Abstract

Variations in illumination on a scene and trichromatic sampling by the eye limit inferences about scene content. The aim of this work was to elucidate these limits in relation to an ideal observer using color signals alone. Simulations were based on 50 hyperspectral images of natural scenes and daylight illuminants with correlated color temperatures 4000K, 6500K, and 25,000K. Estimates were made of the (Shannon) information available from each scene, the redundancies in receptoral and postreceptoral coding, and the information retrieved by an observer identifying corresponding points across image pairs. For the largest illuminant difference, between 25,000K and 4000K, a postreceptoral transformation providing minimum redundancy yielded an efficiency of about 80% in the information retrieved. This increased to about 89% when the transformation was optimized directly for information retrieved, corresponding to an equivalent Gaussian noise amplitude of 3.0% or to a mean of 3.6×104 distinct identifiable points per scene. Using color signals to retrieve information from natural scenes can approach ideal observer efficiency levels.

© 2009 Optical Society of America

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2009

J. J. M. Granzier, E. Brenner, and J. B. J. Smeets, “Reliable identification by color under natural conditions,” J. Vision 9, 1-8 (2009).
[CrossRef]

2008

W. S. Geisler, “Visual perception and the statistical properties of natural scenes,” Annu. Rev. Psychol. 59, 167-192 (2008).
[CrossRef]

2006

D. H. Foster, K. Amano, S. M. C. Nascimento, and M. J. Foster, “Frequency of metamerism in natural scenes,” J. Opt. Soc. Am. A 23, 2359-2372 (2006).
[CrossRef]

D. H. Foster, K. Amano, and S. M. C. Nascimento, “Color constancy in natural scenes explained by global image statistics,” Visual Neurosci. 23, 341-349 (2006).
[CrossRef]

2005

M. N. Goria, N. N. Leonenko, V. V. Mergel, and P. L. Novi Inverardi, “A new class of random vector entropy estimators and its applications in testing statistical hypotheses,” J. Nonparametric Statistics 17, 277-297 (2005).
[CrossRef]

D. H. Foster, S. M. C. Nascimento, and K. Amano, “Information limits on identification of natural surfaces by apparent colour,” Perception 34, 1003-1008 (2005).
[CrossRef] [PubMed]

2004

D. H. Foster, S. M. C. Nascimento, and K. Amano, “Information limits on neural identification of colored surfaces in natural scenes,” Visual Neurosci. 21, 331-336 (2004).
[CrossRef]

K. Amano and D. H. Foster, “Colour constancy under simultaneous changes in surface position and illuminant,” Proc. R. Soc. London, Ser. B 271, 2319-2326 (2004).
[CrossRef]

A. Kraskov, H. Stögbauer, and P. Grassberger, “Estimating mutual information,” Phys. Rev. E 69, 066138 (2004).
[CrossRef]

2003

B. Funt and H. Jiang, “Non-von-Kries 3-parameter color prediction,” Proc. SPIE 5007, 182-189 (2003).
[CrossRef]

N. J. Dominy, J.-C. Svenning, and W.-H. Li, “Historical contingency in the evolution of primate color vision,” J. Human Evolution 44, 25-45 (2003).
[CrossRef]

2002

S. M. C. Nascimento, F. P. Ferreira, and D. H. Foster, “Statistics of spatial cone-excitation ratios in natural scenes,” J. Opt. Soc. Am. A 19, 1484-1490 (2002).
[CrossRef]

R. Steuer, J. Kurths, C. O. Daub, J. Weise, and J. Selbig, “The mutual information: Detecting and evaluating dependencies between variables,” Bioinformatics 18, S231-S240 (2002).
[CrossRef] [PubMed]

J. D. Victor, “Binless strategies for estimation of information from neural data,” Phys. Rev. E 66, 051903 (2002).
[CrossRef]

