Abstract

We modified a two-stage model for color discrimination proposed in a previous study [Color Res. Appl. 25, 105 (2000)]; in order to extend the model to wider conditions, we considered the conditions with luminance modulations in addition to color modulations. Using the modified model, we successfully predicted color discrimination data with test color changes along both the chromatic and luminance axes under a variety of background colors. Both qualitative and quantitative assessments in modeling showed that nonlinearity is required in both the cone and the cone-opponent stages to interpret adaptation effects of both color and luminance on color discrimination. This fact suggests that the nonlinear properties at each stage have different roles in color perception.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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2007

S. Tsujimura, S. Shioiri, and A. Nuruki, “Two distinct cone-opponent processes in the L+M luminance pathway,” Vis. Res. 47, 1839–1854 (2007).
[CrossRef] [PubMed]

2006

B. R. Conway and M. S. Livingstone, “Spatial and temporal properties of cone signals in alert macaque primary visual cortex,” J. Neurosci. 26, 10826–10846 (2006).
[CrossRef] [PubMed]

2003

K. Kawamoto, T. Inamura, H. Yaguchi, and S. Shioiri, “Color discrimination characteristics depending on the background color in the (L, M) plane of a cone space,” Opt. Rev. 10, 391–397(2003).
[CrossRef]

2000

C. F. Stromeyer, III, P. D. Gowdy, A. Chaparro, S. Kladakis, J. D. Willen, and R. E. Kronauer, “Colour adaptation modifies the temporal properties of the long- and middle-wave cone signals in the human luminance mechanism,” J. Physiol. 526, 177–194(2000).
[CrossRef] [PubMed]

S. Shioiri and P. Cavanagh, “Nonlinearity in color space measured by apparent motion,” Percept. Psychophys. 62, 1182–1190(2000).
[CrossRef] [PubMed]

S. Tsujimura, S. Shioiri, Y. Hirai, and H. Yaguchi, “Technique to investigate the temporal phase shift between L- and M-cone inputs to the luminance mechanism,” J. Opt. Soc. Am. A 17, 846–857 (2000).
[CrossRef]

V. C. Smith, J. Pokorny, and V. C. Sun, “Chromatic contrast discrimination: data and prediction for stimuli varying in L and M cone excitation,” Color Res. Appl. 25, 105–115(2000).
[CrossRef]

1999

S. Tsujimura, S. Shioiri, Y. Hirai, and H. Yaguchi, “Selective cone suppression by the L−M- and M−L-cone-opponent mechanisms in the luminance pathway,” J. Opt. Soc. Am. A 16, 1217–1228 (1999).
[CrossRef]

C. F. Stromeyer, III, R. Thabet, A. Chaparro, and R. E. Kronauer, “Spatial masking does not reveal mechanisms selective to combined luminance and red–green color,” Vis. Res. 39, 2099–2112 (1999).
[CrossRef] [PubMed]

1998

1997

1995

A. Chaparro, C. F. Stromeyer, III, G. Chen, and R. E. Kronauer, “Human cones appear to adapt at low light levels: measurements on the red–green detection mechanism,” Vis. Res. 35, 3103–3118 (1995).
[CrossRef] [PubMed]

1994

G. R. Cole, T. J. Hine, and W. McIlhagga, “Estimation of linear detection mechanisms for stimuli of medium spatial frequency,” Vis. Res. 34, 1267–1278 (1994).
[CrossRef] [PubMed]

R. T. Eskew, Jr., C. F. Stromeyer, III, and R. E. Kronauer, “Temporal properties of the red–green chromatic mechanism,” Vis. Res. 34, 3127–3137 (1994).
[CrossRef] [PubMed]

1993

1992

1991

J. Walraven and J. S. Werner, “The invariance of unique white; a possible implication for normalizing cone action spectra,” Vis. Res. 31, 2185–2193 (1991).
[CrossRef] [PubMed]

S. L. Guth, “Model for color vision and light adaptation,” J. Opt. Soc. Am. A 8, 976–993 (1991).
[CrossRef] [PubMed]

1985

C. F. Stromeyer, III, G. R. Cole, and R. E. Kronauer, “Second-site adaptation in the red–green chromatic pathways,” Vis. Res. 25, 219–237 (1985).
[CrossRef] [PubMed]

1983

R. M. Boynton, A. L. Nagy, and C. X. Olson, “A flaw in equations for predicting chromatic differences,” Color Res. Appl. 8, 69–74(1983).
[CrossRef]

J. E. Thornton and E. N. Pugh, Jr., “Red/green color opponency at detection threshold,” Science 219, 191–193(1983).
[CrossRef] [PubMed]

1981

1968

1942

Boynton, R. M.

