Abstract

There are many works in color that assume illumination change can be modeled by multiplying sensor responses by individual scaling factors. The early research in this area is sometimes grouped under the heading “von Kries adaptation”: the scaling factors are applied to the cone responses. In more recent studies, both in psychophysics and in computational analysis, it has been proposed that scaling factors should be applied to linear combinations of the cones that have narrower support: they should be applied to the so-called “sharp sensors.” In this paper, we generalize the computational approach to spectral sharpening in three important ways. First, we introduce spherical sampling as a tool that allows us to enumerate in a principled way all linear combinations of the cones. This allows us to, second, find the optimal sharp sensors that minimize a variety of error measures including CIE Delta E (previous work on spectral sharpening minimized RMS) and color ratio stability. Lastly, we extend the spherical sampling paradigm to the multispectral case. Here the objective is to model the interaction of light and surface in terms of color signal spectra. Spherical sampling is shown to improve on the state of the art.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. T. Maloney and B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
    [CrossRef]
  2. D. H. Marimont and B. A. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913(1992).
    [CrossRef]
  3. J. Park, M. Lee, M. D. Grossberg, and S. K. Nayar, “Multispectral imaging using multiplexed illumination,” International Conference on Computer Vision (2007).
  4. D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–35 (1990).
    [CrossRef]
  5. J. A. Worthey and M. H. Brill, “Heuristic analysis of von Kries color constancy,” J. Opt. Soc. Am. A 3, 1708–1712 (1986).
    [CrossRef]
  6. E. Land, “The retinex,” Am. Sci. 52, 247–264 (1964).
  7. V. C. Smith and J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vis. Res. 15, 161–171 (1975).
    [CrossRef]
  8. G. Wyszecki and W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, 1982).
  9. K. Barnard, L. Martin, B. Funt, and A. Coath, “A data set for colour research,” Color Res. Appl. 27, 147–151 (2002).
    [CrossRef]
  10. G. D. Finlayson, M. S. Drew, and B. V. Funt, “Color constancy: enhancing von Kries adaptation via sensor transformation,” in Human Vision, Visual Processing, and Digital Display IV (1993), pp. 473–484.
  11. G. D. Finlayson, M. S. Drew, and B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
    [CrossRef]
  12. G. D. Finlayson, M. S. Drew, and B. V. Funt, “Color constancy: generalized diagonal transforms suffice,” J. Opt. Soc. Am. A 11, 3011–3019 (1994).
    [CrossRef]
  13. G. Finlayson, “Coefficient colour constancy,” Ph.D. thesis (Simon Fraser University, 1995).
  14. J. Schanda, CIE Colorimetry (Wiley, 2007), pp. 25–78.
  15. J. von Kries, Sources of Color Science (MIT Press, 1970), pp. 110–119.
  16. S. K. Nayar and R. M. Bolle, “Reflectance based object recognition,” Int. J. Comput. Vis. 17, 219–240 (1996).
    [CrossRef]
  17. B. V. Funt and G. D. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Machine Intell. 17, 522–529(1995).
    [CrossRef]
  18. M. S. Drew and G. D. Finlayson, “Multispectral processing without spectra,” J. Opt. Soc. Am. A 20, 1181–1193 (2003).
    [CrossRef]
  19. H. Chong, S. Gortler, and T. Zickler, “The von Kries hypothesis and a basis for color constancy,” in International Conference on Computer Vision (2007).
  20. K. Lam, “Metamerism and colour constancy,” Ph.D. thesis (University of Bradford, 1985).
  21. M. D. Fairchild, Color Appearance Models, 2nd ed (Wiley-IST, 2005).
  22. Y. V. der Haeghen and J. M. Naeyaert, “Consistent cutaneous imaging with commercial digital cameras,” Arch. Dermatol. 142, 42–46 (2006).
    [CrossRef]
  23. E. Chorro, E. Perales, D. de Fez, M. J. Luque, and F. M. Martínez-Verdú, “Application of the S-CIELAB color model to processed and calibrated images with a colorimetric dithering method,” Opt. Express 15, 7810–7817 (2007).
    [CrossRef]
  24. G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).
  25. J. E. Gibson, M. D. Fairchild, and S. L. Wright, “Colorimetric tolerances of various digital image displays,” in Color Imaging Conference’00 (IS&T, 2000), pp. 295–300.
  26. L. Lovisolo and E. da Silva, “Uniform distribution of points on a hyper-sphere with applications to vector bit-plane encoding,” IEE Proc. Vis. Image Signal Process. 148, 187–193 (2001).
    [CrossRef]
  27. J. C. Lagarias, J. Reeds, M. Wright, and P. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
    [CrossRef]
  28. M. S. Drew and G. D. Finlayson, “Spectral sharpening with positivity,” J. Opt. Soc. Am. A 17, 1361–1370 (2000).
    [CrossRef]
  29. M. Vrhel, R. Gershon, and L. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