2001

2000

R. L. De Valois, N. P. Cottaris, S. D. Elfar, L. E. Mahon, and J. A. Wilson, “Some transformations of color information from lateral geniculate nucleus to striate cortex,” Proc. Natl. Acad. Sci. U.S.A. 97, 4997-5002 (2000).
[CrossRef] [PubMed]

A. Stockman and L. T. Sharpe, “The spectral sensitivities of the middle- and long-wavelength-sensitive cones derived from measurements in observers of known genotype,” Vision Res. 40, 1711-1737 (2000).
[CrossRef] [PubMed]

1999

A. Stockman, L. T. Sharpe, and C. Fach, “The spectral sensitivity of the human short-wavelength sensitive cones derived from thresholds and color matches,” Vision Res. 39, 2901-2927 (1999).
[CrossRef] [PubMed]

1998

M. Vorobyev and D. Osorio, “Receptor noise as a determinant of colour thresholds,” Proc. R. Soc. London, Ser. B 265, 351-358 (1998).
[CrossRef]

1997

M. A. Webster and J. D. Mollon, “Adaptation and the color statistics of natural images,” Vision Res. 37, 3283-3298 (1997).
[CrossRef]

1996

A. Lapidoth, “Nearest neighbor decoding for additive non-Gaussian noise channels,” IEEE Trans. Inf. Theory 42, 1520-1529 (1996).
[CrossRef]

1994

1987

L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Probl. Inf. Transm. 23, 95-101 (1987); L. F. Kozachenko and N. N. Leonenko,Russian translation, Probl. Peredachi Inf. 23, 9-16 (1987).

L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Probl. Inf. Transm. 23, 95-101 (1987); L. F. Kozachenko and N. N. Leonenko,Russian translation, Probl. Peredachi Inf. 23, 9-16 (1987).

1983

D. H. Foster and R. S. Snelgar, “Test and field spectral sensitivities of colour mechanisms obtained on small white backgrounds: action of unitary opponent-colour processes?” Vision Res. 23, 787-797 (1983).
[CrossRef] [PubMed]

G. Buchsbaum and A. Gottschalk, “Trichromacy, opponent colours coding and optimum colour information transmission in the retina,” Proc. R. Soc. London, Ser. B 220, 89-113 (1983).
[CrossRef]

1979

W. S. Cleveland, “Robust locally weighted regression and smoothing scatterplots,” J. Am. Stat. Assoc. 74, 829-836 (1979).
[CrossRef]

1964

1961

J. A. Swets, W. P. Tanner, Jr., and T. G. Birdsall, “Decision processes in perception,” Psychol. Rev. 68, 301-340 (1961).
[CrossRef] [PubMed]

1948

C. E. Shannon, “A mathematical theory of communication,” [part 1] Bell Syst. Tech. J. 27, 379-423 (1948).

C. E. Shannon, “A mathematical theory of communication,” [part 2] Bell Syst. Tech. J. 27, 623-656 (1948).

Amano, K.

D. H. Foster, K. Amano, S. M. C. Nascimento, and M. J. Foster, “Frequency of metamerism in natural scenes,” J. Opt. Soc. Am. A 23, 2359-2372 (2006).
[CrossRef]

D. H. Foster, K. Amano, and S. M. C. Nascimento, “Color constancy in natural scenes explained by global image statistics,” Visual Neurosci. 23, 341-349 (2006).
[CrossRef]

D. H. Foster, S. M. C. Nascimento, and K. Amano, “Information limits on identification of natural surfaces by apparent colour,” Perception 34, 1003-1008 (2005).
[CrossRef] [PubMed]

D. H. Foster, S. M. C. Nascimento, and K. Amano, “Information limits on neural identification of colored surfaces in natural scenes,” Visual Neurosci. 21, 331-336 (2004).
[CrossRef]

K. Amano and D. H. Foster, “Colour constancy under simultaneous changes in surface position and illuminant,” Proc. R. Soc. London, Ser. B 271, 2319-2326 (2004).
[CrossRef]

Barlow, H.

