Abstract

A theory is presented to analyze images of anisotropic fine-scale surfaces. We investigate the estimates of illuminance flow by using structure tensors. For anisotropic surfaces, both the gradient-based and the Hessian-based tensors will yield deviations from the true illumination orientation. Our theory predicts this deviation. To show the use of this theory, an algorithm is derived that uses both tensors simultaneously to compensate for small amounts of anisotropy. Experimental results with rendered surfaces are shown to conform well to our theory.

© 2008 Optical Society of America

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  1. A. Gershun, "The light field," P. Moon and G. Timoshenko, translators, J. Math. Phys. (Cambridge, Mass.) 18, 51-151 (1939).
  2. B. M. ter Haar Romeny, L. Florack, J. J. Koenderink, and M. A. Viergever, "Scale-space: its natural operators and differential invariants," in Information Processing in Medical Imaging, A.C. F.Colchester and D.J.Hawkes, eds., Vol. 511 of Lecture Notes in Computer Science (Springer, 1991) pp. 239-255.
    [CrossRef]
  3. D. G. Lowe, "Object recognition from local scale-invariant features," in The Proceedings of the Seventh IEEE International Conference on Computer Vision, 1999 (IEEE, 1999) Vol. 2, pp. 1150-1157.
    [CrossRef]
  4. J. Bigün, T. Bigün, and K. Nilsson, "Recognition by symmetry derivatives and the generalized structure tensor," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1590-1605 (2004).
    [CrossRef] [PubMed]
  5. D. J. Kriegman and P. N. Belhumeur, "What is the set of images of an object under all possible illumination conditions?" Int. J. Comput. Vis. 28, 245-260 (1998).
    [CrossRef]
  6. W. Li, C. Wang, D. Xu, B. Luo, and Z. Chen, A Study on Illumination Invariant Face Recognition Methods Based on Multiple Eigenspaces, Vol. 3947 of Lecture Notes in Computer Science (Springer, 2005), pp. 131-136.
  7. J. J. Koenderink and S. C. Pont, "Irradiation direction from texture," J. Opt. Soc. Am. A 20, 1875-1882 (2003).
    [CrossRef]
  8. S. C. Pont and J. J. Koenderink, "Irradiation orientation from obliquely viewed texture," in Deep Structure, Singularities, and Computer Vision, O.F.Olsen, L.Florack, and A.Kuijper, eds. (Springer, 2005), pp. 205-210.
    [CrossRef]
  9. M. J. Chantler, "Why illuminant direction is fundamental to texture analysis." IEE Proc. Vision Image Signal Process. 142, 199-206 (1995).
    [CrossRef]
  10. P. Kube and A. Pentland, "On the imaging of fractal surfaces," IEEE Trans. Pattern Anal. Mach. Intell. 10, 704-707 (1988).
    [CrossRef]
  11. P. N. Belhumeur, D. Kriegman, and A. Yuille, "The bas-relief ambiguity," Int. J. Comput. Vis. 35, 33-44 (1999).
    [CrossRef]
  12. M. Varma and A. Zisserman, "Estimating illumination direction from textured images," in IEEE Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004 (IEEE, 2004), pp. I-179-I-186.
  13. M. Berry and V. Hannay, "Umbilic points on Gaussian random surfaces," J. Phys. A 10, 1809-1821 (1977).
    [CrossRef]
  14. M. S. Longuet-Higgins, "The statistical analysis of a random moving surface," Philos. Trans. R. Soc. London, Ser. A 249, 321-364 (1956).
  15. J. Bigün and G. H. Granlund, "Optimal orientation detection of linear symmetry," in Proceedings of the IEEE First International Conference on Computer Vision (IEEE, 1987), pp. 433-438.
  16. O. Drbohlav and M. Chantler, "Illumination-invariant texture classification using single training images," in Texture 2005: Proceedings of the 4th International Workshop on Texture Analysis and Synthesis (IEEE, 2005), pp. 31-36.
  17. M. Chantler, M. Petrou, A. Penirsche, M. Schmidt and G. McGunnigle, "Classifying surface texture while simultaneously estimating illumination direction," Int. J. Comput. Vis. 62, 83-96 (2005).
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    [CrossRef]
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  20. Q. Zheng and R. Chellappa, "Estimation of illuminant direction, albedo and shape from shading," IEEE Trans. Pattern Anal. Mach. Intell. 13, 680-702 (1991).
    [CrossRef]
  21. M. Chantler and G. Delguste, "lluminant-tilt estimation from images of isotropic texture," IEE Proc. Vision Image Signal Process. 144, 213-219 (1997).
    [CrossRef]
  22. Y. Zhang and Y. H. Yang, "Illuminant Direction Determination for Multiple Light Sources," in IEEE Conference on Computer Vision and Pattern Recogintion, 2000. Proceedings (IEEE, 2000), Vol. 1, pp. 269-276.
    [CrossRef]
  23. M. Brooks and B. Horn, "Shape and source from shading," Proceedings of the 9th International Joint Conference on Artificial Intelligence, A.K.Joshi, ed. (Morgan Kaufmann, 1985), 932-936.
  24. S. Karlsson and J. Bigun, "Multiscale complex moments of the local power spectrum," J. Opt. Soc. Am. A 24, 618-625 (2007).
    [CrossRef]
  25. M. Chantler, Photex Photometric Image Database, http://www.macs. hw. ac. uk/texturelab/resources/databases/Photex/index. htm (October 2007).
  26. A. Pentland, "The visual inference of shape: computation from local features," Ph.D. dissertation (MIT, 1982).
  27. 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]
  28. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, "Geometrical considerations and nomenclature for reflectance" (National Bureau of Standards, 1977).