2007 (1)

2006 (1)

Y. V. der Haeghen and J. M. Naeyaert, “Consistent cutaneous imaging with commercial digital cameras,” Arch. Dermatol. 142, 42–46 (2006).
[CrossRef]

2003 (1)

2002 (1)

K. Barnard, L. Martin, B. Funt, and A. Coath, “A data set for colour research,” Color Res. Appl. 27, 147–151 (2002).
[CrossRef]

2001 (1)

L. Lovisolo and E. da Silva, “Uniform distribution of points on a hyper-sphere with applications to vector bit-plane encoding,” IEE Proc. Vis. Image Signal Process. 148, 187–193 (2001).
[CrossRef]

2000 (1)

1998 (1)

J. C. Lagarias, J. Reeds, M. Wright, and P. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

1996 (1)

S. K. Nayar and R. M. Bolle, “Reflectance based object recognition,” Int. J. Comput. Vis. 17, 219–240 (1996).
[CrossRef]

1995 (1)

B. V. Funt and G. D. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Machine Intell. 17, 522–529(1995).
[CrossRef]

1994 (3)

1992 (1)

1990 (1)

D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–35 (1990).
[CrossRef]

1986 (2)

1975 (1)

V. C. Smith and J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vis. Res. 15, 161–171 (1975).
[CrossRef]

1964 (1)

E. Land, “The retinex,” Am. Sci. 52, 247–264 (1964).

Barnard, K.

K. Barnard, L. Martin, B. Funt, and A. Coath, “A data set for colour research,” Color Res. Appl. 27, 147–151 (2002).
[CrossRef]

Bolle, R. M.

S. K. Nayar and R. M. Bolle, “Reflectance based object recognition,” Int. J. Comput. Vis. 17, 219–240 (1996).
[CrossRef]

Brill, M. H.

Chong, H.

H. Chong, S. Gortler, and T. Zickler, “The von Kries hypothesis and a basis for color constancy,” in International Conference on Computer Vision (2007).

Chorro, E.

Coath, A.

K. Barnard, L. Martin, B. Funt, and A. Coath, “A data set for colour research,” Color Res. Appl. 27, 147–151 (2002).
[CrossRef]

da Silva, E.

L. Lovisolo and E. da Silva, “Uniform distribution of points on a hyper-sphere with applications to vector bit-plane encoding,” IEE Proc. Vis. Image Signal Process. 148, 187–193 (2001).
[CrossRef]

de Fez, D.

der Haeghen, Y. V.

Y. V. der Haeghen and J. M. Naeyaert, “Consistent cutaneous imaging with commercial digital cameras,” Arch. Dermatol. 142, 42–46 (2006).
[CrossRef]

Drew, M. S.

Fairchild, M. D.

J. E. Gibson, M. D. Fairchild, and S. L. Wright, “Colorimetric tolerances of various digital image displays,” in Color Imaging Conference’00 (IS&T, 2000), pp. 295–300.

M. D. Fairchild, Color Appearance Models, 2nd ed (Wiley-IST, 2005).

Finlayson, G.

G. Finlayson, “Coefficient colour constancy,” Ph.D. thesis (Simon Fraser University, 1995).

Finlayson, G. D.

Forsyth, D. A.

D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–35 (1990).
[CrossRef]

Funt, B.

K. Barnard, L. Martin, B. Funt, and A. Coath, “A data set for colour research,” Color Res. Appl. 27, 147–151 (2002).
[CrossRef]

Funt, B. V.

