Abstract

In this paper we present a parametric model for automatic color naming where each color category is modeled as a fuzzy set with a parametric membership function. The parameters of the functions are estimated in a fitting process using data derived from psychophysical experiments. The name assignments obtained by the model agree with previous psychophysical experiments, and therefore the high-level color-naming information provided can be useful for different computer vision applications where the use of a parametric model will introduce interesting advantages in terms of implementation costs, data representation, model analysis, and model updating.

© 2008 Optical Society of America

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  4. J. Sturges and T. Whitfield, “Salient features of Munsell color space as a function of monolexemic naming and response latencies,” Vision Res. 37, 307-313 (1997).
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  5. S. Guest and D. V. Laar, “The structure of colour naming space,” Vision Res. 40, 723-734 (2000).
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    [CrossRef] [PubMed]
  13. M. Seaborn, L. Hepplewhite, and J. Stonham, “Fuzzy colour category map for the measurement of colour similarity and dissimilarity,” Pattern Recogn. 38, 165-177 (2005).
  14. J. Sturges and T. Whitfield, “Locating basic colours in the Munsell space,” Color Res. Appl. 20, 364-376 (1995).
    [CrossRef]
  15. E. van den Broek, T. Schouten, and P. Kisters, “Modeling human color categorization,” Pattern Recogn. Lett. 29, 1136-1144 (2008).
    [CrossRef]
  16. G. Gagaudakis and P. Rosin, “Incorporating shape into histograms for CBIR,” Pattern Recogn. 35, 81-91 (2002).
    [CrossRef]
  17. P. KaewTrakulPong and R. Bowden, “A real time adaptive visual surveillance system for tracking low-resolution colour targets in dynamically changing scenes,” Image Vis. Comput. 21, 913-929 (2003).
    [CrossRef]
  18. A. Mojsilovic, J. Gomes, and B. Rogowitz, “Semantic-friendly indexing and quering of images based on the extraction of the objective semantic cues,” Int. J. Comput. Vis. 56, 79-107 (2004).
    [CrossRef]
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    [CrossRef]
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  24. R. Benavente and M. Vanrell, “Fuzzy colour naming based on sigmoid membership functions,” in Proceedings of the 2nd European Conference on Colour in Graphics, Imaging, and Vision (CGIV'2004) (Society for Imaging Science and Technology, IS&T, 2004), pp. 135-139.
  25. R. Benavente, M. Vanrell, and R. Baldrich, “Estimation of fuzzy sets for computational colour categorization,” Color Res. Appl. 29, 342-353 (2004).
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    [CrossRef]
  30. R. MacLaury, “From brightness to hue: an explanatory model of color-category evolution,” Curr. Anthropol. 33, 137-186 (1992).
    [CrossRef]
  31. D. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5-36 (1990).
    [CrossRef]
  32. G. Finlayson, S. Hordley, and P. Hubel, “Color by correltation: a simple, unifying framework for color constancy,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 1209-1221 (2001).
    [CrossRef]
  33. G. Finlayson, S. Hordley, and I. Tastl, “Gamut constrained illuminant estimation,” Int. J. Comput. Vis. 67, 93-109 (2006).
    [CrossRef]
  34. M. Vanrell, R. Baldrich, A. Salvatella, R. Benavente, and F. Tous, “Induction operators for a computational colour texture representation,” Comput. Vis. Image Underst. 94, 92-114 (2004).
    [CrossRef]
  35. X. Otazu and M. Vanrell, “Building perceived colour images,” in Proceedings of the 2nd European Conference on Colour in Graphics, Imaging, and Vision (CGIV'2004) (Society for Imaging Science and Technology, IS&T, 2004), pp. 140-145.
  36. R. Boynton, “Insights gained from naming the OSA colors,” in Color Categories in Thought and Language, C.L.Hardin, and L.Maffi, eds. (Cambridge U. Press, 1997), pp. 135-150.
    [CrossRef]

2008 (1)

E. van den Broek, T. Schouten, and P. Kisters, “Modeling human color categorization,” Pattern Recogn. Lett. 29, 1136-1144 (2008).
[CrossRef]

2006 (2)

R. Benavente, M. Vanrell, and R. Baldrich, “A data set for fuzzy colour naming,” Color Res. Appl. 31, 48-56 (2006).
[CrossRef]

G. Finlayson, S. Hordley, and I. Tastl, “Gamut constrained illuminant estimation,” Int. J. Comput. Vis. 67, 93-109 (2006).
[CrossRef]

2005 (2)

A. Mojsilovic, “A computational model for color naming and describing color composition of images,” IEEE Trans. Image Process. 14, 690-699 (2005).
[CrossRef] [PubMed]

M. Seaborn, L. Hepplewhite, and J. Stonham, “Fuzzy colour category map for the measurement of colour similarity and dissimilarity,” Pattern Recogn. 38, 165-177 (2005).

