Abstract

Wide generalizability of the CIE-recommended gray-scale calculation for self-luminous devices suggests that its parameters, the extrema of the calculation’s derivatives, and its limiting behavior are fundamental. The calculation has a negative-contrast point of inflection that is not predicted by any other gray-scale calculation, but that is consistent with data and with the terrestrial luminance histogram. The parameters of the calculation are analyzed, and their significance is explained. High-positive-contrast behavior of the calculation is shown and related to scientific literature. This knowledge represents a clearer understanding of daylight suprathreshold vision, and it enables optimal luminance-coding of contemporary high-resolution, high-contrast, high-luminance displays.

© 2019 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
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  39. R. C. Carter, “Gray-scale perceptions calculated: optimum display background luminance,” Appl. Opt. 36, 1705–1717 (1997).
    [Crossref]

2018 (1)

2017 (1)

M. E. Rudd, “Lightness computation by the human visual system,” J. Electron. Imaging 26, 031209 (2017).
[Crossref]

2016 (1)

T. Seim and A. Valberg, “A neurophysiologically-based analysis of lightness and brightness perception,” Col. Res. Appl. 41, 339–351 (2016).
[Crossref]

2015 (2)

A. B. Watson, “Computing human optical point spread functions,” J. Vis. 15(2):26 (2015).
[Crossref]

J. M. Bosten, R. D. Beer, and D. I. A. MacLeod, “What is white?” J. Vis. 15(16):5 (2015).
[Crossref]

2014 (1)

R. C. Carter and M. H. Brill, “Calculation of self-luminous neutral-scale: how many neutral steps can you see on that display?” J. Soc. Inf. Display. 22, 177–186 (2014).
[Crossref]

2013 (1)

2012 (1)

2011 (3)

M. Melgosa, P. A. Garcia, L. Gomez-Robledo, R. Shamey, D. Hinks, G. Cui, and M. R. Luo, “Notes on the application of the standardized residual sum of squares index for the assessment of intra- and inter-observer variability in color-difference experiments,” J. Opt. Soc. Am. A 28, 949–953 (2011).
[Crossref]

F. A. A. Kingdom, “Lightness, brightness and transparency: a quarter century of new ideas, captivating demonstrations and unrelenting controversy,” Vis. Res. 51, 652–673 (2011).
[Crossref]

A. Radonjic, A. Gilchrist, S. Allred, and D. Brainard, “The dynamic range of human lightness perception,” Curr. Biol. 21, 1931–1936 (2011).
[Crossref]

2010 (2)

R. C. Carter and R. Huertas, “Ultra-large color difference and small subtense,” Col. Res. Appl. 35, 4–17 (2010).
[Crossref]

D. Laming, “Fechner’s law: where does the log transform come from?” Seeing Perceiving 23, 155–171 (2010).

2005 (1)

R. C. Carter, “Biological gray scale for digital imagery,” J. Electron. Imaging 14, 023004 (2005).
[Crossref]

2004 (1)

M. E. Rudd and I. K. Zemach, “Quantitative properties of achromatic color induction: an edge integration analysis,” Vis. Res. 44, 971–981 (2004).
[Crossref]

1999 (1)

A. Gilchrist, C. Kossifydis, F. Bonato, T. Agostini, J. Cataliotti, X. Li, B. Spehar, V. Annan, and E. Economou, “An anchoring theory of lightness perception,” Psychol. Rev. 106, 795–834 (1999).
[Crossref]

1997 (1)

1996 (1)

F. A. A. Kingdom and P. Whittle, “Contrast discrimination at high contrasts reveals the influence of local light adaptation on contrast processing,” Vis. Res. 36, 817–829 (1996).
[Crossref]

1992 (2)

S. K. Shevell, I. Holliday, and P. Whittle, “Two separate neural mechanisms of brightness induction,” Vis. Res. 32, 2331–2340 (1992).
[Crossref]

P. Whittle, “Brightness, discriminability and the ‘crispening effect’,” Vis. Res. 32, 1493–1507 (1992).
[Crossref]

1989 (1)

E. G. Heinemann, “Brightness contrast, brightness constancy, and the ratio principle,” Percept. Psychophys. 45, 89–91 (1989).
[Crossref]

1986 (1)

P. Whittle, “Increments and decrements: luminance discrimination,” Vis. Res. 26, 1677–1691 (1986).
[Crossref]

1982 (1)

1978 (1)

1974 (1)

1961 (1)

E. G. Heinemann, “The relation of apparent brightness to the threshold for differences in luminance,” J. Exp. Psychol. 61, 389–399 (1961).
[Crossref]

1953 (1)

H. Leibowitz, F. A. Mote, and W. R. Thurlow, “Simultaneous contrast as a function of separation between test and inducing fields,” J. Exp. Psychol. 46, 453–456 (1953).
[Crossref]

1946 (1)

Agostini, T.