B. V. Funt and G. D. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Machine Intell. 17, 522–529(1995).
[CrossRef]

G. D. Finlayson, M. S. Drew, and B. V. Funt, “Color constancy: generalized diagonal transforms suffice,” J. Opt. Soc. Am. A 11, 3011–3019 (1994).
[CrossRef]

G. D. Finlayson, M. S. Drew, and B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
[CrossRef]

G. D. Finlayson, M. S. Drew, and B. V. Funt, “Color constancy: enhancing von Kries adaptation via sensor transformation,” in Human Vision, Visual Processing, and Digital Display IV (1993), pp. 473–484.

Gershon, R.

M. Vrhel, R. Gershon, and L. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

Gibson, J. E.

J. E. Gibson, M. D. Fairchild, and S. L. Wright, “Colorimetric tolerances of various digital image displays,” in Color Imaging Conference’00 (IS&T, 2000), pp. 295–300.

Golub, G. H.

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).

Gortler, S.

H. Chong, S. Gortler, and T. Zickler, “The von Kries hypothesis and a basis for color constancy,” in International Conference on Computer Vision (2007).

Grossberg, M. D.

J. Park, M. Lee, M. D. Grossberg, and S. K. Nayar, “Multispectral imaging using multiplexed illumination,” International Conference on Computer Vision (2007).

Iwan, L.

M. Vrhel, R. Gershon, and L. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

Lagarias, J. C.

J. C. Lagarias, J. Reeds, M. Wright, and P. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Lam, K.

K. Lam, “Metamerism and colour constancy,” Ph.D. thesis (University of Bradford, 1985).

Land, E.

E. Land, “The retinex,” Am. Sci. 52, 247–264 (1964).

Lee, M.

J. Park, M. Lee, M. D. Grossberg, and S. K. Nayar, “Multispectral imaging using multiplexed illumination,” International Conference on Computer Vision (2007).

Loan, C. F. V.

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).

Lovisolo, L.

L. Lovisolo and E. da Silva, “Uniform distribution of points on a hyper-sphere with applications to vector bit-plane encoding,” IEE Proc. Vis. Image Signal Process. 148, 187–193 (2001).
[CrossRef]

Luque, M. J.

Maloney, L. T.

Marimont, D. H.

Martin, L.

K. Barnard, L. Martin, B. Funt, and A. Coath, “A data set for colour research,” Color Res. Appl. 27, 147–151 (2002).
[CrossRef]

Martínez-Verdú, F. M.

Naeyaert, J. M.

Y. V. der Haeghen and J. M. Naeyaert, “Consistent cutaneous imaging with commercial digital cameras,” Arch. Dermatol. 142, 42–46 (2006).
[CrossRef]

Nayar, S. K.

S. K. Nayar and R. M. Bolle, “Reflectance based object recognition,” Int. J. Comput. Vis. 17, 219–240 (1996).
[CrossRef]

J. Park, M. Lee, M. D. Grossberg, and S. K. Nayar, “Multispectral imaging using multiplexed illumination,” International Conference on Computer Vision (2007).

Park, J.

J. Park, M. Lee, M. D. Grossberg, and S. K. Nayar, “Multispectral imaging using multiplexed illumination,” International Conference on Computer Vision (2007).

Perales, E.

Pokorny, J.

V. C. Smith and J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vis. Res. 15, 161–171 (1975).
[CrossRef]

Reeds, J.

J. C. Lagarias, J. Reeds, M. Wright, and P. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Schanda, J.

J. Schanda, CIE Colorimetry (Wiley, 2007), pp. 25–78.

Smith, V. C.

V. C. Smith and J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vis. Res. 15, 161–171 (1975).
[CrossRef]

Stiles, W.

G. Wyszecki and W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, 1982).

von Kries, J.

J. von Kries, Sources of Color Science (MIT Press, 1970), pp. 110–119.

Vrhel, M.

M. Vrhel, R. Gershon, and L. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

Wandell, B. A.

Worthey, J. A.

Wright, M.

J. C. Lagarias, J. Reeds, M. Wright, and P. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Wright, P.

J. C. Lagarias, J. Reeds, M. Wright, and P. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Wright, S. L.

J. E. Gibson, M. D. Fairchild, and S. L. Wright, “Colorimetric tolerances of various digital image displays,” in Color Imaging Conference’00 (IS&T, 2000), pp. 295–300.