2004 (3)

R. Benavente, M. Vanrell, and R. Baldrich, “Estimation of fuzzy sets for computational colour categorization,” Color Res. Appl. 29, 342-353 (2004).
[CrossRef]

A. Mojsilovic, J. Gomes, and B. Rogowitz, “Semantic-friendly indexing and quering of images based on the extraction of the objective semantic cues,” Int. J. Comput. Vis. 56, 79-107 (2004).
[CrossRef]

M. Vanrell, R. Baldrich, A. Salvatella, R. Benavente, and F. Tous, “Induction operators for a computational colour texture representation,” Comput. Vis. Image Underst. 94, 92-114 (2004).
[CrossRef]

2003 (3)

P. KaewTrakulPong and R. Bowden, “A real time adaptive visual surveillance system for tracking low-resolution colour targets in dynamically changing scenes,” Image Vis. Comput. 21, 913-929 (2003).
[CrossRef]

N. Moroney, “Unconstrained Web-based color naming experiment,” Proc. SPIE 5008, 36-46 (2003).
[CrossRef]

K. Gegenfurtner and D. Kiper, “Color vision,” Annu. Rev. Neurosci. 26, 181-206 (2003).
[CrossRef] [PubMed]

2002 (1)

G. Gagaudakis and P. Rosin, “Incorporating shape into histograms for CBIR,” Pattern Recogn. 35, 81-91 (2002).
[CrossRef]

2001 (2)

H. Lin, M. Luo, L. MacDonald, and A. Tarrant, “A cross-cultural colour-naming study. Part III--A colour-naming model,” Color Res. Appl. 26, 270-277 (2001).
[CrossRef]

G. Finlayson, S. Hordley, and P. Hubel, “Color by correltation: a simple, unifying framework for color constancy,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 1209-1221 (2001).
[CrossRef]

2000 (1)

S. Guest and D. V. Laar, “The structure of colour naming space,” Vision Res. 40, 723-734 (2000).
[CrossRef] [PubMed]

1998 (1)

J. 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]

1997 (1)

J. Sturges and T. Whitfield, “Salient features of Munsell color space as a function of monolexemic naming and response latencies,” Vision Res. 37, 307-313 (1997).
[CrossRef] [PubMed]

1995 (1)

J. Sturges and T. Whitfield, “Locating basic colours in the Munsell space,” Color Res. Appl. 20, 364-376 (1995).
[CrossRef]

1992 (1)

R. MacLaury, “From brightness to hue: an explanatory model of color-category evolution,” Curr. Anthropol. 33, 137-186 (1992).
[CrossRef]

1990 (2)

D. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5-36 (1990).
[CrossRef]

R. Boynton and C. Olson, “Salience of chromatic basic color terms confirmed by three measures,” Vision Res. 30, 1311-1317 (1990).
[CrossRef] [PubMed]

1978 (1)

P. Kay and C. McDaniel, “The linguistic significance of the meanings of basic color terms,” Language 3, 610-646 (1978).
[CrossRef]

Alexander, D.

D. Alexander, “Statistical modelling of colour data and model selection for region tracking,” Ph.D. thesis (Department of Computer Science, University College London, 1997).

Baldrich, R.

R. Benavente, M. Vanrell, and R. Baldrich, “A data set for fuzzy colour naming,” Color Res. Appl. 31, 48-56 (2006).
[CrossRef]

R. Benavente, M. Vanrell, and R. Baldrich, “Estimation of fuzzy sets for computational colour categorization,” Color Res. Appl. 29, 342-353 (2004).
[CrossRef]

M. Vanrell, R. Baldrich, A. Salvatella, R. Benavente, and F. Tous, “Induction operators for a computational colour texture representation,” Comput. Vis. Image Underst. 94, 92-114 (2004).
[CrossRef]

R. Benavente, F. Tous, R. Baldrich, and M. Vanrell, “Statistical modelling of a colour naming space,” in Proceedings of the 1st European Conference on Colour in Graphics, Imaging, and Vision (CGIV'2002) (Society for Imaging Science and Technology, IS&T, 2002), pp. 406-411.