R. M. Boynton, A. L. Nagy, and C. X. Olson, “A flaw in equations for predicting chromatic differences,” Color Res. Appl. 8, 69–74(1983).
[CrossRef]

Cavanagh, P.

S. Shioiri and P. Cavanagh, “Nonlinearity in color space measured by apparent motion,” Percept. Psychophys. 62, 1182–1190(2000).
[CrossRef] [PubMed]

Chaparro, A.

C. F. Stromeyer, III, P. D. Gowdy, A. Chaparro, S. Kladakis, J. D. Willen, and R. E. Kronauer, “Colour adaptation modifies the temporal properties of the long- and middle-wave cone signals in the human luminance mechanism,” J. Physiol. 526, 177–194(2000).
[CrossRef] [PubMed]

C. F. Stromeyer, III, R. Thabet, A. Chaparro, and R. E. Kronauer, “Spatial masking does not reveal mechanisms selective to combined luminance and red–green color,” Vis. Res. 39, 2099–2112 (1999).
[CrossRef] [PubMed]

A. Chaparro, C. F. Stromeyer, III, G. Chen, and R. E. Kronauer, “Human cones appear to adapt at low light levels: measurements on the red–green detection mechanism,” Vis. Res. 35, 3103–3118 (1995).
[CrossRef] [PubMed]

Chen, G.

A. Chaparro, C. F. Stromeyer, III, G. Chen, and R. E. Kronauer, “Human cones appear to adapt at low light levels: measurements on the red–green detection mechanism,” Vis. Res. 35, 3103–3118 (1995).
[CrossRef] [PubMed]

Cole, G. R.

G. R. Cole, T. J. Hine, and W. McIlhagga, “Estimation of linear detection mechanisms for stimuli of medium spatial frequency,” Vis. Res. 34, 1267–1278 (1994).
[CrossRef] [PubMed]

G. R. Cole, T. H. Hine, and W. McIlhagga, “Detection mechanism in L-, M- and S-cone contrast space,” J. Opt. Soc. Am. A 10, 38–51(1993).
[CrossRef] [PubMed]

C. F. Stromeyer, III, G. R. Cole, and R. E. Kronauer, “Second-site adaptation in the red–green chromatic pathways,” Vis. Res. 25, 219–237 (1985).
[CrossRef] [PubMed]

Conway, B. R.

B. R. Conway and M. S. Livingstone, “Spatial and temporal properties of cone signals in alert macaque primary visual cortex,” J. Neurosci. 26, 10826–10846 (2006).
[CrossRef] [PubMed]

Eisner, A.

Eskew, R. T.

R. T. Eskew, Jr., C. F. Stromeyer, III, and R. E. Kronauer, “Temporal properties of the red–green chromatic mechanism,” Vis. Res. 34, 3127–3137 (1994).
[CrossRef] [PubMed]

R. T. Eskew, Jr., J. S. McLellan, and F. Giulianini, “Chromatic detection and discrimination,” in Color Vision: From Genes to Perception, K.Gegenfurtner and L.T.Sharpe, eds. (Cambridge University, 1999), pp. 345–368.

Gegenfurtner, K. R.

Giulianini, F.

R. T. Eskew, Jr., J. S. McLellan, and F. Giulianini, “Chromatic detection and discrimination,” in Color Vision: From Genes to Perception, K.Gegenfurtner and L.T.Sharpe, eds. (Cambridge University, 1999), pp. 345–368.

Gowdy, P. D.

C. F. Stromeyer, III, P. D. Gowdy, A. Chaparro, S. Kladakis, J. D. Willen, and R. E. Kronauer, “Colour adaptation modifies the temporal properties of the long- and middle-wave cone signals in the human luminance mechanism,” J. Physiol. 526, 177–194(2000).
[CrossRef] [PubMed]

Guth, S. L.

Hine, T. H.

Hine, T. J.

