Michael E. Rudd and Dorin Popa, "Stevens's brightness law, contrast gain control, and edge integration in achromatic color perception: a unified model," J. Opt. Soc. Am. A 24, 2766-2782 (2007)

The brightness of an isolated test patch is related to its luminance by a power law having an exponent of about $1\u22153$, a result known as Stevens’s brightness law. The brightness law exponent characterizes the rate at which brightness grows with luminance and can thus be thought of as an “exponential” gain factor. We studied changes in this gain factor for incremental and decremental test squares as a function of the size of a surrounding frame of homogeneous luminance. For incremental targets, the gain decreased as an approximately linear function of the frame width. For decremental targets, the gain increased as an approximately linear function of the frame width. We modeled the brightness of the frame-embedded target with a quantitative theory based on the assumption that the target brightness is determined by the sum of achromatic color induction signals originating from the inner and outer edges of the surround, a theory that has previously been used to account for the results of several other brightness matching experiments. To account for the frame-width-dependent gain changes observed in the present study, we elaborate this edge integration theory by proposing the existence of a cortical contrast gain control mechanism by which the gains applied to neural edge detectors are influenced by the responses of other edge detectors responding to the nearby edges.

Michael E. Rudd and Dorin Popa, "Stevens's brightness law, contrast gain control, and edge integration in achromatic color perception: a unified model: errata," J. Opt. Soc. Am. A 24, 3335-3335 (2007) https://opg.optica.org/josaa/abstract.cfm?uri=josaa-24-10-3335

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Results of the Linear Regression Analyses of the log Matching Square Luminance versus log Test Square Luminance Data from Experiment 1 (Decremental Squares)

Frame Width (deg)

Slope

s.e.^{
a
}

Intercept

s.e.

${R}^{2}$

JA

0.12

0.8240

0.0133

0.1260

0.0056

0.9893

0.35

0.8232

0.0114

0.0563

0.0048

0.9935

0.58

0.8827

0.0109

0.0406

0.0045

0.9977

0.82

0.9106

0.0132

0.0341

0.0055

0.9990

1.05

0.9054

0.0128

0.0271

0.0054

0.9960

AH

0.12

0.7583

0.0076

0.1769

0.0032

0.9983

0.35

0.7819

0.0094

0.1012

0.0039

0.9977

0.58

0.8441

0.0085

0.0919

0.0036

0.9988

0.82

0.8825

0.0113

0.0780

0.0047

0.9992

1.05

0.8470

0.0079

0.0930

0.0033

0.9974

Standard error of the coefficient.

Table 2

Summary of the Least-Squares Linear and Second-Order Polynomial Regression Models Relating the Slopes and Intercepts of the $\mathrm{log}\phantom{\rule{0.2em}{0ex}}{S}_{M}$ vs $\mathrm{log}\phantom{\rule{0.2em}{0ex}}{S}_{T}$ Plots shown in Fig. 2 to the Test Frame Width (Experiment 1, Decremental Squares)^{
a
}

Dependent Variable: Slope

AH

JA

Model

b

s.e.

t

Significance

Model

b

s.e.

t

Significance

Linear

Linear

Intercept

0.753

0.027

27.601

${0.000}^{*}$

Intercept

0.807

0.017

46.240

${0.000}^{*}$

Width

0.119

0.041

2.928

${0.061}^{\u2020}$

Width

0.107

0.026

4.121

${0.026}^{*}$

2nd order

2nd order

Intercept

0.710

0.037

19.392

${0.003}^{*}$

Intercept

0.794

0.032

24.545

${0.002}^{*}$

Width

0.337

0.148

2.271

0.151

Width

0.169

0.131

1.290

0.326

Width^{2}

0.186

0.124

$-1.507$

0.271

Width^{2}

$-0.053$

0.109

0.484

0.676

Dependent Variable: Intercept

AH

JA

Model

b

s.e.

t

Significance

Model

b

s.e.

t

Significance

Linear

Linear

Intercept

0.156

0.026

5.910

${0.010}^{*}$

Intercept

0.112

0.021

5.317

${0.013}^{*}$

Width

0.082

0.039

$-2.080$

0.129

Width

$-0.094$

0.031

$-3.006$

${0.057}^{\u2020}$

2nd order

2nd order

Intercept

0.210

0.018

11.828

0.007

Intercept

0.153

0.018

8.379

${0.014}^{*}$

Width

0.353

0.072

$-4.916$

${0.039}^{*}$

Width

0.300

0.074

$-4.074$

${0.055}^{\u2020}$

Width^{2}

0.232

0.060

3.879

${0.060}^{\u2020}$

Width^{2}

0.177

0.062

2.872

0.103

Symbols: $b=\text{regression}$ coefficient; $\mathrm{s.e.}=\text{standard}$ error of the coefficient; ${\text{\hspace{0.17em}}}^{*}$ indicates $p<0.05$; ${\text{\hspace{0.17em}}}^{\u2020}$ indicates a borderline significant result.

Table 3

Results of the Linear Regression Analyses of the log Matching Square Luminance versus log Test Square Luminance Data from Experiment 2 (Incremental Squares)

Frame Width (deg)

Slope

s.e.^{
a
}

Intercept

s.e.