2007 (1)

2005 (1)

M. Chantler, M. Petrou, A. Penirsche, M. Schmidt and G. McGunnigle, "Classifying surface texture while simultaneously estimating illumination direction," Int. J. Comput. Vis. 62, 83-96 (2005).

2004 (1)

J. Bigün, T. Bigün, and K. Nilsson, "Recognition by symmetry derivatives and the generalized structure tensor," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1590-1605 (2004).
[CrossRef] [PubMed]

2003 (1)

1999 (1)

P. N. Belhumeur, D. Kriegman, and A. Yuille, "The bas-relief ambiguity," Int. J. Comput. Vis. 35, 33-44 (1999).
[CrossRef]

1998 (1)

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

1997 (1)

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

1995 (1)

M. J. Chantler, "Why illuminant direction is fundamental to texture analysis." IEE Proc. Vision Image Signal Process. 142, 199-206 (1995).
[CrossRef]

1991 (1)

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

1990 (1)

1988 (1)

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

1983 (1)

1982 (1)

1977 (1)

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

1956 (1)

M. S. Longuet-Higgins, "The statistical analysis of a random moving surface," Philos. Trans. R. Soc. London, Ser. A 249, 321-364 (1956).

1939 (1)

A. Gershun, "The light field," P. Moon and G. Timoshenko, translators, J. Math. Phys. (Cambridge, Mass.) 18, 51-151 (1939).

IEE Proc. Vision Image Signal Process. (2)

M. J. Chantler, "Why illuminant direction is fundamental to texture analysis." IEE Proc. Vision Image Signal Process. 142, 199-206 (1995).
[CrossRef]

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

IEEE Trans. Pattern Anal. Mach. Intell. (3)

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

J. Bigün, T. Bigün, and K. Nilsson, "Recognition by symmetry derivatives and the generalized structure tensor," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1590-1605 (2004).
[CrossRef] [PubMed]

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

Int. J. Comput. Vis. (3)

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

M. Chantler, M. Petrou, A. Penirsche, M. Schmidt and G. McGunnigle, "Classifying surface texture while simultaneously estimating illumination direction," Int. J. Comput. Vis. 62, 83-96 (2005).

P. N. Belhumeur, D. Kriegman, and A. Yuille, "The bas-relief ambiguity," Int. J. Comput. Vis. 35, 33-44 (1999).
[CrossRef]

J. Math. Phys. (Cambridge, Mass.) (1)

A. Gershun, "The light field," P. Moon and G. Timoshenko, translators, J. Math. Phys. (Cambridge, Mass.) 18, 51-151 (1939).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

J. Phys. A (1)

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 (1)

M. S. Longuet-Higgins, "The statistical analysis of a random moving surface," Philos. Trans. R. Soc. London, Ser. A 249, 321-364 (1956).

Other (12)

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

O. Drbohlav and M. Chantler, "Illumination-invariant texture classification using single training images," in Texture 2005: Proceedings of the 4th International Workshop on Texture Analysis and Synthesis (IEEE, 2005), pp. 31-36.

M. Varma and A. Zisserman, "Estimating illumination direction from textured images," in IEEE Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004 (IEEE, 2004), pp. I-179-I-186.