G. R. Cole, T. J. Hine, and W. McIlhagga, “Estimation of linear detection mechanisms for stimuli of medium spatial frequency,” Vis. Res. 34, 1267–1278 (1994).
[CrossRef] [PubMed]

Hirai, Y.

Ikeda, M.

Inamura, T.

K. Kawamoto, T. Inamura, H. Yaguchi, and S. Shioiri, “Color discrimination characteristics depending on the background color in the (L, M) plane of a cone space,” Opt. Rev. 10, 391–397(2003).
[CrossRef]

Kawamoto, K.

K. Kawamoto, T. Inamura, H. Yaguchi, and S. Shioiri, “Color discrimination characteristics depending on the background color in the (L, M) plane of a cone space,” Opt. Rev. 10, 391–397(2003).
[CrossRef]

Kiper, D. C.

Kladakis, S.

C. F. Stromeyer, III, P. D. Gowdy, A. Chaparro, S. Kladakis, J. D. Willen, and R. E. Kronauer, “Colour adaptation modifies the temporal properties of the long- and middle-wave cone signals in the human luminance mechanism,” J. Physiol. 526, 177–194(2000).
[CrossRef] [PubMed]

Kronauer, R. E.

C. F. Stromeyer, III, P. D. Gowdy, A. Chaparro, S. Kladakis, J. D. Willen, and R. E. Kronauer, “Colour adaptation modifies the temporal properties of the long- and middle-wave cone signals in the human luminance mechanism,” J. Physiol. 526, 177–194(2000).
[CrossRef] [PubMed]

C. F. Stromeyer, III, R. Thabet, A. Chaparro, and R. E. Kronauer, “Spatial masking does not reveal mechanisms selective to combined luminance and red–green color,” Vis. Res. 39, 2099–2112 (1999).
[CrossRef] [PubMed]

A. Chaparro, C. F. Stromeyer, III, G. Chen, and R. E. Kronauer, “Human cones appear to adapt at low light levels: measurements on the red–green detection mechanism,” Vis. Res. 35, 3103–3118 (1995).
[CrossRef] [PubMed]

R. T. Eskew, Jr., C. F. Stromeyer, III, and R. E. Kronauer, “Temporal properties of the red–green chromatic mechanism,” Vis. Res. 34, 3127–3137 (1994).
[CrossRef] [PubMed]

C. F. Stromeyer, III, G. R. Cole, and R. E. Kronauer, “Second-site adaptation in the red–green chromatic pathways,” Vis. Res. 25, 219–237 (1985).
[CrossRef] [PubMed]

Kuriki, I.

Livingstone, M. S.

B. R. Conway and M. S. Livingstone, “Spatial and temporal properties of cone signals in alert macaque primary visual cortex,” J. Neurosci. 26, 10826–10846 (2006).
[CrossRef] [PubMed]

MacAdam, D. L.

Macleod, D. I.

McIlhagga, W.

G. R. Cole, T. J. Hine, and W. McIlhagga, “Estimation of linear detection mechanisms for stimuli of medium spatial frequency,” Vis. Res. 34, 1267–1278 (1994).
[CrossRef] [PubMed]

G. R. Cole, T. H. Hine, and W. McIlhagga, “Detection mechanism in L-, M- and S-cone contrast space,” J. Opt. Soc. Am. A 10, 38–51(1993).
[CrossRef] [PubMed]

McLellan, J. S.

R. T. Eskew, Jr., J. S. McLellan, and F. Giulianini, “Chromatic detection and discrimination,” in Color Vision: From Genes to Perception, K.Gegenfurtner and L.T.Sharpe, eds. (Cambridge University, 1999), pp. 345–368.

Mullen, K. T.

Nagy, A. L.

R. M. Boynton, A. L. Nagy, and C. X. Olson, “A flaw in equations for predicting chromatic differences,” Color Res. Appl. 8, 69–74(1983).
[CrossRef]

Nuruki, A.

S. Tsujimura, S. Shioiri, and A. Nuruki, “Two distinct cone-opponent processes in the L+M luminance pathway,” Vis. Res. 47, 1839–1854 (2007).
[CrossRef] [PubMed]

Olson, C. X.