Wyszecki, G.

G. Wyszecki and W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, 1982).

Zickler, T.

H. Chong, S. Gortler, and T. Zickler, “The von Kries hypothesis and a basis for color constancy,” in International Conference on Computer Vision (2007).

Am. Sci. (1)

E. Land, “The retinex,” Am. Sci. 52, 247–264 (1964).

Arch. Dermatol. (1)

Y. V. der Haeghen and J. M. Naeyaert, “Consistent cutaneous imaging with commercial digital cameras,” Arch. Dermatol. 142, 42–46 (2006).
[CrossRef]

Color Res. Appl. (2)

M. Vrhel, R. Gershon, and L. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

K. Barnard, L. Martin, B. Funt, and A. Coath, “A data set for colour research,” Color Res. Appl. 27, 147–151 (2002).
[CrossRef]

IEE Proc. Vis. Image Signal Process. (1)

L. Lovisolo and E. da Silva, “Uniform distribution of points on a hyper-sphere with applications to vector bit-plane encoding,” IEE Proc. Vis. Image Signal Process. 148, 187–193 (2001).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell. (1)

B. V. Funt and G. D. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Machine Intell. 17, 522–529(1995).
[CrossRef]

Int. J. Comput. Vis. (2)

D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5–35 (1990).
[CrossRef]

S. K. Nayar and R. M. Bolle, “Reflectance based object recognition,” Int. J. Comput. Vis. 17, 219–240 (1996).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Express (1)

SIAM J. Optim. (1)

J. C. Lagarias, J. Reeds, M. Wright, and P. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Vis. Res. (1)

V. C. Smith and J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vis. Res. 15, 161–171 (1975).
[CrossRef]

Other (11)

G. Wyszecki and W. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, 1982).

G. D. Finlayson, M. S. Drew, and B. V. Funt, “Color constancy: enhancing von Kries adaptation via sensor transformation,” in Human Vision, Visual Processing, and Digital Display IV (1993), pp. 473–484.

J. Park, M. Lee, M. D. Grossberg, and S. K. Nayar, “Multispectral imaging using multiplexed illumination,” International Conference on Computer Vision (2007).

G. Finlayson, “Coefficient colour constancy,” Ph.D. thesis (Simon Fraser University, 1995).

J. Schanda, CIE Colorimetry (Wiley, 2007), pp. 25–78.

J. von Kries, Sources of Color Science (MIT Press, 1970), pp. 110–119.

H. Chong, S. Gortler, and T. Zickler, “The von Kries hypothesis and a basis for color constancy,” in International Conference on Computer Vision (2007).

K. Lam, “Metamerism and colour constancy,” Ph.D. thesis (University of Bradford, 1985).

M. D. Fairchild, Color Appearance Models, 2nd ed (Wiley-IST, 2005).

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).

J. E. Gibson, M. D. Fairchild, and S. L. Wright, “Colorimetric tolerances of various digital image displays,” in Color Imaging Conference’00 (IS&T, 2000), pp. 295–300.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1.

Pokorny and Smith cones sensitivities (top), XYZ sensitivities (middle), SONY DXC-930 sensitivities (bottom).

Fig. 2.
Fig. 2.

Schema of the spherical sampling method. From a original sensor, we find the set of points P in the sphere. We sample the sphere around those points, and then return to the sensor domain.

Fig. 3.
Fig. 3.

X Y Z curves (solid lines) versus X Y Z perturbed 1° (dashed lines).

Fig. 4.
Fig. 4.

Sharp sensors for the experiment minimizing CIE Delta E error for the four different datasets. Dataset 1: 102 illuminants (straight line). Dataset 2: eight daylights and three fluorescent lights (dashed line). Dataset 3: eight daylights (dotted line). Dataset 4: 10 illuminants (dashed-dotted line). X Y Z sensitivities are plotted in black.

Fig. 5.
Fig. 5.

Sharp sensors for the experiment maximizing color ratio stability for the four different datasets. X Y Z sensitivities are plotted in black.

Fig. 6.
Fig. 6.

