of the British Machine Vision Conference, Norwich, England (2003).

Pentland, A.

P. Kube and A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704-707 (1988).
[CrossRef]

A. Pentland, “The visual inference of shape: computation from local features,” Ph.D. thesis (Massachusetts Institute of Technology, 1982).

Petrou, M.

X. Lladó, A. Oliver, M. Petrou, J. Freixenet, and J. Martí, “Simultaneous surface texture classification and illumination tilt angle prediction,” in Proceedings of the British Machine Vision Conference, Norwich, England (2003).

Pont, S. C.

S. Karlsson, S. C. Pont, and J. J. Koenderink, “Illuminance flow over anisotropic surfaces,” J. Opt. Soc. Am. A 25, 282-291 (2008).
[CrossRef]

J. J. Koenderink and S. C. Pont, “Irradiation direction from texture,” J. Opt. Soc. Am. A 20, 1875-1882 (2003).
[CrossRef]

S. C. Pont and J. J. Koenderink, “Irradiation orientation from obliquely viewed texture,” in Proceedings of the Conference on Deep Structure, Singularities, and Computer Vision (EU,DSSCV05) (Springer, 2005), pp. 205-210.
[CrossRef]

Varma, M.

M. Varma and A. Zisserman, “Estimating illumination direction from textured images,” in Proceedings of Computer Vision and Pattern Recognition Workshop (IEEE, 2004) pp. 179-186.

Zisserman, A.

M. Varma and A. Zisserman, “Estimating illumination direction from textured images,” in Proceedings of Computer Vision and Pattern Recognition Workshop (IEEE, 2004) pp. 179-186.

IEE Proc. Vision Image Signal Process.

M. Chantler and G. Delguste, “Illuminant-tilt estimation from images of isotropic texture,” IEE Proc. Vision Image Signal Process. 144, 213-219 (1997).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

P. Kube and A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704-707 (1988).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

M. Berry and V. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809-1821 (1977).
[CrossRef]

Philos. Trans. R. Soc. London, Ser. A

M. S. Longuet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London, Ser. A 249, 321-64 (1956).

Other

X. Lladó, A. Oliver, M. Petrou, J. Freixenet, and J. Martí, “Simultaneous surface texture classification and illumination tilt angle prediction,” in Proceedings of the British Machine Vision Conference, Norwich, England (2003).

M. Varma and A. Zisserman, “Estimating illumination direction from textured images,” in Proceedings of Computer Vision and Pattern Recognition Workshop (IEEE, 2004) pp. 179-186.

J. Bigun and G. H. Granlund, “Optimal orientation detection of linear symmetry,” in Proceedings of International Conference on Computer Vision (IEEE, 1987), pp. 433-438.

S. C. Pont and J. J. Koenderink, “Irradiation orientation from obliquely viewed texture,” in Proceedings of the Conference on Deep Structure, Singularities, and Computer Vision (EU,DSSCV05) (Springer, 2005), pp. 205-210.
[CrossRef]

A. Pentland, “The visual inference of shape: computation from local features,” Ph.D. thesis (Massachusetts Institute of Technology, 1982).

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Figures (4)

Fig. 1
Fig. 1

Left: illustration of the modeling of oblique viewing by affine transform V. Dotted line illustrates illuminance flow. Right: illuminance flow across the surface of a textured sphere illuminated by a distant point source. Flow lines are the projection of the light vector into the local tangential plane of the surface.

Fig. 2
Fig. 2

Predictions based on the theory. Each curve corresponds to a specific illuminance flow direction ( φ ̂ = 90 , 85 , , 90 ) . Left, ( ξ h = 0 ); center, ( ξ h = 0.2 ) ; right, ( ξ h = 0.4 ) .

Fig. 3
Fig. 3

Rendered Gaussian surface used in the experiments with ξ = 0.4 and θ = 15 . Left, φ ̂ = 90 ° ; middle, φ ̂ = 10 ° ; right, φ ̂ = 10 ° and viewed from another direction.

Fig. 4
Fig. 4

Results on renderings. Each row is the output for a different illuminant incidence angle. The rows are all predicted by the theoretical curves of Fig. 2. Curves are for ( φ ̂ = 80 , 60 , , 80 ) .

Equations (29)

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ρ ( r ) = σ h 2 f ( r T r ),
ρ ( r ) = σ h 2 A f ( r T A T A r ) = σ h 2 G h 12 f ( r T G h r )
G h = h x h x h x h y h x h y h y h y .
ξ h = λ G h max λ G h min λ G h max + λ G h min [ 0 , 1 ) .
μ h = v G h max [ 0 ° , 180 ° ) .
σ h 2 ( G h ) = σ h 2 G h 1 2 ,
σ h 2 ( G h ) = σ h 2 G h 1 2 tr G h 2.
G h = ( 1 ξ h 2 ) a R μ h T D 1 ξ h , 1 + ξ h R μ h ,
l 3 D = { cos θ cos φ , cos θ sin φ , sin θ } ,
n ( h x , h y ) = { h x ( r ) , h y ( r ) , 1 } T ( h x 2 ( r ) + h y 2 ( r ) + 1 ) 1 2 .
q ( ε h x , ε h y ) = ε g T l cos θ + ε 2 2 g T g sin θ + O ( ε 3 ) ,
I ( r ) = r T { h ( r ) } l .
r I ( r ) = r T { h ( r ) } l = H r l .
G I = r { I } r T { I } = H r l l T H r .
G I ( 3 l l T + l l T ) ,
G I ( 2 + cos 2 φ sin 2 φ sin 2 φ 2 cos 2 φ ) .
r I V ( r ) = V T H V r l .
H V r = H k = ( 2 h ( u , v ) u 2 2 h ( u , v ) u v 2 h ( u , v ) u v 2 h ( u , v ) v 2 ) .
G I V = V T H V r l l T H V r V .
G I V = V T G I V = V T ( 3 l l T + l l T ) V .
r T { h ( A r ) } = A r T { h ( A r ) } A ,
r T { h ( A r ) } = A T H A r A .
r I A ( r ) = r T { h ( A r ) } l = A T H A r A l .
G I A = A T H A r ( A l ) ( A l ) T H A r A .
G I A = A T ( 3 l ̃ l ̃ T + l ̃ l ̃ T ) A ,
G I A = 3 G h l l T G h + G h l l T .
G I ( A , V ) = V T ( 3 G h l l T G h + G h l l T ) V .
μ I ( ξ , μ , ω , φ ) = 1 2 arg [ 3 ξ 2 cos ( 4 μ 2 φ ) ( sec 2 ω + 1 ) ( ( ξ 2 + 2 ) cos 2 φ 6 ξ cos 2 μ ) ( sec 2 ω + 1 ) + 2 ξ 2 2 ( ξ 2 + 2 ) sec 2 ω + 4 + i 6 ξ sec ω ( ξ sin ( 4 μ 2 φ ) 2 sin 2 μ ) + i 2 ( ξ 2 + 2 ) sec ω sin 2 φ ] .
I ( r ) = g T ( r ) ( l cos θ + g ( r ) 2 sin θ ) .

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