A. Gilchrist, C. Kossifydis, F. Bonato, T. Agostini, J. Cataliotti, X. Li, B. Spehar, V. Annan, and E. Economou, “An anchoring theory of lightness perception,” Psychol. Rev. 106, 795–834 (1999).
[Crossref]

Allred, S.

A. Radonjic, A. Gilchrist, S. Allred, and D. Brainard, “The dynamic range of human lightness perception,” Curr. Biol. 21, 1931–1936 (2011).
[Crossref]

Annan, V.

A. Gilchrist, C. Kossifydis, F. Bonato, T. Agostini, J. Cataliotti, X. Li, B. Spehar, V. Annan, and E. Economou, “An anchoring theory of lightness perception,” Psychol. Rev. 106, 795–834 (1999).
[Crossref]

Beer, R. D.

J. M. Bosten, R. D. Beer, and D. I. A. MacLeod, “What is white?” J. Vis. 15(16):5 (2015).
[Crossref]

Blackwell, H. R.

Bonato, F.

A. Gilchrist, C. Kossifydis, F. Bonato, T. Agostini, J. Cataliotti, X. Li, B. Spehar, V. Annan, and E. Economou, “An anchoring theory of lightness perception,” Psychol. Rev. 106, 795–834 (1999).
[Crossref]

Bosten, J. M.

J. M. Bosten, R. D. Beer, and D. I. A. MacLeod, “What is white?” J. Vis. 15(16):5 (2015).
[Crossref]

Brainard, D.

A. Radonjic, A. Gilchrist, S. Allred, and D. Brainard, “The dynamic range of human lightness perception,” Curr. Biol. 21, 1931–1936 (2011).
[Crossref]

Brill, M. H.

R. C. Carter and M. H. Brill, “Calculation of self-luminous neutral-scale: how many neutral steps can you see on that display?” J. Soc. Inf. Display. 22, 177–186 (2014).
[Crossref]

Carter, R. C.

R. C. Carter, “Suprathreshold gray scale is implied by thresholds,” Appl. Opt. 57, 8751–8756 (2018).
[Crossref]

R. C. Carter and M. H. Brill, “Calculation of self-luminous neutral-scale: how many neutral steps can you see on that display?” J. Soc. Inf. Display. 22, 177–186 (2014).
[Crossref]

R. C. Carter and L. D. Silverstein, “Perceiving color across scale: great and small, discrete and continuous,” J. Opt. Soc. Am. A 29, 1346–1355 (2012).
[Crossref]

R. C. Carter and R. Huertas, “Ultra-large color difference and small subtense,” Col. Res. Appl. 35, 4–17 (2010).
[Crossref]

R. C. Carter, “Biological gray scale for digital imagery,” J. Electron. Imaging 14, 023004 (2005).
[Crossref]

R. C. Carter, “Gray-scale perceptions calculated: optimum display background luminance,” Appl. Opt. 36, 1705–1717 (1997).
[Crossref]

G. Yoon and R. C. Carter, “How gray-scale appearance is affected by intraocular optics” (manuscript in preparation).

Cataliotti, J.

A. Gilchrist, C. Kossifydis, F. Bonato, T. Agostini, J. Cataliotti, X. Li, B. Spehar, V. Annan, and E. Economou, “An anchoring theory of lightness perception,” Psychol. Rev. 106, 795–834 (1999).
[Crossref]

Cornsweet, T. N.

T. N. Cornsweet, Visual Perception (Academic, 1970).

Cui, G.

Economou, E.

A. Gilchrist, C. Kossifydis, F. Bonato, T. Agostini, J. Cataliotti, X. Li, B. Spehar, V. Annan, and E. Economou, “An anchoring theory of lightness perception,” Psychol. Rev. 106, 795–834 (1999).
[Crossref]

Garcia, P. A.