Benavente, R.

R. Benavente, M. Vanrell, and R. Baldrich, “A data set for fuzzy colour naming,” Color Res. Appl. 31, 48-56 (2006).
[CrossRef]

R. Benavente, M. Vanrell, and R. Baldrich, “Estimation of fuzzy sets for computational colour categorization,” Color Res. Appl. 29, 342-353 (2004).
[CrossRef]

M. Vanrell, R. Baldrich, A. Salvatella, R. Benavente, and F. Tous, “Induction operators for a computational colour texture representation,” Comput. Vis. Image Underst. 94, 92-114 (2004).
[CrossRef]

R. Benavente, F. Tous, R. Baldrich, and M. Vanrell, “Statistical modelling of a colour naming space,” in Proceedings of the 1st European Conference on Colour in Graphics, Imaging, and Vision (CGIV'2002) (Society for Imaging Science and Technology, IS&T, 2002), pp. 406-411.

R. Benavente and M. Vanrell, “Fuzzy colour naming based on sigmoid membership functions,” in Proceedings of the 2nd European Conference on Colour in Graphics, Imaging, and Vision (CGIV'2004) (Society for Imaging Science and Technology, IS&T, 2004), pp. 135-139.

Berlin, B.

B. Berlin and P. Kay, Basic Color Terms: Their Universality and Evolution (University of California Press, 1969).

Bowden, R.

P. KaewTrakulPong and R. Bowden, “A real time adaptive visual surveillance system for tracking low-resolution colour targets in dynamically changing scenes,” Image Vis. Comput. 21, 913-929 (2003).
[CrossRef]

Boynton, R.

R. Boynton and C. Olson, “Salience of chromatic basic color terms confirmed by three measures,” Vision Res. 30, 1311-1317 (1990).
[CrossRef] [PubMed]

R. Boynton, “Insights gained from naming the OSA colors,” in Color Categories in Thought and Language, C.L.Hardin, and L.Maffi, eds. (Cambridge U. Press, 1997), pp. 135-150.
[CrossRef]

Choh, H.

Z. Wang, M. Luo, B. Kang, H. Choh, and C. Kim, “An algorithm for categorising colours into universal colour names,” in Proceedings of the 3rd European Conference on Colour in Graphics, Imaging, and Vision (Society for Imaging Science and Technology, IS&T, 2006), pp. 426-430.

Finlayson, G.

G. Finlayson, S. Hordley, and I. Tastl, “Gamut constrained illuminant estimation,” Int. J. Comput. Vis. 67, 93-109 (2006).
[CrossRef]

G. Finlayson, S. Hordley, and P. Hubel, “Color by correltation: a simple, unifying framework for color constancy,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 1209-1221 (2001).
[CrossRef]

Forsyth, D.

D. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vis. 5, 5-36 (1990).
[CrossRef]

Gagaudakis, G.

G. Gagaudakis and P. Rosin, “Incorporating shape into histograms for CBIR,” Pattern Recogn. 35, 81-91 (2002).
[CrossRef]

Gegenfurtner, K.

K. Gegenfurtner and D. Kiper, “Color vision,” Annu. Rev. Neurosci. 26, 181-206 (2003).
[CrossRef] [PubMed]

Gomes, J.

A. Mojsilovic, J. Gomes, and B. Rogowitz, “Semantic-friendly indexing and quering of images based on the extraction of the objective semantic cues,” Int. J. Comput. Vis. 56, 79-107 (2004).
[CrossRef]

Graustein, W.

W. Graustein, Homogeneous Cartesian Coordinates. Linear Dependence of Points and Lines (Macmillan, 1930), Chap. 3, pp. 29-49.

Guest, S.

S. Guest and D. V. Laar, “The structure of colour naming space,” Vision Res. 40, 723-734 (2000).
[CrossRef] [PubMed]

Hepplewhite, L.

M. Seaborn, L. Hepplewhite, and J. Stonham, “Fuzzy colour category map for the measurement of colour similarity and dissimilarity,” Pattern Recogn. 38, 165-177 (2005).

Hordley, S.