${R}^{2}$

JA

0.12

1.1649

0.0222

$-0.0805$

0.0127

0.9968

0.35

1.0419

0.0173

$-0.0072$

0.0099

0.9971

0.58

0.9916

0.0137

0.0427

0.0078

0.9993

0.82

0.9600

0.0154

0.0590

0.0088

0.9954

1.05

0.9286

0.0154

0.0828

0.0088

0.9990

AH

0.12

0.9219

0.0145

0.0737

0.0082

0.9963

0.35

0.8671

0.0126

0.0924

0.0072

0.9995

0.58

0.8566

0.0114

0.0965

0.0065

0.9954

0.82

0.8033

0.0133

0.1229

0.0076

0.9931

1.05

0.7995

0.0119

0.1220

0.0068

0.9874

Standard error of the coefficient.

Table 4

Summary of the Least-Square Linear and Second-Order Polynomial Regression Models Relating the Slopes and Intercepts of the $\mathrm{log}\phantom{\rule{0.2em}{0ex}}{S}_{M}$ vs $\mathrm{log}\phantom{\rule{0.2em}{0ex}}{S}_{T}$ Plots Shown in Fig. 5 to the Test Frame Width (Experiment 2, Incremental Squares)^{
a
}

Dependent Variable: Slope

AH

JA

Model

b

s.e.

t

Significance

Model

b

s.e.

t

Significance

Linear

Linear

Intercept

0.927

0.014

66.204

$0.000*$

Intercept

1.156

0.031

37.495

${0.000}^{*}$

Width

$-0.132$

0.021

$-6.331$

${0.008}^{*}$

Width

$-0.238$

0.046

$-5.165$

${0.014}^{*}$

2nd order

2nd order

Intercept

0.945

0.022

42.667

${0.001}^{*}$

Intercept

1.217

0.024

50.676

${0.000}^{*}$

Width

$-0.223$

0.090

$-2.485$

0.131

Width

$-0.547$

0.097

$-5.623$

${0.030}^{*}$

Width^{2}

0.077

0.075

1.037

0.409

Width^{2}

0.265

0.081

3.267

0.082

Dependent Variable: Intercept

AH

JA

Model

b

s.e.

t

Significance

Model

b

s.e.

t

Significance

Linear

Linear

Intercept

0.070

0.006

11.133

${0.002}^{*}$

Intercept

$-0.079$

0.020

$-4.034$

${0.027}^{*}$

Width

0.054

0.009

5.818

${0.010}^{*}$

Width

0.168

0.029

5.773

${0.010}^{*}$

2nd order

2nd order

Intercept

0.065

0.011

5.683

0.030

Intercept

$-0.119$

0.012

$-10.058$

${0.010}^{*}$

Width

0.080

0.046

1.747

0.223

Width

0.371

0.048

7.750

${0.016}^{*}$

Width^{2}

$-0.022$

0.038

$-0.580$

0.621

Width^{2}

$-0.174$

0.040

$-4.354$

${0.049}^{*}$

Symbol: $b=\text{regression}$ coefficient; $\mathrm{s.e.}=\text{Standard}$ error of the coefficient; ${\text{\hspace{0.17em}}}^{*}$ indicates $p<0.05$.

Tables (4)

Table 1

Results of the Linear Regression Analyses of the log Matching Square Luminance versus log Test Square Luminance Data from Experiment 1 (Decremental Squares)

Frame Width (deg)

Slope

s.e.^{
a
}

Intercept

s.e.

${R}^{2}$

JA

0.12

0.8240

0.0133

0.1260

0.0056

0.9893

0.35

0.8232

0.0114

0.0563

0.0048

0.9935

0.58

0.8827

0.0109

0.0406

0.0045

0.9977

0.82

0.9106

0.0132

0.0341

0.0055

0.9990

1.05

0.9054

0.0128

0.0271

0.0054

0.9960

AH

0.12

0.7583

0.0076

0.1769

0.0032

0.9983

0.35

0.7819

0.0094

0.1012

0.0039

0.9977

0.58

0.8441

0.0085

0.0919

0.0036

0.9988

0.82

0.8825

0.0113

0.0780

0.0047

0.9992

1.05

0.8470

0.0079

0.0930

0.0033

0.9974

Standard error of the coefficient.