Gilchrist, A.

A. Radonjic, A. Gilchrist, S. Allred, and D. Brainard, “The dynamic range of human lightness perception,” Curr. Biol. 21, 1931–1936 (2011).
[Crossref]

A. Gilchrist, C. Kossifydis, F. Bonato, T. Agostini, J. Cataliotti, X. Li, B. Spehar, V. Annan, and E. Economou, “An anchoring theory of lightness perception,” Psychol. Rev. 106, 795–834 (1999).
[Crossref]

A. Gilchrist, “Perceptual organization in lightness,” in Handbook of Perceptual Organization, J. Wagemans, ed. (Oxford University, 2014), pp. 391–412.

Gomez-Robledo, L.

Heinemann, E. G.

E. G. Heinemann, “Brightness contrast, brightness constancy, and the ratio principle,” Percept. Psychophys. 45, 89–91 (1989).
[Crossref]

E. G. Heinemann, “The relation of apparent brightness to the threshold for differences in luminance,” J. Exp. Psychol. 61, 389–399 (1961).
[Crossref]

E. G. Heinemann, “Simultaneous brightness induction,” in Visual Psychophysics, D. Jameson and L. M. Hurvich, eds. (Springer-Verlag, 1972), pp. 146–169.

Hinks, D.

Holliday, I.

S. K. Shevell, I. Holliday, and P. Whittle, “Two separate neural mechanisms of brightness induction,” Vis. Res. 32, 2331–2340 (1992).
[Crossref]

Huertas, R.

R. C. Carter and R. Huertas, “Ultra-large color difference and small subtense,” Col. Res. Appl. 35, 4–17 (2010).
[Crossref]

Kingdom, F. A. A.

F. A. A. Kingdom, “Lightness, brightness and transparency: a quarter century of new ideas, captivating demonstrations and unrelenting controversy,” Vis. Res. 51, 652–673 (2011).
[Crossref]

F. A. A. Kingdom and P. Whittle, “Contrast discrimination at high contrasts reveals the influence of local light adaptation on contrast processing,” Vis. Res. 36, 817–829 (1996).
[Crossref]

Kossifydis, C.

A. Gilchrist, C. Kossifydis, F. Bonato, T. Agostini, J. Cataliotti, X. Li, B. Spehar, V. Annan, and E. Economou, “An anchoring theory of lightness perception,” Psychol. Rev. 106, 795–834 (1999).
[Crossref]

Laming, D.

D. Laming, “Visual adaptation—a reinterpretation: discussion,” J. Opt. Soc. Am. A 30, 2066–2078 (2013).
[Crossref]

D. Laming, “Fechner’s law: where does the log transform come from?” Seeing Perceiving 23, 155–171 (2010).

D. Laming, Sensory Analysis (Academic, 1986).

Leibowitz, H.

H. Leibowitz, F. A. Mote, and W. R. Thurlow, “Simultaneous contrast as a function of separation between test and inducing fields,” J. Exp. Psychol. 46, 453–456 (1953).
[Crossref]

Li, X.

A. Gilchrist, C. Kossifydis, F. Bonato, T. Agostini, J. Cataliotti, X. Li, B. Spehar, V. Annan, and E. Economou, “An anchoring theory of lightness perception,” Psychol. Rev. 106, 795–834 (1999).
[Crossref]

Lotto, R. B.

D. Purves and R. B. Lotto, Why We See What We Do: An Empirical Theory of Vision (Sinauer Associates, 2003).

Luo, M. R.

MacAdam, D. L.

MacLeod, D. I. A.

J. M. Bosten, R. D. Beer, and D. I. A. MacLeod, “What is white?” J. Vis. 15(16):5 (2015).
[Crossref]

McCann, J.

J. McCann and A. Rizzi, The Art and Science of HDR Imaging (Wiley, 2012).

Melgosa, M.

Mote, F. A.

H. Leibowitz, F. A. Mote, and W. R. Thurlow, “Simultaneous contrast as a function of separation between test and inducing fields,” J. Exp. Psychol. 46, 453–456 (1953).
[Crossref]

Purves, D.

D. Purves and R. B. Lotto, Why We See What We Do: An Empirical Theory of Vision (Sinauer Associates, 2003).

Radonjic, A.