H. Barlow, “Redundancy reduction revisited,” Network Comput. Neural Syst. 12, 241-253 (2001).
[CrossRef]

Birdsall, T. G.

J. A. Swets, W. P. Tanner, Jr., and T. G. Birdsall, “Decision processes in perception,” Psychol. Rev. 68, 301-340 (1961).
[CrossRef] [PubMed]

Brenner, E.

J. J. M. Granzier, E. Brenner, and J. B. J. Smeets, “Reliable identification by color under natural conditions,” J. Vision 9, 1-8 (2009).
[CrossRef]

Buchsbaum, G.

G. Buchsbaum and A. Gottschalk, “Trichromacy, opponent colours coding and optimum colour information transmission in the retina,” Proc. R. Soc. London, Ser. B 220, 89-113 (1983).
[CrossRef]

Cleveland, W. S.

W. S. Cleveland, “Robust locally weighted regression and smoothing scatterplots,” J. Am. Stat. Assoc. 74, 829-836 (1979).
[CrossRef]

Cottaris, N. P.

R. L. De Valois, N. P. Cottaris, S. D. Elfar, L. E. Mahon, and J. A. Wilson, “Some transformations of color information from lateral geniculate nucleus to striate cortex,” Proc. Natl. Acad. Sci. U.S.A. 97, 4997-5002 (2000).
[CrossRef] [PubMed]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Creelman, C. D.

N. A. Macmillan and C. D. Creelman, Detection Theory: A User's Guide, 2nd ed. (Erlbaum, 2005).

Daub, C. O.

R. Steuer, J. Kurths, C. O. Daub, J. Weise, and J. Selbig, “The mutual information: Detecting and evaluating dependencies between variables,” Bioinformatics 18, S231-S240 (2002).
[CrossRef] [PubMed]

De Valois, R. L.

R. L. De Valois, N. P. Cottaris, S. D. Elfar, L. E. Mahon, and J. A. Wilson, “Some transformations of color information from lateral geniculate nucleus to striate cortex,” Proc. Natl. Acad. Sci. U.S.A. 97, 4997-5002 (2000).
[CrossRef] [PubMed]

Dominy, N. J.

N. J. Dominy, J.-C. Svenning, and W.-H. Li, “Historical contingency in the evolution of primate color vision,” J. Human Evolution 44, 25-45 (2003).
[CrossRef]

Drew, M. S.

Elfar, S. D.

R. L. De Valois, N. P. Cottaris, S. D. Elfar, L. E. Mahon, and J. A. Wilson, “Some transformations of color information from lateral geniculate nucleus to striate cortex,” Proc. Natl. Acad. Sci. U.S.A. 97, 4997-5002 (2000).
[CrossRef] [PubMed]

Fach, C.

A. Stockman, L. T. Sharpe, and C. Fach, “The spectral sensitivity of the human short-wavelength sensitive cones derived from thresholds and color matches,” Vision Res. 39, 2901-2927 (1999).
[CrossRef] [PubMed]

Ferreira, F. P.

Finlayson, G. D.

Foster, D. H.

D. H. Foster, K. Amano, S. M. C. Nascimento, and M. J. Foster, “Frequency of metamerism in natural scenes,” J. Opt. Soc. Am. A 23, 2359-2372 (2006).
[CrossRef]

D. H. Foster, K. Amano, and S. M. C. Nascimento, “Color constancy in natural scenes explained by global image statistics,” Visual Neurosci. 23, 341-349 (2006).
[CrossRef]

D. H. Foster, S. M. C. Nascimento, and K. Amano, “Information limits on identification of natural surfaces by apparent colour,” Perception 34, 1003-1008 (2005).
[CrossRef] [PubMed]

D. H. Foster, S. M. C. Nascimento, and K. Amano, “Information limits on neural identification of colored surfaces in natural scenes,” Visual Neurosci. 21, 331-336 (2004).
[CrossRef]