B. M. ter Haar Romeny, L. Florack, J. J. Koenderink, and M. A. Viergever, "Scale-space: its natural operators and differential invariants," in Information Processing in Medical Imaging, A.C. F.Colchester and D.J.Hawkes, eds., Vol. 511 of Lecture Notes in Computer Science (Springer, 1991) pp. 239-255.
[CrossRef]

D. G. Lowe, "Object recognition from local scale-invariant features," in The Proceedings of the Seventh IEEE International Conference on Computer Vision, 1999 (IEEE, 1999) Vol. 2, pp. 1150-1157.
[CrossRef]

W. Li, C. Wang, D. Xu, B. Luo, and Z. Chen, A Study on Illumination Invariant Face Recognition Methods Based on Multiple Eigenspaces, Vol. 3947 of Lecture Notes in Computer Science (Springer, 2005), pp. 131-136.

S. C. Pont and J. J. Koenderink, "Irradiation orientation from obliquely viewed texture," in Deep Structure, Singularities, and Computer Vision, O.F.Olsen, L.Florack, and A.Kuijper, eds. (Springer, 2005), pp. 205-210.
[CrossRef]

Y. Zhang and Y. H. Yang, "Illuminant Direction Determination for Multiple Light Sources," in IEEE Conference on Computer Vision and Pattern Recogintion, 2000. Proceedings (IEEE, 2000), Vol. 1, pp. 269-276.
[CrossRef]

M. Brooks and B. Horn, "Shape and source from shading," Proceedings of the 9th International Joint Conference on Artificial Intelligence, A.K.Joshi, ed. (Morgan Kaufmann, 1985), 932-936.

M. Chantler, Photex Photometric Image Database, http://www.macs. hw. ac. uk/texturelab/resources/databases/Photex/index. htm (October 2007).

A. Pentland, "The visual inference of shape: computation from local features," Ph.D. dissertation (MIT, 1982).

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, "Geometrical considerations and nomenclature for reflectance" (National Bureau of Standards, 1977).

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Figures (7)

Fig. 1
Fig. 1

Left, illustration of illuminance flow on a sphere. Two scales of interest are the mesoscale, surface variation due to texture, and object scale, defining the geometry of the object (here a sphere). Right, imaging geometry. We assume an object that is flat on the global scale for this paper. θ and ϕ define the direction of illumination.

Fig. 2
Fig. 2

Theoretical predictions for deviations of the three estimates, dev H (dashed), dev G (dotted), and dev C (solid), as a function of true azimuthal direction of illumination ( ϕ ) , all three for ξ { G h } = 0.1 .

Fig. 3
Fig. 3

Rendered Gaussian surface with ξ = 0.6 , σ = 100 , and θ = 30 . Left, ϕ = 90 ° . Middle, ϕ = 0 ° . Right, ϕ = 0 ° and viewed from another direction.

Fig. 4
Fig. 4

Typical outputs from the renderings for dev H (dashed), dev G (dotted), and dev C (solid). X and Y axes, same as Fig. 2 (note that the ranges of the Y axes differ in the figure). First graph (Top), close to the assumptions of the theory ( σ = 30 , θ = 30 ° , ξ = 0.1 ). Second, higher relief ( σ = 60 , θ = 30 ° , ξ = 0.1 ). Third, strongly anisotropic surface ( σ = 30 , θ = 30 ° , ξ = 0.6 ). Fourth (bottom), high light elevation (nonoblique lighting, σ = 30 , θ = 70 ° , ξ = 0.1 ).

Fig. 5
Fig. 5

Contour plots of the average magnitude deviations. The absolute value of each curve of the type in Fig. 4 is averaged (one nonnegative scalar for each curve). The gray value of each region indicates an upper bound on the error, specified by the bars on the right of each figure. Top, θ versus σ with ξ = 0.1 . Middle, ξ versus σ with θ = 30 ° . Bottom, ξ versus θ with σ = 30 .

Fig. 6
Fig. 6

Example images corresponding to ϕ = 90 ° (top illumination), each one corresponding to one out of 360 images used to draw the graphs in Fig. 4. The images have been histogram normalized for visualizing details. Top left, σ = 30 , θ = 30 ° , ξ = 0.1 . Top right, σ = 60 , θ = 30 ° , ξ = 0.1 . Bottom left, σ = 30 , θ = 70 ° , ξ = 0.1 . Bottom right, σ = 30 , θ = 30 ° , ξ = 0.6 .