A. Radonjic, A. Gilchrist, S. Allred, and D. Brainard, “The dynamic range of human lightness perception,” Curr. Biol. 21, 1931–1936 (2011).
[Crossref]

Richards, W. A.

Rizzi, A.

J. McCann and A. Rizzi, The Art and Science of HDR Imaging (Wiley, 2012).

Rudd, M. E.

M. E. Rudd, “Lightness computation by the human visual system,” J. Electron. Imaging 26, 031209 (2017).
[Crossref]

M. E. Rudd and I. K. Zemach, “Quantitative properties of achromatic color induction: an edge integration analysis,” Vis. Res. 44, 971–981 (2004).
[Crossref]

Seim, T.

T. Seim and A. Valberg, “A neurophysiologically-based analysis of lightness and brightness perception,” Col. Res. Appl. 41, 339–351 (2016).
[Crossref]

Shamey, R.

Shevell, S. K.

S. K. Shevell, I. Holliday, and P. Whittle, “Two separate neural mechanisms of brightness induction,” Vis. Res. 32, 2331–2340 (1992).
[Crossref]

Silverstein, L. D.

Spehar, B.

A. Gilchrist, C. Kossifydis, F. Bonato, T. Agostini, J. Cataliotti, X. Li, B. Spehar, V. Annan, and E. Economou, “An anchoring theory of lightness perception,” Psychol. Rev. 106, 795–834 (1999).
[Crossref]

Thurlow, W. R.

H. Leibowitz, F. A. Mote, and W. R. Thurlow, “Simultaneous contrast as a function of separation between test and inducing fields,” J. Exp. Psychol. 46, 453–456 (1953).
[Crossref]

Valberg, A.

T. Seim and A. Valberg, “A neurophysiologically-based analysis of lightness and brightness perception,” Col. Res. Appl. 41, 339–351 (2016).
[Crossref]

Watson, A. B.

A. B. Watson, “Computing human optical point spread functions,” J. Vis. 15(2):26 (2015).
[Crossref]

Whittle, P.

F. A. A. Kingdom and P. Whittle, “Contrast discrimination at high contrasts reveals the influence of local light adaptation on contrast processing,” Vis. Res. 36, 817–829 (1996).
[Crossref]

S. K. Shevell, I. Holliday, and P. Whittle, “Two separate neural mechanisms of brightness induction,” Vis. Res. 32, 2331–2340 (1992).
[Crossref]

P. Whittle, “Brightness, discriminability and the ‘crispening effect’,” Vis. Res. 32, 1493–1507 (1992).
[Crossref]

P. Whittle, “Increments and decrements: luminance discrimination,” Vis. Res. 26, 1677–1691 (1986).
[Crossref]

P. Whittle, “The psychophysics of contrast brightness,” in Lightness, Brightness, and Transparency, A. Gilchrist, ed. (Erlbaum, 1994), pp. 35–110.

P. Whittle, “Contrast brightness and ordinary seeing,” in Lightness, Brightness, and Transparency, A. Gilchrist, ed. (Erlbaum, 1994), pp. 111–157.

Yoon, G.

G. Yoon and R. C. Carter, “How gray-scale appearance is affected by intraocular optics” (manuscript in preparation).

Zemach, I. K.

M. E. Rudd and I. K. Zemach, “Quantitative properties of achromatic color induction: an edge integration analysis,” Vis. Res. 44, 971–981 (2004).
[Crossref]

Appl. Opt. (3)

Col. Res. Appl. (2)

R. C. Carter and R. Huertas, “Ultra-large color difference and small subtense,” Col. Res. Appl. 35, 4–17 (2010).
[Crossref]

T. Seim and A. Valberg, “A neurophysiologically-based analysis of lightness and brightness perception,” Col. Res. Appl. 41, 339–351 (2016).
[Crossref]

Curr. Biol. (1)

A. Radonjic, A. Gilchrist, S. Allred, and D. Brainard, “The dynamic range of human lightness perception,” Curr. Biol. 21, 1931–1936 (2011).
[Crossref]

J. Electron. Imaging (2)

R. C. Carter, “Biological gray scale for digital imagery,” J. Electron. Imaging 14, 023004 (2005).
[Crossref]

M. E. Rudd, “Lightness computation by the human visual system,” J. Electron. Imaging 26, 031209 (2017).
[Crossref]

J. Exp. Psychol. (2)

E. G. Heinemann, “The relation of apparent brightness to the threshold for differences in luminance,” J. Exp. Psychol. 61, 389–399 (1961).
[Crossref]