K. Amano and D. H. Foster, “Colour constancy under simultaneous changes in surface position and illuminant,” Proc. R. Soc. London, Ser. B 271, 2319-2326 (2004).
[CrossRef]

S. M. C. Nascimento, F. P. Ferreira, and D. H. Foster, “Statistics of spatial cone-excitation ratios in natural scenes,” J. Opt. Soc. Am. A 19, 1484-1490 (2002).
[CrossRef]

D. H. Foster and R. S. Snelgar, “Test and field spectral sensitivities of colour mechanisms obtained on small white backgrounds: action of unitary opponent-colour processes?” Vision Res. 23, 787-797 (1983).
[CrossRef] [PubMed]

D. H. Foster and K. Żychaluk, “Is there a better non-parametric alternative to von Kries scaling?” in CGIV 2008 (Society for Imaging Science and Technology, 2008), pp. 41-44.

I. Marín-Franch and D. H. Foster, “Distribution of information within and across colour spaces,” in CGIV 2008 (Society for Imaging Science and Technology, 2008), pp. 303-306.

Foster, M. J.

Funt, B.

B. Funt and H. Jiang, “Non-von-Kries 3-parameter color prediction,” Proc. SPIE 5007, 182-189 (2003).
[CrossRef]

Funt, B. V.

Geisler, W. S.

W. S. Geisler, “Visual perception and the statistical properties of natural scenes,” Annu. Rev. Psychol. 59, 167-192 (2008).
[CrossRef]

Goria, M. N.

M. N. Goria, N. N. Leonenko, V. V. Mergel, and P. L. Novi Inverardi, “A new class of random vector entropy estimators and its applications in testing statistical hypotheses,” J. Nonparametric Statistics 17, 277-297 (2005).
[CrossRef]

Gottschalk, A.

G. Buchsbaum and A. Gottschalk, “Trichromacy, opponent colours coding and optimum colour information transmission in the retina,” Proc. R. Soc. London, Ser. B 220, 89-113 (1983).
[CrossRef]

Granzier, J. J. M.

J. J. M. Granzier, E. Brenner, and J. B. J. Smeets, “Reliable identification by color under natural conditions,” J. Vision 9, 1-8 (2009).
[CrossRef]

Grassberger, P.

A. Kraskov, H. Stögbauer, and P. Grassberger, “Estimating mutual information,” Phys. Rev. E 69, 066138 (2004).
[CrossRef]

P. Grassberger, “Entropy estimates from insufficient samplings,” (2008), retrieved http://arxiv.org/abs/physics/0307138v2.

Hernández-Andrés, J.

Jiang, H.

B. Funt and H. Jiang, “Non-von-Kries 3-parameter color prediction,” Proc. SPIE 5007, 182-189 (2003).
[CrossRef]

Judd, D. B.

Kozachenko, L. F.

L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Probl. Inf. Transm. 23, 95-101 (1987); L. F. Kozachenko and N. N. Leonenko,Russian translation, Probl. Peredachi Inf. 23, 9-16 (1987).

L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Probl. Inf. Transm. 23, 95-101 (1987); L. F. Kozachenko and N. N. Leonenko,Russian translation, Probl. Peredachi Inf. 23, 9-16 (1987).

Kraskov, A.

A. Kraskov, H. Stögbauer, and P. Grassberger, “Estimating mutual information,” Phys. Rev. E 69, 066138 (2004).
[CrossRef]

Kurths, J.

R. Steuer, J. Kurths, C. O. Daub, J. Weise, and J. Selbig, “The mutual information: Detecting and evaluating dependencies between variables,” Bioinformatics 18, S231-S240 (2002).
[CrossRef] [PubMed]

Lapidoth, A.

A. Lapidoth, “Nearest neighbor decoding for additive non-Gaussian noise channels,” IEEE Trans. Inf. Theory 42, 1520-1529 (1996).
[CrossRef]

Leonenko, N. N.