R. M. Boynton, A. L. Nagy, and C. X. Olson, “A flaw in equations for predicting chromatic differences,” Color Res. Appl. 8, 69–74(1983).
[CrossRef]

Pokorny, J.

V. C. Smith, J. Pokorny, and V. C. Sun, “Chromatic contrast discrimination: data and prediction for stimuli varying in L and M cone excitation,” Color Res. Appl. 25, 105–115(2000).
[CrossRef]

Pugh, E. N.

J. E. Thornton and E. N. Pugh, Jr., “Red/green color opponency at detection threshold,” Science 219, 191–193(1983).
[CrossRef] [PubMed]

Sankeralli, M. J.

Shioiri, S.

S. Tsujimura, S. Shioiri, and A. Nuruki, “Two distinct cone-opponent processes in the L+M luminance pathway,” Vis. Res. 47, 1839–1854 (2007).
[CrossRef] [PubMed]

K. Kawamoto, T. Inamura, H. Yaguchi, and S. Shioiri, “Color discrimination characteristics depending on the background color in the (L, M) plane of a cone space,” Opt. Rev. 10, 391–397(2003).
[CrossRef]

S. Tsujimura, S. Shioiri, Y. Hirai, and H. Yaguchi, “Technique to investigate the temporal phase shift between L- and M-cone inputs to the luminance mechanism,” J. Opt. Soc. Am. A 17, 846–857 (2000).
[CrossRef]

S. Shioiri and P. Cavanagh, “Nonlinearity in color space measured by apparent motion,” Percept. Psychophys. 62, 1182–1190(2000).
[CrossRef] [PubMed]

S. Tsujimura, S. Shioiri, Y. Hirai, and H. Yaguchi, “Selective cone suppression by the L−M- and M−L-cone-opponent mechanisms in the luminance pathway,” J. Opt. Soc. Am. A 16, 1217–1228 (1999).
[CrossRef]

Smith, V. C.

V. C. Smith, J. Pokorny, and V. C. Sun, “Chromatic contrast discrimination: data and prediction for stimuli varying in L and M cone excitation,” Color Res. Appl. 25, 105–115(2000).
[CrossRef]

Stiles, W. S.

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

Stromeyer, C. F.

C. F. Stromeyer, III, P. D. Gowdy, A. Chaparro, S. Kladakis, J. D. Willen, and R. E. Kronauer, “Colour adaptation modifies the temporal properties of the long- and middle-wave cone signals in the human luminance mechanism,” J. Physiol. 526, 177–194(2000).
[CrossRef] [PubMed]

C. F. Stromeyer, III, R. Thabet, A. Chaparro, and R. E. Kronauer, “Spatial masking does not reveal mechanisms selective to combined luminance and red–green color,” Vis. Res. 39, 2099–2112 (1999).
[CrossRef] [PubMed]

A. Chaparro, C. F. Stromeyer, III, G. Chen, and R. E. Kronauer, “Human cones appear to adapt at low light levels: measurements on the red–green detection mechanism,” Vis. Res. 35, 3103–3118 (1995).
[CrossRef] [PubMed]

R. T. Eskew, Jr., C. F. Stromeyer, III, and R. E. Kronauer, “Temporal properties of the red–green chromatic mechanism,” Vis. Res. 34, 3127–3137 (1994).
[CrossRef] [PubMed]

C. F. Stromeyer, III, G. R. Cole, and R. E. Kronauer, “Second-site adaptation in the red–green chromatic pathways,” Vis. Res. 25, 219–237 (1985).
[CrossRef] [PubMed]

Sun, V. C.

V. C. Smith, J. Pokorny, and V. C. Sun, “Chromatic contrast discrimination: data and prediction for stimuli varying in L and M cone excitation,” Color Res. Appl. 25, 105–115(2000).
[CrossRef]

Thabet, R.

C. F. Stromeyer, III, R. Thabet, A. Chaparro, and R. E. Kronauer, “Spatial masking does not reveal mechanisms selective to combined luminance and red–green color,” Vis. Res. 39, 2099–2112 (1999).
[CrossRef] [PubMed]

Thornton, J. E.

J. E. Thornton and E. N. Pugh, Jr., “Red/green color opponency at detection threshold,” Science 219, 191–193(1983).
[CrossRef] [PubMed]

Tsujimura, S.

Uchikawa, K.

Urakubo, M.