G. Finlayson, S. Hordley, and I. Tastl, “Gamut constrained illuminant estimation,” Int. J. Comput. Vis. 67, 93-109 (2006).
[CrossRef]

G. Finlayson, S. Hordley, and P. Hubel, “Color by correltation: a simple, unifying framework for color constancy,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 1209-1221 (2001).
[CrossRef]

Hubel, P.

G. Finlayson, S. Hordley, and P. Hubel, “Color by correltation: a simple, unifying framework for color constancy,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 1209-1221 (2001).
[CrossRef]

KaewTrakulPong, P.

P. KaewTrakulPong and R. Bowden, “A real time adaptive visual surveillance system for tracking low-resolution colour targets in dynamically changing scenes,” Image Vis. Comput. 21, 913-929 (2003).
[CrossRef]

Kang, B.

Z. Wang, M. Luo, B. Kang, H. Choh, and C. Kim, “An algorithm for categorising colours into universal colour names,” in Proceedings of the 3rd European Conference on Colour in Graphics, Imaging, and Vision (Society for Imaging Science and Technology, IS&T, 2006), pp. 426-430.

Kay, P.

P. Kay and C. McDaniel, “The linguistic significance of the meanings of basic color terms,” Language 3, 610-646 (1978).
[CrossRef]

B. Berlin and P. Kay, Basic Color Terms: Their Universality and Evolution (University of California Press, 1969).

Kim, C.

Z. Wang, M. Luo, B. Kang, H. Choh, and C. Kim, “An algorithm for categorising colours into universal colour names,” in Proceedings of the 3rd European Conference on Colour in Graphics, Imaging, and Vision (Society for Imaging Science and Technology, IS&T, 2006), pp. 426-430.

Kiper, D.

K. Gegenfurtner and D. Kiper, “Color vision,” Annu. Rev. Neurosci. 26, 181-206 (2003).
[CrossRef] [PubMed]

Kisters, P.

E. van den Broek, T. Schouten, and P. Kisters, “Modeling human color categorization,” Pattern Recogn. Lett. 29, 1136-1144 (2008).
[CrossRef]

Klir, G.

G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications (Prentice Hall, 1995).

Laar, D. V.

S. Guest and D. V. Laar, “The structure of colour naming space,” Vision Res. 40, 723-734 (2000).
[CrossRef] [PubMed]

Lagarias, J.

J. 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]

Lammens, J.

J. Lammens, “A computational model of color perception and color naming,” Ph.D. thesis (State University of New York, 1994).

Lin, H.

H. Lin, M. Luo, L. MacDonald, and A. Tarrant, “A cross-cultural colour-naming study. Part III--A colour-naming model,” Color Res. Appl. 26, 270-277 (2001).
[CrossRef]

Luo, M.

H. Lin, M. Luo, L. MacDonald, and A. Tarrant, “A cross-cultural colour-naming study. Part III--A colour-naming model,” Color Res. Appl. 26, 270-277 (2001).
[CrossRef]

Z. Wang, M. Luo, B. Kang, H. Choh, and C. Kim, “An algorithm for categorising colours into universal colour names,” in Proceedings of the 3rd European Conference on Colour in Graphics, Imaging, and Vision (Society for Imaging Science and Technology, IS&T, 2006), pp. 426-430.

MacDonald, L.

H. Lin, M. Luo, L. MacDonald, and A. Tarrant, “A cross-cultural colour-naming study. Part III--A colour-naming model,” Color Res. Appl. 26, 270-277 (2001).
[CrossRef]

MacLaury, R.

R. MacLaury, “From brightness to hue: an explanatory model of color-category evolution,” Curr. Anthropol. 33, 137-186 (1992).
[CrossRef]

McDaniel, C.

P. Kay and C. McDaniel, “The linguistic significance of the meanings of basic color terms,” Language 3, 610-646 (1978).
[CrossRef]

Mojsilovic, A.

A. Mojsilovic, “A computational model for color naming and describing color composition of images,” IEEE Trans. Image Process. 14, 690-699 (2005).
[CrossRef] [PubMed]

A. Mojsilovic, J. Gomes, and B. Rogowitz, “Semantic-friendly indexing and quering of images based on the extraction of the objective semantic cues,” Int. J. Comput. Vis. 56, 79-107 (2004).
[CrossRef]

Moroney, N.