M. N. Goria, N. N. Leonenko, V. V. Mergel, and P. L. Novi Inverardi, “A new class of random vector entropy estimators and its applications in testing statistical hypotheses,” J. Nonparametric Statistics 17, 277-297 (2005).
[CrossRef]

L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Probl. Inf. Transm. 23, 95-101 (1987); L. F. Kozachenko and N. N. Leonenko,Russian translation, Probl. Peredachi Inf. 23, 9-16 (1987).

L. F. Kozachenko and N. N. Leonenko, “Sample estimate of the entropy of a random vector,” Probl. Inf. Transm. 23, 95-101 (1987); L. F. Kozachenko and N. N. Leonenko,Russian translation, Probl. Peredachi Inf. 23, 9-16 (1987).

Li, W.-H.

N. J. Dominy, J.-C. Svenning, and W.-H. Li, “Historical contingency in the evolution of primate color vision,” J. Human Evolution 44, 25-45 (2003).
[CrossRef]

MacAdam, D. L.

MacLeod, D. I. A.

D. I. A. MacLeod and T. von der Twer, “The pleistochrome: optimal opponent codes for natural colours,” in Colour Perception: Mind and the Physical World, R.Mausfeld and D.Heyer, eds. (Oxford Univ. Press, 2003), pp. 155-184.

Macmillan, N. A.

N. A. Macmillan and C. D. Creelman, Detection Theory: A User's Guide, 2nd ed. (Erlbaum, 2005).

Mahon, L. E.

R. L. De Valois, N. P. Cottaris, S. D. Elfar, L. E. Mahon, and J. A. Wilson, “Some transformations of color information from lateral geniculate nucleus to striate cortex,” Proc. Natl. Acad. Sci. U.S.A. 97, 4997-5002 (2000).
[CrossRef] [PubMed]

Marín-Franch, I.

I. Marín-Franch and D. H. Foster, “Distribution of information within and across colour spaces,” in CGIV 2008 (Society for Imaging Science and Technology, 2008), pp. 303-306.

I. Marín-Franch, “Information-theoretic analysis of trichromatic images of natural scenes under different phases of daylight,” Ph.D. thesis (The University of Manchester, Manchester, 2009).

Mergel, V. V.

M. N. Goria, N. N. Leonenko, V. V. Mergel, and P. L. Novi Inverardi, “A new class of random vector entropy estimators and its applications in testing statistical hypotheses,” J. Nonparametric Statistics 17, 277-297 (2005).
[CrossRef]

Mollon, J. D.

M. A. Webster and J. D. Mollon, “Adaptation and the color statistics of natural images,” Vision Res. 37, 3283-3298 (1997).
[CrossRef]

Nascimento, S. M. C.

D. H. Foster, K. Amano, S. M. C. Nascimento, and M. J. Foster, “Frequency of metamerism in natural scenes,” J. Opt. Soc. Am. A 23, 2359-2372 (2006).
[CrossRef]

D. H. Foster, K. Amano, and S. M. C. Nascimento, “Color constancy in natural scenes explained by global image statistics,” Visual Neurosci. 23, 341-349 (2006).
[CrossRef]

D. H. Foster, S. M. C. Nascimento, and K. Amano, “Information limits on identification of natural surfaces by apparent colour,” Perception 34, 1003-1008 (2005).
[CrossRef] [PubMed]

D. H. Foster, S. M. C. Nascimento, and K. Amano, “Information limits on neural identification of colored surfaces in natural scenes,” Visual Neurosci. 21, 331-336 (2004).
[CrossRef]

S. M. C. Nascimento, F. P. Ferreira, and D. H. Foster, “Statistics of spatial cone-excitation ratios in natural scenes,” J. Opt. Soc. Am. A 19, 1484-1490 (2002).
[CrossRef]

Nieves, J. L.

Novi Inverardi, P. L.