Walraven, J.

J. Walraven and J. S. Werner, “The invariance of unique white; a possible implication for normalizing cone action spectra,” Vis. Res. 31, 2185–2193 (1991).
[CrossRef] [PubMed]

Werner, J. S.

J. Walraven and J. S. Werner, “The invariance of unique white; a possible implication for normalizing cone action spectra,” Vis. Res. 31, 2185–2193 (1991).
[CrossRef] [PubMed]

Willen, J. D.

C. F. Stromeyer, III, P. D. Gowdy, A. Chaparro, S. Kladakis, J. D. Willen, and R. E. Kronauer, “Colour adaptation modifies the temporal properties of the long- and middle-wave cone signals in the human luminance mechanism,” J. Physiol. 526, 177–194(2000).
[CrossRef] [PubMed]

Wyszecki, G.

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

Yaguchi, H.

Color Res. Appl.

R. M. Boynton, A. L. Nagy, and C. X. Olson, “A flaw in equations for predicting chromatic differences,” Color Res. Appl. 8, 69–74(1983).
[CrossRef]

V. C. Smith, J. Pokorny, and V. C. Sun, “Chromatic contrast discrimination: data and prediction for stimuli varying in L and M cone excitation,” Color Res. Appl. 25, 105–115(2000).
[CrossRef]

J. Neurosci.

B. R. Conway and M. S. Livingstone, “Spatial and temporal properties of cone signals in alert macaque primary visual cortex,” J. Neurosci. 26, 10826–10846 (2006).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Physiol.

C. F. Stromeyer, III, P. D. Gowdy, A. Chaparro, S. Kladakis, J. D. Willen, and R. E. Kronauer, “Colour adaptation modifies the temporal properties of the long- and middle-wave cone signals in the human luminance mechanism,” J. Physiol. 526, 177–194(2000).
[CrossRef] [PubMed]

Opt. Rev.

K. Kawamoto, T. Inamura, H. Yaguchi, and S. Shioiri, “Color discrimination characteristics depending on the background color in the (L, M) plane of a cone space,” Opt. Rev. 10, 391–397(2003).
[CrossRef]

Percept. Psychophys.

S. Shioiri and P. Cavanagh, “Nonlinearity in color space measured by apparent motion,” Percept. Psychophys. 62, 1182–1190(2000).
[CrossRef] [PubMed]

Science

J. E. Thornton and E. N. Pugh, Jr., “Red/green color opponency at detection threshold,” Science 219, 191–193(1983).
[CrossRef] [PubMed]

Vis. Res.

G. R. Cole, T. J. Hine, and W. McIlhagga, “Estimation of linear detection mechanisms for stimuli of medium spatial frequency,” Vis. Res. 34, 1267–1278 (1994).
[CrossRef] [PubMed]

R. T. Eskew, Jr., C. F. Stromeyer, III, and R. E. Kronauer, “Temporal properties of the red–green chromatic mechanism,” Vis. Res. 34, 3127–3137 (1994).
[CrossRef] [PubMed]

C. F. Stromeyer, III, R. Thabet, A. Chaparro, and R. E. Kronauer, “Spatial masking does not reveal mechanisms selective to combined luminance and red–green color,” Vis. Res. 39, 2099–2112 (1999).
[CrossRef] [PubMed]

S. Tsujimura, S. Shioiri, and A. Nuruki, “Two distinct cone-opponent processes in the L+M luminance pathway,” Vis. Res. 47, 1839–1854 (2007).
[CrossRef] [PubMed]

C. F. Stromeyer, III, G. R. Cole, and R. E. Kronauer, “Second-site adaptation in the red–green chromatic pathways,” Vis. Res. 25, 219–237 (1985).
[CrossRef] [PubMed]

A. Chaparro, C. F. Stromeyer, III, G. Chen, and R. E. Kronauer, “Human cones appear to adapt at low light levels: measurements on the red–green detection mechanism,” Vis. Res. 35, 3103–3118 (1995).
[CrossRef] [PubMed]

J. Walraven and J. S. Werner, “The invariance of unique white; a possible implication for normalizing cone action spectra,” Vis. Res. 31, 2185–2193 (1991).
[CrossRef] [PubMed]

Other

R. T. Eskew, Jr., J. S. McLellan, and F. Giulianini, “Chromatic detection and discrimination,” in Color Vision: From Genes to Perception, K.Gegenfurtner and L.T.Sharpe, eds. (Cambridge University, 1999), pp. 345–368.