Table 2

Summary of the Least-Squares Linear and Second-Order Polynomial Regression Models Relating the Slopes and Intercepts of the $\mathrm{log}\phantom{\rule{0.2em}{0ex}}{S}_{M}$ vs $\mathrm{log}\phantom{\rule{0.2em}{0ex}}{S}_{T}$ Plots shown in Fig. 2 to the Test Frame Width (Experiment 1, Decremental Squares)^{
a
}

Dependent Variable: Slope

AH

JA

Model

b

s.e.

t

Significance

Model

b

s.e.

t

Significance

Linear

Linear

Intercept

0.753

0.027

27.601

${0.000}^{*}$

Intercept

0.807

0.017

46.240

${0.000}^{*}$

Width

0.119

0.041

2.928

${0.061}^{\u2020}$

Width

0.107

0.026

4.121

${0.026}^{*}$

2nd order

2nd order

Intercept

0.710

0.037

19.392

${0.003}^{*}$

Intercept

0.794

0.032

24.545

${0.002}^{*}$

Width

0.337

0.148

2.271

0.151

Width

0.169

0.131

1.290

0.326

Width^{2}

0.186

0.124

$-1.507$

0.271

Width^{2}

$-0.053$

0.109

0.484

0.676

Dependent Variable: Intercept

AH

JA

Model

b

s.e.

t

Significance

Model

b

s.e.

t

Significance

Linear

Linear

Intercept

0.156

0.026

5.910

${0.010}^{*}$

Intercept

0.112

0.021

5.317

${0.013}^{*}$

Width

0.082

0.039

$-2.080$

0.129

Width

$-0.094$

0.031

$-3.006$

${0.057}^{\u2020}$

2nd order

2nd order

Intercept

0.210

0.018

11.828

0.007

Intercept

0.153

0.018

8.379

${0.014}^{*}$

Width

0.353

0.072

$-4.916$

${0.039}^{*}$

Width

0.300

0.074

$-4.074$

${0.055}^{\u2020}$

Width^{2}

0.232

0.060

3.879

${0.060}^{\u2020}$

Width^{2}

0.177

0.062

2.872

0.103

Symbols: $b=\text{regression}$ coefficient; $\mathrm{s.e.}=\text{standard}$ error of the coefficient; ${\text{\hspace{0.17em}}}^{*}$ indicates $p<0.05$; ${\text{\hspace{0.17em}}}^{\u2020}$ indicates a borderline significant result.

Table 3

Results of the Linear Regression Analyses of the log Matching Square Luminance versus log Test Square Luminance Data from Experiment 2 (Incremental Squares)

Frame Width (deg)

Slope

s.e.^{
a
}

Intercept

s.e.

${R}^{2}$

JA

0.12

1.1649

0.0222

$-0.0805$

0.0127

0.9968

0.35

1.0419

0.0173

$-0.0072$

0.0099

0.9971

0.58

0.9916

0.0137

0.0427

0.0078

0.9993

0.82

0.9600

0.0154

0.0590

0.0088

0.9954

1.05

0.9286

0.0154

0.0828

0.0088

0.9990

AH

0.12

0.9219

0.0145

0.0737

0.0082

0.9963

0.35

0.8671

0.0126

0.0924

0.0072

0.9995

0.58

0.8566

0.0114

0.0965

0.0065

0.9954

0.82

0.8033

0.0133

0.1229

0.0076

0.9931

1.05

0.7995

0.0119

0.1220

0.0068

0.9874

Standard error of the coefficient.

Table 4

Summary of the Least-Square Linear and Second-Order Polynomial Regression Models Relating the Slopes and Intercepts of the $\mathrm{log}\phantom{\rule{0.2em}{0ex}}{S}_{M}$ vs $\mathrm{log}\phantom{\rule{0.2em}{0ex}}{S}_{T}$ Plots Shown in Fig. 5 to the Test Frame Width (Experiment 2, Incremental Squares)^{
a
}

Dependent Variable: Slope

AH

JA

Model

b

s.e.

t

Significance

Model

b

s.e.

t

Significance

Linear

Linear

Intercept

0.927

0.014

66.204

$0.000*$

Intercept

1.156

0.031

37.495

${0.000}^{*}$

Width

$-0.132$

0.021

$-6.331$

${0.008}^{*}$

Width

$-0.238$

0.046

$-5.165$

${0.014}^{*}$

2nd order

2nd order

Intercept

0.945

0.022

42.667

${0.001}^{*}$

Intercept

1.217

0.024

50.676

${0.000}^{*}$

Width

$-0.223$

0.090

$-2.485$

0.131

Width

$-0.547$

0.097

$-5.623$

${0.030}^{*}$

Width^{2}

0.077

0.075

1.037

0.409

Width^{2}

0.265

0.081

3.267

0.082

Dependent Variable: Intercept

AH

JA

Model

b

s.e.

t

Significance

Model

b

s.e.

t

Significance

Linear

Linear

Intercept

0.070

0.006

11.133

${0.002}^{*}$

Intercept

$-0.079$

0.020

$-4.034$

${0.027}^{*}$

Width

0.054

0.009

5.818

${0.010}^{*}$

Width

0.168

0.029

5.773

${0.010}^{*}$

2nd order

2nd order

Intercept

0.065

0.011

5.683

0.030

Intercept

$-0.119$

0.012

$-10.058$

${0.010}^{*}$

Width

0.080

0.046

1.747

0.223

Width

0.371

0.048

7.750

${0.016}^{*}$

Width^{2}

$-0.022$

0.038

$-0.580$

0.621

Width^{2}

$-0.174$

0.040

$-4.354$

${0.049}^{*}$

Symbol: $b=\text{regression}$ coefficient; $\mathrm{s.e.}=\text{Standard}$ error of the coefficient; ${\text{\hspace{0.17em}}}^{*}$ indicates $p<0.05$.