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J. Opt. Soc. Am. A

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D. H. Foster and R. S. Snelgar, “Test and field spectral sensitivities of colour mechanisms obtained on small white backgrounds: action of unitary opponent-colour processes?” Vision Res. 23, 787-797 (1983).
[CrossRef] [PubMed]

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[CrossRef]

A. Stockman, L. T. Sharpe, and C. Fach, “The spectral sensitivity of the human short-wavelength sensitive cones derived from thresholds and color matches,” Vision Res. 39, 2901-2927 (1999).
[CrossRef] [PubMed]

A. Stockman and L. T. Sharpe, “The spectral sensitivities of the middle- and long-wavelength-sensitive cones derived from measurements in observers of known genotype,” Vision Res. 40, 1711-1737 (2000).
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[CrossRef]

D. I. A. MacLeod and T. von der Twer, “The pleistochrome: optimal opponent codes for natural colours,” in Colour Perception: Mind and the Physical World, R.Mausfeld and D.Heyer, eds. (Oxford Univ. Press, 2003), pp. 155-184.

J. von Kries, “Theoretische Studien über die Umstimmung des Sehorgans,” in Festschrift der Albrecht-Ludwigs-Universität (Freiburg, 1902), pp. 145-158.

J. von Kries, “Die Gesichtsempfindungen,” in Handbuch der Physiologie des Menschen, W.Nagel, ed. (Vieweg, 1905), pp. 109-282.

B. W. Silverman, Density Estimation for Statistics and Data Analysis (Chapman & Hall, London, 1986).

M. Studený and J. Vejnarová, “The multi-information function as a tool for measuring stochastic dependence,” in Learning in Graphical Models, M.I.Jordan, ed. (MIT Press, 1999), pp. 261-298.

I. Marín-Franch and D. H. Foster, “Distribution of information within and across colour spaces,” in CGIV 2008 (Society for Imaging Science and Technology, 2008), pp. 303-306.

D. H. Foster and K. Żychaluk, “Is there a better non-parametric alternative to von Kries scaling?” in CGIV 2008 (Society for Imaging Science and Technology, 2008), pp. 41-44.

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Figures (6)

Fig. 1
Fig. 1

Image statistics. The image in panel a is of a scene under a daylight illuminant with correlated color temperature 6500 K . The histograms in panels b, c, and d are naïve estimates of the response probability density functions of L, M, and S cones based on a sample of 1000 points drawn randomly from the image a. The small gray sphere at the bottom left of the image was excluded from the analysis by a mask. Responses were normalized to a mean of 1.0 (equivalent to von Kries scaling; abscissas and ordinates have been truncated).The continuous smooth curves are exponential model density functions fitted by maximum likelihood to responses > 1.0 , with dotted lines indicating the fit over the remainder of the range.

Fig. 2
Fig. 2

Identification errors across images of the scene in Fig. 1a under a daylight illuminant with correlated color temperature a, 25,000 K and b, 4000 K . The points marked 1, 2, 3, 4 (not all are distinguishable) in a are candidate trichromatic matches to point 1 in b, after cone responses were scaled to a mean of 1.0 (von Kries scaling) for the sample (see text). For the purposes of illustration, the images in a and b have not been normalized. The histogram in c is the naïve estimate of the probability mass function for identification errors based on a sample of 1000 points drawn randomly from the image. The smooth curve is an exponential model density function fitted to identification errors > 5.0 , with a dotted line indicating the fit over the remainder of the range.

Fig. 3
Fig. 3

Decomposition of the mean estimated information available into individual information components and redundancies at a, receptoral and b, postreceptoral levels. The postreceptoral transformation T [Eq. (4)] was optimized for minimum higher-order redundancy R 12 # (b). Means were taken over 50 scenes and for daylight illuminants with correlated color temperatures of 25,000 K and 4000 K (solid symbols) and 4000 K and 6500 K (open symbols). Horizontal bars show ± 1 sample SD.