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

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Figures (9)

Fig. 1
Fig. 1

Color discrimination thresholds of Kawamoto et al. [17] with model predictions. Each square indicates the color that is just discriminable from the color indicated by the cross (the background color). The open square is the one in the L + M direction, which is not used for modeling. The lines show the prediction of threshold (a), (b), (c) for observers IN, YK, and KS, respectively. Threshold values (distances from the background colors) are magnified by three times so that the results can be clearly seen.

Fig. 2
Fig. 2

Two-stage color vision model proposed by Smith et al. [6]. Cones have nonlinear outputs expressed by a gain control, and the opponent process output OPP also has a nonlinear output expressed by a Naka–Rushton type function. Adaptation to the background color is modeled by subtracting a certain percentage of the OPP activity in response to the background color ( OPP A ). OPP A represents the activity of the opponent mechanism after adapting to the background color.

Fig. 3
Fig. 3

Threshold contour that corresponds to the line expressed by Eq. (5). The thick solid line shows a threshold contour, which is determined by the threshold along the L M direction (star) and a slope in Eq. (5). Threshold in other color directions [Eqs. (6, 7, 8)] are also shown (triangles).

Fig. 4
Fig. 4

(a) Slope of the threshold contour as a function of L 2 M value (activity of opponent color). Open circles represent model prediction, and solid diamonds represent the slope of the line fitted to experimental results (YK). (b) Threshold expressed by the distance of the threshold contour from the background color. The data are again shown as a function of L 2 M .

Fig. 5
Fig. 5

Model prediction of (a) slope and (b) threshold but without nonlinearity of the cone process. Experimental results are from YK.

Fig. 6
Fig. 6

Model prediction of (a) slope and (b) threshold but with out nonlinearity of the opponent process. Experimental results are from YK.

Fig. 7
Fig. 7

Comparison of model predictions for five background colors. The solid line and the dashed line in each panel indicate the prediction of the present model and that of SPS model, respectively. Solid squares represent experimental results (KS). Units are cd / m 2 as in Fig. 1.

Fig. 8
Fig. 8

Parameter k 1 estimated from unique yellow measurements (squares) and model predictions (line).

Fig. 9
Fig. 9

The open circle indicates the threshold contour slope predicted by the model with 0.8 of k 2 , which value was used in Smith et al. The solid diamond shows the experimental results [YK as in Fig. 4a].

Tables (1)

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Table 1 Fitting Parameters and Coefficients of Determination ( R 2 ) for the Present Model, SPS Model, and the Present Model with Variable k 1 a

Equations (11)

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G ( L ) = 1 ( 1 + k 3 L ) k 4 , L = L A l nor ,
OPP ( + LWS MWS ) = L T l nor G ( L A / l nor ) k 2 M T m nor G ( M A / m nor ) ,
OPP A = ( 1 k 1 ) · OPP A ,
R = R max · OPP OPP + SAT ,
Δ L L M = Δ OPP G ( L ) l nor + G ( M ) m nor , Sl = G ( L ) / G ( M ) , where    Δ OPP = δ R max · ( OPP A + SAT ) 2 SAT and L = L A l nor , M = M A m nor ,
Δ L L = Δ OPP G ( L ) l nor ,
Δ M M = Δ OPP G ( M ) m nor ,
Δ L L + M = Δ OPP G ( L ) l nor G ( M ) m nor .
δ Δ OPP d R d OPP = R max · SAT ( OPP A + SAT ) 2 , Δ OPP = δ R max · ( OPP A + SAT ) 2 SAT .
{ Δ OPP = Δ L l nor G ( L A / l nor ) k 2 Δ M m nor G ( M A / m nor ) Δ L + Δ M = 0 .
Δ OPP = OPP t OPP A = ( L A + Δ L ) l nor G ( L A / l nor ) k 2 ( M A + Δ M ) m nor G ( M A / m nor ) ( L A l nor G ( L A / l nor ) k 2 M A m nor G ( M A / m nor ) ) = Δ L l nor G ( L A / l nor ) k 2 Δ M m nor G ( M A / m nor ) .

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