Fig. 7
Fig. 7

Example of results from real-world textures. Top row, two images of the texture aab with light elevation ( θ ) = 45 ° ; they vary in azimuthal light angle ( ϕ ) . The dotted line is the combined measure; the black line the gradient, and the white line the Hessian. The black arrows indicate global light direction: top left, ϕ = 90 ° ; top right, ϕ = 60 ° . Bottom graph, results for the 36 different images corresponding to equidistant sampling of ϕ.

Equations (57)

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h p q = p + q h ( x , y ) x p y q ,
H h = ( h 20 h 11 h 11 h 02 ) .
m h p q = u p v q ρ ̂ h .
h p 1 q 1 h p 2 q 2 = m h ( p 1 + p 2 ) , ( q 1 + q 2 ) ( 1 ) ( p 1 p 2 + q 1 q 2 ) 2 .
h p q ( j u ) p ( j v ) q h ̂ ( k ) ,
f ( r ) g ( r ) = f ̂ ( k ) g ̂ * ( k ) ,
G h = ( m h 20 m h 11 m h 11 m h 02 ) = g h g h T .
ξ { G h } = λ G h max λ G h min λ G h max + λ G h min [ 0 , 1 ] .
μ { G h } = v G h max [ 0 ° , 180 ° ) .
T h = ( m h 40 + m h 22 m h 31 + m h 13 m h 31 + m h 13 m h 04 + m h 22 ) = H h H h T .
n = { h 10 , h 01 , 1 } T h 10 2 + h 01 2 + 1 .
i = c { cos θ cos ϕ , cos θ sin ϕ , sin θ } ,
I ( h 01 , h 10 , θ , ϕ ) = i n = c sin θ cos θ ( h 10 cos ϕ + h 01 sin ϕ ) h 10 2 + h 01 2 + 1 .
I ( h 01 , h 10 , θ , ϕ ) = c sin θ c cos θ ( h 10 cos ϕ + h 01 sin ϕ ) + O ( g T g ) ,
h ( r ) = x h 10 + y h 01 + x 2 h 11 2 + .
I ( r , θ , ϕ ) c { sin ( θ ) 1 2 cos ( θ ) [ sin ( ϕ ) ( h 21 x 2 + 2 h 11 x + 2 y h 12 x + 2 h 01 + 2 y h 02 + y 2 h 03 ) + cos ( ϕ ) ( h 30 x 2 + 2 h 20 x + 2 y h 21 x + 2 h 10 + 2 y h 11 + y 2 h 12 ) ] } .
G I ( 1 , 1 ) = c 2 2 cos 2 θ [ m h 22 + m h 40 + ( m h 40 m h 22 ) cos 2 ϕ + 2 m h 31 sin 2 ϕ ] ,
G I ( 1 , 2 ) = c 2 2 cos 2 θ [ m h 31 + m h 13 + ( m h 31 m h 13 ) cos 2 ϕ + 2 m h 22 sin 2 ϕ ] ,
G I ( 2 , 2 ) = c 2 2 cos 2 θ [ m h 22 + m h 04 + ( m h 22 m h 04 ) cos 2 ϕ + 2 m h 13 sin 2 ϕ ] ,
T I ( 1 , 1 ) = c 2 2 cos 2 θ [ m h 60 + 2 m h 42 + m h 24 + ( m h 60 m h 24 ) cos 2 ϕ + 2 ( m h 51 + m h 33 ) sin 2 ϕ ] ,
T I ( 1 , 2 ) = c 2 2 cos 2 θ [ m h 51 + 2 m h 33 + m h 15 + ( m h 51 m h 15 ) cos 2 ϕ + 2 ( m h 42 + m h 24 ) sin 2 ϕ ] ,