Fig. 4
Fig. 4

Information retrieved. The mean estimate of the information retrieved receptorally I K ( N ) (circles) and postreceptorally I K # ( N ) (squares) is shown as a function of the logarithm of the size N of the sample for two daylight illuminants with correlated color temperatures of a, 25,000 K and 4000 K and b, 4000 K and 6500 K . The postreceptoral transformation was optimized for minimum higher-order redundancy R 12 # . The bounding value of the discrete entropy of a sample of size N is shown by the dashed line and the mean estimate of the information available I L M S by the horizontal line (different ordinates in a and b). The smooth curves are locally weighted linear regressions derived from a best-fitting template from which asymptotic values were estimated, shown arrowed. Means were taken over 50 scenes.

Fig. 5
Fig. 5

Information retrieved. Details as for Fig. 4, except that the postreceptoral transformation was optimized for maximum information retrieved I K # ( N ) . Data for information retrieved receptorally were identical to those in Fig. 4 and have been omitted.

Fig. 6
Fig. 6

Information estimates for Gaussian trichromatic images. The mean estimate of the information available I L M S from the offset Kozachenko–Leonenko estimator [22, 39] (squares) and the mean estimate of the information retrieved I K ( N ) by trichromatic matching with a sample of size N from the Grassberger estimator ([42], Eqs. (23) and (27)) (circles) are shown as a function of the logarithm of the size N of the sample. The smooth curves are loess fits [43]. The bounding value of the discrete entropy of a sample of size N is shown by the dashed line and the true value I true of the information available for Gaussian images by the horizontal line. Means were taken over 100 samples each of size N. Sample SDs at log 2 N = 18 were 0.008   bits for I L M S and 0.002   bits for I K ( N ) .

Equations (36)