N. Moroney, “Unconstrained Web-based color naming experiment,” Proc. SPIE 5008, 36-46 (2003).
[CrossRef]

Olson, C.

R. Boynton and C. Olson, “Salience of chromatic basic color terms confirmed by three measures,” Vision Res. 30, 1311-1317 (1990).
[CrossRef] [PubMed]

Otazu, X.

X. Otazu and M. Vanrell, “Building perceived colour images,” in Proceedings of the 2nd European Conference on Colour in Graphics, Imaging, and Vision (CGIV'2004) (Society for Imaging Science and Technology, IS&T, 2004), pp. 140-145.

Reeds, J.

J. 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]

Rogowitz, B.

A. Mojsilovic, J. Gomes, and B. Rogowitz, “Semantic-friendly indexing and quering of images based on the extraction of the objective semantic cues,” Int. J. Comput. Vis. 56, 79-107 (2004).
[CrossRef]

Rosin, P.

G. Gagaudakis and P. Rosin, “Incorporating shape into histograms for CBIR,” Pattern Recogn. 35, 81-91 (2002).
[CrossRef]

Salvatella, A.

M. Vanrell, R. Baldrich, A. Salvatella, R. Benavente, and F. Tous, “Induction operators for a computational colour texture representation,” Comput. Vis. Image Underst. 94, 92-114 (2004).
[CrossRef]

Schouten, T.

E. van den Broek, T. Schouten, and P. Kisters, “Modeling human color categorization,” Pattern Recogn. Lett. 29, 1136-1144 (2008).
[CrossRef]

E. van den Broek, E. van Rikxoort, and T. Schouten, “Human-centered object-based image retrieval,” in Advances in Pattern Recognition, Vol. 3687 of Lecture Notes in Computer Science, S.Singh, M.Singh, C.Apte, and P.Perner (Springer-Verlag, 2005), pp. 492-501.

Seaborn, M.

M. Seaborn, L. Hepplewhite, and J. Stonham, “Fuzzy colour category map for the measurement of colour similarity and dissimilarity,” Pattern Recogn. 38, 165-177 (2005).

Stonham, J.

M. Seaborn, L. Hepplewhite, and J. Stonham, “Fuzzy colour category map for the measurement of colour similarity and dissimilarity,” Pattern Recogn. 38, 165-177 (2005).

Sturges, J.

J. Sturges and T. Whitfield, “Salient features of Munsell color space as a function of monolexemic naming and response latencies,” Vision Res. 37, 307-313 (1997).
[CrossRef] [PubMed]

J. Sturges and T. Whitfield, “Locating basic colours in the Munsell space,” Color Res. Appl. 20, 364-376 (1995).
[CrossRef]

Tarrant, A.

H. Lin, M. Luo, L. MacDonald, and A. Tarrant, “A cross-cultural colour-naming study. Part III--A colour-naming model,” Color Res. Appl. 26, 270-277 (2001).
[CrossRef]

Tastl, I.

G. Finlayson, S. Hordley, and I. Tastl, “Gamut constrained illuminant estimation,” Int. J. Comput. Vis. 67, 93-109 (2006).
[CrossRef]

Tominaga, S.

S. Tominaga, “A color-naming method for computer color vision,” in Proceedings of IEEE International Conference on Cybernetics and Society (IEEE, 1985), pp. 573-577.

Tous, F.

M. Vanrell, R. Baldrich, A. Salvatella, R. Benavente, and F. Tous, “Induction operators for a computational colour texture representation,” Comput. Vis. Image Underst. 94, 92-114 (2004).
[CrossRef]

R. Benavente, F. Tous, R. Baldrich, and M. Vanrell, “Statistical modelling of a colour naming space,” in Proceedings of the 1st European Conference on Colour in Graphics, Imaging, and Vision (CGIV'2002) (Society for Imaging Science and Technology, IS&T, 2002), pp. 406-411.

van den Broek, E.

E. van den Broek, T. Schouten, and P. Kisters, “Modeling human color categorization,” Pattern Recogn. Lett. 29, 1136-1144 (2008).
[CrossRef]

E. van den Broek, E. van Rikxoort, and T. Schouten, “Human-centered object-based image retrieval,” in Advances in Pattern Recognition, Vol. 3687 of Lecture Notes in Computer Science, S.Singh, M.Singh, C.Apte, and P.Perner (Springer-Verlag, 2005), pp. 492-501.

van Rikxoort, E.