H. Leibowitz, F. A. Mote, and W. R. Thurlow, “Simultaneous contrast as a function of separation between test and inducing fields,” J. Exp. Psychol. 46, 453–456 (1953).
[Crossref]

J. Opt. Soc. Am. (3)

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

J. Soc. Inf. Display. (1)

R. C. Carter and M. H. Brill, “Calculation of self-luminous neutral-scale: how many neutral steps can you see on that display?” J. Soc. Inf. Display. 22, 177–186 (2014).
[Crossref]

J. Vis. (2)

A. B. Watson, “Computing human optical point spread functions,” J. Vis. 15(2):26 (2015).
[Crossref]

J. M. Bosten, R. D. Beer, and D. I. A. MacLeod, “What is white?” J. Vis. 15(16):5 (2015).
[Crossref]

Percept. Psychophys. (1)

E. G. Heinemann, “Brightness contrast, brightness constancy, and the ratio principle,” Percept. Psychophys. 45, 89–91 (1989).
[Crossref]

Psychol. Rev. (1)

A. Gilchrist, C. Kossifydis, F. Bonato, T. Agostini, J. Cataliotti, X. Li, B. Spehar, V. Annan, and E. Economou, “An anchoring theory of lightness perception,” Psychol. Rev. 106, 795–834 (1999).
[Crossref]

Seeing Perceiving (1)

D. Laming, “Fechner’s law: where does the log transform come from?” Seeing Perceiving 23, 155–171 (2010).

Vis. Res. (6)

P. Whittle, “Increments and decrements: luminance discrimination,” Vis. Res. 26, 1677–1691 (1986).
[Crossref]

F. A. A. Kingdom and P. Whittle, “Contrast discrimination at high contrasts reveals the influence of local light adaptation on contrast processing,” Vis. Res. 36, 817–829 (1996).
[Crossref]

M. E. Rudd and I. K. Zemach, “Quantitative properties of achromatic color induction: an edge integration analysis,” Vis. Res. 44, 971–981 (2004).
[Crossref]

S. K. Shevell, I. Holliday, and P. Whittle, “Two separate neural mechanisms of brightness induction,” Vis. Res. 32, 2331–2340 (1992).
[Crossref]

P. Whittle, “Brightness, discriminability and the ‘crispening effect’,” Vis. Res. 32, 1493–1507 (1992).
[Crossref]

F. A. A. Kingdom, “Lightness, brightness and transparency: a quarter century of new ideas, captivating demonstrations and unrelenting controversy,” Vis. Res. 51, 652–673 (2011).
[Crossref]

Other (11)

CIE, Grey-Scale Calculation for Self-Luminous Devices (2018).

D. Purves and R. B. Lotto, Why We See What We Do: An Empirical Theory of Vision (Sinauer Associates, 2003).

G. Yoon and R. C. Carter, “How gray-scale appearance is affected by intraocular optics” (manuscript in preparation).

A. Gilchrist, “Perceptual organization in lightness,” in Handbook of Perceptual Organization, J. Wagemans, ed. (Oxford University, 2014), pp. 391–412.

D. Laming, Sensory Analysis (Academic, 1986).

E. G. Heinemann, “Simultaneous brightness induction,” in Visual Psychophysics, D. Jameson and L. M. Hurvich, eds. (Springer-Verlag, 1972), pp. 146–169.

D. L. MacAdam, “Color-order systems,” in Color Measurement (Springer-Verlag, 1985), pp. 165–177.

P. Whittle, “The psychophysics of contrast brightness,” in Lightness, Brightness, and Transparency, A. Gilchrist, ed. (Erlbaum, 1994), pp. 35–110.

P. Whittle, “Contrast brightness and ordinary seeing,” in Lightness, Brightness, and Transparency, A. Gilchrist, ed. (Erlbaum, 1994), pp. 111–157.