T I ( 2 , 2 ) = c 2 2 cos 2 θ [ m h 42 + 2 m h 24 + m h 06 + ( m h 42 m h 06 ) cos 2 ϕ + 2 ( m h 15 + m h 33 ) sin 2 ϕ ] .
ρ ̂ h ( u , v ) = g ( u , s 1 ) g ( v , 1 + ξ { G h } 1 ξ { G h } s 1 ) ,
m h 40 = 12 π 2 ( ξ + 1 ) 5 s 4 ( ξ 1 ) 4 ,
m h 22 = 4 π 2 ( ξ + 1 ) 3 s 4 ( ξ 1 ) 2 ,
m h 04 = 12 π 2 ( ξ + 1 ) 5 s 4 ( ξ 1 ) 4 ,
m h 60 = 60 π 2 ( ξ + 1 ) s 6 ,
m h 42 = 12 π 2 ( ξ + 1 ) 3 s 6 ( ξ 1 ) 2 ,
m h 24 = 12 π 2 ( ξ + 1 ) 5 s 6 ( ξ 1 ) 4 ,
m h 06 = 60 π 2 ( ξ + 1 ) 7 s 6 ( ξ 1 ) 6 ,
G I ( 1 , 1 ) = C 1 2 ( ξ 3 + 1 ) + ( ξ + 1 ) ( ( ξ 4 ) ξ + 1 ) cos 2 ϕ ( ξ 1 ) 2 ,
G I ( 1 , 2 ) = C 1 ( ξ + 1 ) 3 sin 2 ϕ ( ξ 1 ) 2 ,
G I ( 2 , 2 ) = C 1 [ ( ξ + 1 ) 3 2 ( ξ 2 + ξ + 1 ) ( ξ 1 ) 4 ( ξ ( ξ + 4 ) + 1 ) cos 2 ϕ ( ξ 1 ) 4 ] ,
T I ( 1 , 1 ) = C 2 [ 2 ξ 5 2 ξ 4 + 4 ξ 3 + 4 ξ 2 2 ξ + 2 ( ξ 1 ) 4 + ( ξ 5 5 ξ 4 5 ξ + 1 ) cos 2 ϕ ( ξ 1 ) 4 ] ,
T I ( 1 , 2 ) = C 2 ( ξ + 1 ) 3 ( ξ 2 + 1 ) sin 2 ϕ ( ξ 1 ) 4 ,
T I ( 2 , 2 ) = C 2 a + 2 ( b + 1 ) cos 2 ϕ ( ξ 1 ) 6 , a = 2 ( ξ 7 + 5 ξ 6 + 13 ξ 5 + 21 ξ 4 + 21 ξ 3 + 13 ξ 2 + 5 ξ ) , b = ( ξ 7 + 9 ξ 6 + 27 ξ 5 + 43 ξ 4 + 43 ξ 3 + 27 ξ 2 + 9 ξ ) ,
μ { G I } = tan 1 ( a b + c ) ,
a = 2 ( ξ 2 1 ) sin ( 2 ϕ ) ,
b = ( 4 ξ 2 + 2 ) cos ( 2 ϕ ) 6 ξ ,
c = 10 ( ξ 2 + 4 ) ξ 2 + 4 + 6 ( ξ 2 + 2 ) ξ 2 cos ( 4 ϕ ) 24 ( 2 ξ 3 + ξ ) cos ( 2 ϕ ) ,
μ { T I } = tan 1 ( a b + c ) ,
a = 2 ( ξ 2 1 ) sin ( 2 ϕ ) ,
b = ( 8 ξ 2 + 2 ) cos ( 2 ϕ ) 2 ξ 3 8 ξ ,
c = 4 ξ 6 + 66 ξ 4 + 76 ξ 2 + 4 + 10 ( 3 ξ 2 + 2 ) ξ 2 cos ( 4 ϕ ) 8 ( ξ 2 + 4 ) ( 4 ξ 2 + 1 ) ξ cos ( 2 ϕ ) .
ξ { G I } = a b c ,
a = 5 ( ξ 2 + 4 ) ξ 2 + 2 + 3 ( ξ 2 + 2 ) ξ 2 cos ( 4 ϕ ) ,
b = 12 ( 2 ξ 3 + ξ ) cos ( 2 ϕ ) ,
c = 2 ( ξ 2 3 ξ cos ( 2 ϕ ) + 2 ) ,
ξ { T I } = a b c ,
a = 2 ξ 6 + 33 ξ 4 + 38 ξ 2 + 2 + 5 ( 3 ξ 2 + 2 ) ξ 2 cos ( 4 ϕ ) ,
b = 4 ξ ( ξ 2 + 4 ) ( 4 ξ 2 + 1 ) cos ( 2 ϕ ) ,
c = 2 [ 3 ξ 2 + 2 ( ξ 2 + 4 ) ξ cos ( 2 ϕ ) ] .
dev G ( ϕ ) = ( μ { G I } ϕ ) mod 180 °
dev G ( ϕ ) = 3 ξ sin 2 ϕ + O ( ϕ 2 ) ,
dev H ( ϕ ) = 4 ξ sin 2 ϕ + O ( ϕ 2 ) .
μ C = 4 μ { G I } 3 μ { H I } .
I rend ( r ) = I ( h 01 ( r ) , h 10 ( r ) , θ , ϕ ) S ( r ) ,

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