Color signal basis functions (top), DF corresponding sharpened counterparts (middle), spherical sampling sharpened counterparts (bottom)

Fig. 7.
Fig. 7.

Points sampled with the Lovisolo and da Silva algorithm for three dimensions.

Tables (3)

Tables Icon

Table 1. Results of the Experiment Minimizing CIE Delta E Error for the Four Different Illuminant Datasets (Measures: Median, Mean, and 75% of Δ E )

Tables Icon

Table 2. Average Ratio Error for Four Illuminant Datasets and Three Methods (See Text for Description)

Tables Icon

Table 3. Results of the Experiment for Spectra Recovering. Measure: Mean Δ E

Equations (39)

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

p k = ω R k ( λ ) E ( λ ) S ( λ ) d λ , k = { R , G , B } .
p ̲ i 1 M 1 , 2 p ̲ i 2 ,
p ̲ i 1 D 1 , 2 p ̲ i 2 ,
D i , i 1 , 2 = w i 1 / w i 2 .
T p ̲ i 1 D 1 , 2 T p ̲ i 2 ,
p ̲ i 1 T 1 D 1 , 2 T p ̲ i 2 .
f ( p ̲ ref i ) f ( T 1 D ref , j T p ̲ i j )
min T err = j f ( p ̲ i ref ) f ( T 1 D T ref , j T p ̲ i j ) ,
min t ̲ k err = j t ̲ k . p ̲ u j t ̲ k . p ̲ v j t ̲ k . p ̲ u ref t ̲ k . p ̲ v ref k { r , g , b } .
min t ̲ k 1 N j = 1 N c ̲ D 65 ( t ̲ k ) c ̲ j ( t ̲ k ) c ̲ D 65 ( t ̲ k ) .
S ( λ ) = i = 1 3 σ i S i ( λ ) , E ( λ ) = i = 1 2 ϵ i E i ( λ ) .
T 1 D T = M .
min x err = j f ( p ̲ i ref ) f ( T x 1 D x ref , k T x p ̲ i j ) , x 1 , 2 , , M ,
min x 1 N j = 1 N c ̲ x D 65 c ̲ x j c ̲ x D 65 .
[ p ̲ p ̲ ] t R t R [ p ̲ p ̲ ] δ .
R = U Σ V t .
[ p ̲ p ̲ ] t U t U [ p ̲ p ̲ ] = δ ;
R = U P , P = [ p ̲ 1 , p ̲ 2 , p ̲ 3 ] .
R = U P = U Σ V t ( Σ V t ) 1 P = R ( Σ V t ) 1 P .
T = ( ( Σ V t ) 1 P ) t .
l ̲ = f ( ρ ̲ ) = Lab ( ρ ̲ ) , ρ ̲ = ω S ( λ ) D 65 ( λ ) X ̲ ( λ ) d λ .
ρ ̲ = ω S ( λ ) E ( λ ) X ̲ ( λ ) d λ .
ρ ̲ ^ = T - 1 D D 65 , E T ρ ̲ .
Δ E = l ̲ l ̲ ^ .
C ̲ = diag ( E ̲ ) S ̲ ,
C = U Σ V t ,
B ^ = B T .
e ̲ = B ^ t E ̲ .
s ̲ = B ^ t S ̲ .
E ̲ ^ = B ^ + e ̲ ,
S ̲ ^ = B ^ + s ̲ ,
c ̲ = diag ( e ̲ ) s ̲ .
C ̲ ^ = B ^ + c ̲ .
L * = 116 f ( Y Y w ) 16 , a * = 500 ( f ( X X w ) f ( Y Y w ) ) ,
b * = 500 ( f ( Y Y w ) f ( Z Z w ) ) ,
f ( x ) = { x 1 3 if x > ( 6 29 ) 3 , 1 3 ( 29 6 ) 2 x + 4 29 elsewhere .
δ = Δ ω 1 , δ = Δ ω j i = 1 j 1 sin ω j , j = 2 : n 1.
A n = K δ n 1 ,
A n = n π n / 2 ( n / 2 ) ! for n even, A n = n 2 n π ( n 1 ) / 2 ( n 1 2 ) ! n ! for n odd .

Metrics