T. N. Cornsweet, Visual Perception (Academic, 1970).

J. McCann and A. Rizzi, The Art and Science of HDR Imaging (Wiley, 2012).

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Figures (7)

Fig. 1.
Fig. 1. Plot of nEPD, the number of equal perceptible differences between ${L_b}$ and $L$, drawn as a function of $L$, with ${L_b}$ as a parameter. In this plot, $k=0.055$, corresponding to a 2 deg disk contrast; other values of $k$ alter the trajectory of nEPD (e.g., see Fig. 4).
Fig. 2.
Fig. 2. nEPD plotted versus contrast for bipartite-field data used by Seim and Valberg [17] to derive a Naka–Rushton equation. Thanks to T. Seim (personal communication, 2015) for his analysis and plotting, and for recent permission to publish this figure.
Fig. 3.
Fig. 3. Numerical simulation of the experiment of Leibowitz et al. [19]. Retinal background luminances (in various stimulus configurations) of a visual target (${\textbf T}$) were calculated two ways: (1) convolution of the intraocular PSF with the stimulus image and (2) ${L_b}$ from the Whittle formula, nEPD. The similarity of the two independent calculations (${L_b}$ from nEPD computed by the author versus ${L_b}$ from convolution computed by Professor Geunyoung Yoon at Flaum Eye Institute, University of Rochester) suggests that the effects of the stimulus configuration on the gray scale can be elegantly described as due to intraocular scattering of light within the image. The parameters of the graph are the angular separation between ${\textbf T}$ and an inducing field (${\textbf I}$) and the luminance of ${\textbf I}$. ${\textbf I}$ and ${\textbf T}$ are squares with sides 30 min (30’) of arc.
Fig. 4.
Fig. 4. Negative-contrast domain of nEPD, corresponding to $L \lt {L_b}$ and ${\rm nEPD} \lt {0}$ in Fig. 1. Background luminance, ${L_b}$, is nominally ${100}\;{{\rm cd/m}^2}$ in this figure. See the text for discussion of $k$; contrasts with larger visual subtense (smaller ${k}$) have a steeper and more accelerated nEPD.
Fig. 5.
Fig. 5. Plot of Eq. (7), the derivative with respect to $L$ of negative-contrast nEPD, ${\rm dnEPD}/{\rm d}L$, parameterized by $k$ (= 0, 0.055, 0.1, 0.25, 0.5 in order from top to bottom of the figure); $k$ is the proportion of external contrast (${L_b} - L$) lost to intraocular scattering on the way to becoming retinal contrast.
Fig. 6.
Fig. 6. Graph of Eq. (9) showing how nEPD changes with small changes of $b$, for positive contrasts in the CIE [1] self-luminous gray-scale calculation. The parameter $b$ most affects changes of positive-contrast nEPD for suprathreshold values of $W$ and for smaller values of $b$, associated with low background luminance. The parameter $k$ (= 0.055 in this figure) has little effect on the plot.
Fig. 7.
Fig. 7. Plot of nEPD calculated for positive contrasts versus logarithmic visual-contrast luminance (${{\log}_{10}}\, L$). The background luminance, ${L_b}$, for each curve is the parameter indicated in the legend. In this figure, $k = {0.055}$ as appropriate for a 2 deg disk contrast; an arbitrary choice, but it illustrates that the Line of Hipparchus is not just applicable to star (i.e., small-subtense) brightness.

Equations (9)

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n E P D = a l o g 10 [ 1 + b ( 1 k ) W ] ,
W = Δ L / [ 0.39 + m i n i m u m ( L , L b ) ] ,
b = 0.26 + 0.3095 L b .
n E P D = l o g [ b ( L b L ) / L ] ,
d n E P D / d L = ( L b / L 2 ) / [ ( L b / L ) 1 ] = 1 / [ L ( 1 L / L b ) ] ,
d / d L ( d n E P D / d L ) = ( 1 2 L / L b ) / ( L L 2 / L b ) 2 .
d n E P D / d L = 7.07 l o g 10 ( e ) { ( V / U ) [ ( V d U U d V ) / V 2 ] } = 3.0705 [ d U / U d V / V ] = 3.0705 { [ 1 ( k + b ( 1 k ) ] / { L + 0.39 + ( L b L ) × [ k + b ( 1 k ) ] } ( 1 k ) / [ L + 0.39 + k ( L b L ) ] } .
d n E P D / d W = a log 10 ( e ) b ( 1 k ) / [ 1 + b ( 1 k ) W ] = a l o g 10 ( e ) / { W + [ 1 / b ( 1 k ) ] } .
d n E P D / d b = a log 10 ( e ) / { b + 1 / [ ( 1 k ) W ] } .

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