E. van den Broek, E. van Rikxoort, and T. Schouten, “Human-centered object-based image retrieval,” in Advances in Pattern Recognition, Vol. 3687 of Lecture Notes in Computer Science, S.Singh, M.Singh, C.Apte, and P.Perner (Springer-Verlag, 2005), pp. 492-501.

Vanrell, M.

R. Benavente, M. Vanrell, and R. Baldrich, “A data set for fuzzy colour naming,” Color Res. Appl. 31, 48-56 (2006).
[CrossRef]

R. Benavente, M. Vanrell, and R. Baldrich, “Estimation of fuzzy sets for computational colour categorization,” Color Res. Appl. 29, 342-353 (2004).
[CrossRef]

M. Vanrell, R. Baldrich, A. Salvatella, R. Benavente, and F. Tous, “Induction operators for a computational colour texture representation,” Comput. Vis. Image Underst. 94, 92-114 (2004).
[CrossRef]

R. Benavente, F. Tous, R. Baldrich, and M. Vanrell, “Statistical modelling of a colour naming space,” in Proceedings of the 1st European Conference on Colour in Graphics, Imaging, and Vision (CGIV'2002) (Society for Imaging Science and Technology, IS&T, 2002), pp. 406-411.

R. Benavente and M. Vanrell, “Fuzzy colour naming based on sigmoid membership functions,” in Proceedings of the 2nd European Conference on Colour in Graphics, Imaging, and Vision (CGIV'2004) (Society for Imaging Science and Technology, IS&T, 2004), pp. 135-139.

X. Otazu and M. Vanrell, “Building perceived colour images,” in Proceedings of the 2nd European Conference on Colour in Graphics, Imaging, and Vision (CGIV'2004) (Society for Imaging Science and Technology, IS&T, 2004), pp. 140-145.

Wang, Z.

Z. Wang, M. Luo, B. Kang, H. Choh, and C. Kim, “An algorithm for categorising colours into universal colour names,” in Proceedings of the 3rd European Conference on Colour in Graphics, Imaging, and Vision (Society for Imaging Science and Technology, IS&T, 2006), pp. 426-430.

Whitfield, T.

J. Sturges and T. Whitfield, “Salient features of Munsell color space as a function of monolexemic naming and response latencies,” Vision Res. 37, 307-313 (1997).
[CrossRef] [PubMed]

J. Sturges and T. Whitfield, “Locating basic colours in the Munsell space,” Color Res. Appl. 20, 364-376 (1995).
[CrossRef]

Wright, M.

J. 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. 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]

Yuan, B.

G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications (Prentice Hall, 1995).

Annu. Rev. Neurosci. (1)

K. Gegenfurtner and D. Kiper, “Color vision,” Annu. Rev. Neurosci. 26, 181-206 (2003).
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Figures (12)

Fig. 1
Fig. 1

Scheme of the model. The color space is divided into N L levels along the lightness axis.

Fig. 2
Fig. 2

Desirable properties of the membership function for chromatic categories. In this case, on the blue category.

Fig. 3
Fig. 3

(a) Sigmoid function in one dimension. The value of β determines the slope of the function. (b) Sigmoid function in two dimensions. Vector u i determines the axis in which the function is oriented.

Fig. 4
Fig. 4

Two-dimensional sigmoid functions. (a) S 1 , sigmoid function oriented in the x-axis direction (b) S 2 , sigmoid function oriented in the y-axis direction. (c) D S , product of two differently oriented sigmoid functions generates a plateau with some of the properties needed for the membership function.

Fig. 5
Fig. 5

Elliptic-sigmoid function E S ( p , t , θ E S ) . (a) ES for β e < 0 and (b) ES for β e > 0 .

Fig. 6
Fig. 6

TSE function.

Fig. 7
Fig. 7

TSE function fitted to the chromatic categories defined on a given lightness level. In this case, only six categories have memberships different from zero.

Fig. 8
Fig. 8

Sigmoid functions are used to differentiate among the three achromatic categories

Fig. 9
Fig. 9

Histogram of the learning set samples used to determine the values that define the lightness levels of the model.

Fig. 10
Fig. 10

Membership maps for the six chromaticity planes of the model.

Fig. 11
Fig. 11

Comparison between our model’s Munsell categorization and Berlin and Kay’s boundaries. Samples named differently by our model are marked with a cross.