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l ( x , y ) = l ¯ ( λ ) c ( λ ; x , y ) d λ ,
m ( x , y ) = m ¯ ( λ ) c ( λ ; x , y ) d λ ,
s ( x , y ) = s ¯ ( λ ) c ( λ ; x , y ) d λ ,
h ( L , M , S ) = f ( l , m , s ) log f ( l , m , s ) d l d m d s ,
I ( L 1 , M 1 , S 1 ; L 2 , M 2 , S 2 ) = h ( L 1 , M 1 , S 1 ) + h ( L 2 , M 2 , S 2 ) h ( L 1 , M 1 , S 1 , L 2 , M 2 , S 2 ) .
( L 1 # , M 1 # , S 1 # ) = T ( L 1 , M 1 , S 1 ) ,
( L 2 # , M 2 # , S 2 # ) = T ( L 2 , M 2 , S 2 ) .
I ( L 1 # , M 1 # , S 1 # ; L 2 # , M 2 # , S 2 # ) = I ( L 1 , M 1 , S 1 ; L 2 , M 2 , S 2 ) .
R 1 = h ( L 1 ) + h ( M 1 ) + h ( S 1 ) h ( L 1 , M 1 , S 1 ) ,
R 2 = h ( L 2 ) + h ( M 2 ) + h ( S 2 ) h ( L 2 , M 2 , S 2 ) .
R 12 = h ( L 1 , L 2 ) + h ( M 1 , M 2 ) + h ( S 1 , S 2 ) h ( L 1 , L 2 , M 1 , M 2 , S 1 , S 2 ) .
I L = h ( L 1 ) + h ( L 2 ) h ( L 1 , L 2 ) ,
I M = h ( M 1 ) + h ( M 2 ) h ( M 1 , M 2 ) ,
I S = h ( S 1 ) + h ( S 2 ) h ( S 1 , S 2 ) .
I L M S = I L + I M + I S R 1 R 2 + R 12 .
I L M S # = I L # + I M # + I S # R 1 # R 2 # + R 12 # .
H N ( K ) = p k log p k ,
I K ( N ) = log N H N ( K ) .
H N ( K # ) = p k # log p k # ,
I K # ( N ) = log N H N ( K # ) .
lim N I K ( N ) I L M S
lim N I K # ( N ) I L M S # .
h ( L , M , S ) = h ( L * , M * , S * ) + 1 2 log | Var ( L , M , S ) | ,
T = ( 1 0.931 ( 0.060 ) 0.066 ( 0.025 ) 0.259 ( 0.096 ) 1 0.156 ( 0.032 ) 0.003 ( 0.054 ) 0.035 ( 0.087 ) 1 ) .
T = ( 1 0.971 ( 0.048 ) 0.093 ( 0.028 ) 0.249 ( 0.056 ) 1 0.173 ( 0.034 ) 0.005 ( 0.041 ) 0.034 ( 0.070 ) 1 ) .
h ( L 1 , M 1 , S 1 ) = f ( l 1 , m 1 , s 1 ) log f ( l 1 , m 1 , s 1 ) d l 1 d m 1 d s 1 ,
h ( L 2 , M 2 , S 2 ) = f ( l 2 , m 2 , s 2 ) log f ( l 2 , m 2 , s 2 ) d l 2 d m 2 d s 2 ,
h ( L 1 , M 1 , S 1 , L 2 , M 2 , S 2 ) = f ( l 1 , m 1 , s 1 , l 2 , m 2 , s 2 ) × log f ( l 1 , m 1 , s 1 , l 2 , m 2 , s 2 ) d l 1 d m 1 d s 1 d l 2 d m 2 d s 2 ,
I ( L 1 , M 1 , S 1 ; L 2 M 2 , S 2 ) = f 12 ( l 1 , m 1 , s 1 , l 2 , m 2 , s 2 ) × log f 12 ( l 1 , m 1 , s 1 , l 2 , m 2 , s 2 ) f 1 ( l 1 , m 1 , s 1 ) f 2 ( l 2 , m 2 , s 2 ) d l 1 d m 1 d s 1 d l 2 d m 2 d s 2 .
I ( L 1 , M 1 , S 1 ; L 2 , M 2 , S 2 ) = h ( L 1 , M 1 , S 1 ) + h ( L 2 , M 2 , S 2 ) h ( L 1 , M 1 , S 1 , L 2 , M 2 , S 2 ) .
I ( L 1 # , M 1 # , S 1 # ; L 2 # , M 2 # , S 2 # ) = f 12 # ( l 1 # , m 1 # , s 1 # , l 2 # , m 2 # , s 2 # ) × log f 12 # ( l 1 # , m 1 # , s 1 # , l 2 # , m 2 # , s 2 # ) f 1 # ( l 1 # , m 1 # , s 1 # ) f 2 # ( l 2 # , m 2 # , s 2 # ) d l 1 # d m 1 # d s 1 # d l 2 # d m 2 # d s 2 # .
I ( L 1 # , M 1 # , S 1 # ; L 2 # , M 2 # , S 2 # ) = I ( L 1 , M 1 , S 1 ; L 2 , M 2 , S 2 ) .
I LMS I L I M I S = h ( L 1 , M 1 , S 1 ) + h ( L 2 , M 2 , S 2 ) h ( L 1 , M 1 , S 1 , L 2 , M 2 , S 2 ) h ( L 1 ) h ( L 2 ) + h ( L 1 , L 2 ) h ( M 1 ) h ( M 2 ) + h ( M 1 , M 2 ) h ( S 1 ) h ( S 2 ) + h ( S 1 , S 2 ) .
I LMS = I L + I M + I S [ h ( L 1 ) + h ( M 1 ) + h ( S 1 ) h ( L 1 , M 1 , S 1 ) ] [ h ( L 2 ) + h ( M 2 ) + h ( S 2 ) h ( L 2 , M 2 , S 2 ) ] + [ h ( L 1 , L 2 ) + h ( M 1 , M 2 ) + h ( S 1 , S 2 ) h ( L 1 , M 1 , S 1 , L 2 , M 2 , S 2 ) ] .
I LMS = I L + I M + I S R 1 R 2 + R 12 ,
I true = 3 2 log ( 1 ρ 2 ) .

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