Fig. 12
Fig. 12

Consensus areas and focus from Sturges and Whitfield’s experiment superimposed on our model’s categorization of the Munsell array.

Tables (2)

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Table 1 Parameters of the Triple-Sigmoid with Elliptical Center Model a

Tables Icon

Table 2 Comparison of Different Munsell Categorizations to the Results from Color-Naming Experiments of Berlin and Kay [2] and Sturges and Whitfield [14]

Equations (29)

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k = 1 n μ C k ( s ) = 1 with μ C k ( s ) [ 0 , 1 ] , k = 1 , , n .
C D ( s ) = [ μ C 1 ( s ) , , μ C n ( s ) ] ,
N ( s ) = t k max k max = arg max k = 1 , , n { μ C k ( s ) } ,
C k { Red , Orange , Brown , Yellow , Green , Blue , Purple , Pink , Black , Gray , White } .
S 1 ( x , β ) = 1 1 + exp ( β x ) ,
S ( p , β ) = 1 1 + exp ( β u i p ) , i = 1 , 2 ,
S i ( p , t , α , β ) = 1 1 + exp ( β u i R α T t p ) , i = 1 , 2 ,
T t = ( 1 0 t x 0 1 t y 0 0 1 ) , R α = ( cos ( α ) sin ( α ) 0 sin ( α ) cos ( α ) 0 0 0 1 ) .
DS ( p , t , θ DS ) = S 1 ( p , t , α y , β y ) S 2 ( p , t , α x , β x ) ,
ES ( p , t , θ ES ) = 1 1 + exp { β e [ ( u 1 R ϕ T t p e x ) 2 + ( u 2 R ϕ T t p e y ) 2 1 ] } ,
TSE ( p , θ ) = DS ( p , t , θ DS ) ES ( p , t , θ ES ) ,
μ C k ( s ) = { μ C k 1 = TSE ( c 1 , c 2 , θ C k 1 ) if I I 1 μ C k 2 = TSE ( c 1 , c 2 , θ C k 2 ) if I 1 < I I 2 , μ C k N L = TSE ( c 1 , c 2 , θ C k N L ) if I N L 1 < I }
μ A i ( c 1 , c 2 ) = 1 k = 1 n c μ C k i ( c 1 , c 2 ) ,
μ A Black ( I , θ Black ) = 1 1 + exp [ β b ( I t b ) ] ,
μ A Gray ( I , θ Gray ) = 1 1 + exp [ β b ( I t b ) ] 1 1 + exp [ β w ( I t w ) ] ,
μ A White ( I , θ White ) = 1 1 + exp [ β w ( I t w ) ] ,
μ C k ( s , θ C k ) = μ A i ( c 1 , c 2 ) μ A C k ( I , θ C k ) ,
9 k 11 , I i < I I i + 1 ,
k = 1 11 μ C k i ( s ) = 1 , i = 1 , , N L ,
D = { s i , m 1 i , , m 11 i } , i = 1 , , n s ,
θ ̂ j = arg min θ j 1 n cp i = 1 n cp k = 1 n c ( μ C k j ( s i , θ C k j ) m k i ) 2 , j = 1 , , N L ,
k = 1 11 μ C k j ( s , θ C k j ) = 1 , s = ( I , c 1 , c 2 ) I j 1 < I I j ,
θ ES C p j = θ ES C q j , t C p j = t C q j , p , q { 1 , , n c } ,
β y C p = β x C q , α y C p = α x C q ( π 2 ) ,
( t ̂ j , θ ̂ ES j ) = arg min t j , θ ES j 1 n cp i = 1 n cp ( ES ( s i , t j , θ ES j ) k = 9 11 m k i ) 2 ,
( θ ̂ DS C p j , θ ̂ DS C q j ) = argmin θ DS C p j , θ DS C q j i = 1 n cp ( ( μ C p j ( s i , θ C p j ) m p i ) 2 + ( μ C q j ( s i , θ C q j ) m q i ) 2 ) ,
θ ̂ A = arg min θ A i = 1 n s k = 9 11 ( μ C k ( s i , θ C k ) m k i ) 2 ,
MAE fit = 1 n s 1 11 i = 1 n s k = 1 11 m k i μ C k ( s i ) ,
MAE unitsum = 1 n p i = 1 n p 1 k = 1 11 μ C k ( p i ) ,

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