Robert C. Carter^{1,}^{*}
and Louis D. Silverstein^{2,}^{†}

^{1}21 Castle Drive, Pennsville, New Jersey 08070–2419, USA

^{2}VCD Sciences, 9695 East Yucca Street, Scottsdale, Arizona 85260–6201, USA

^{†}Louis D. Silverstein died on May 1, 2012, while working on the final revision of this paper at his home in Scottsdale, Arizona. Lou has been a long-time member of the Optical Society of America, and a contributor to JOSA A in particular, in addition to his commitment to the Society for Information Display (where he was a fellow and 2008 Otto Schade Award recipient), the Society for Imaging Science and Technology, and the Inter Society Color Council, which presented him its 2004 Macbeth Award.

Robert C. Carter and Louis D. Silverstein, "Perceiving color across scale: great and small, discrete and continuous," J. Opt. Soc. Am. A 29, 1346-1355 (2012)

We generalize, to images with continuously varying colors, our previously published model for comparing color differences of spatially discrete visual fields (icons, symbols) of disparate sizes. Our model is structural, including scattering of light by the intraocular media, followed by sparse retinal cone cell sampling of each physiological color primary. We use our model to show that small subtense of less than half a degree drastically reduces the number of discriminable colors available within a color gamut. The proposed generalization predicts and explains appearance of color fields having a wide range of subtenses (from $1/8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{deg}$ to 44 deg in examples given).

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Multivariate Correlations of Cone Primary Attenuations Derived from Experiments of Jacobsen, Highnote, and Zhang–Wandell (for ${120}^{\prime}$, ${60}^{\prime}$, ${30}^{\prime}$, ${15}^{\prime}$ and ${7.5}^{\prime}$; normalized to ${120}^{\prime}\text{\hspace{0.17em}}\mathbf{\text{threshold}}=1$)

Cone Primary Attenuations

CS2010 Highnote

Zhang

dJacobsen

CS2010 Highnote

1.0000

0.9500

0.8885

Zhang

0.9500

1.0000

0.9050

dJacobsen

0.8885

0.9050

1.0000

Table 2.

Proportion of Target Contrast ($|\mathbf{\text{Target Intensity}}-\mathbf{\text{Surround Intensity}}|$) That Is Lost to Scatteringa

Subtense (Arc Min)

${120}^{\prime}$

${60}^{\prime}$

${30}^{\prime}$

${15}^{\prime}$

${7.5}^{\prime}$

Positive contrasts

0.059

0.108

0.175

0.256

0.346

Negative contrasts

0.055

0.103

0.171

0.252

0.341

Out of the target for positive contrasts, in for negative—based upon the van den Berg point spread function convolved with targets having specified subtenses.

Table 3.

Relation between Inverse Search Time ($1/\mathrm{s}$) and CIEDE2000, as Mediated by Various Models of Effects of Subtense

Models of $\mathrm{\Delta}E$

Model ${\mathrm{SS}}_{\mathrm{error}}$

Proportion of Inverse Search Time Variance Explained by the Model, ${R}^{2}$

Statistics Associated with Fig 3, Inverse Search Time versus CIEDE2000 Calculated from Tristimulus Values Preprocessed with the GM Using Zhang–Wandell Kernels and the van den Berg PSF

SSE

DFE

MSE

RMSE

12.357187449

117

0.105617

0.3249877

Parameter

Estimate

Approx. Std. Err.

Lower CL

Upper CL

Intercept

0.2838482652

0.10077022

0.0670307

0.49100222

RT

1.7924461538

0.04030976

1.71141942

1.87799599

Knee

14.028720798

0.86922935

12.1972424

16.6558905

Table 5.

Display Primary Tristimulus Values

$X$

$Y$

$Z$

R

84.65455

43.65

3.968182

G

71.725

143.45

23.90833

B

32.25

12.9

169.85

Table 6.

Example of Optimal Selection of Four Colors Corresponding to the 4-Color 2 Deg Result in Table 7

Color Number & Display Primary

Display Primary Intensity, Proportion of Maximum ($=1$)

Color 1 R

0

Color 1 G

1

Color 1 B

0

Color 2 R

0

Color 2 G

0.01052

Color 2 B

0

Color 3 R

1

Color 3 G

0.42987

Color 3 B

1

Color 4 R

0.00799

Color 4 G

0.02360

Color 4 B

0.57355

Table 7.

Maximized Minimum Color Difference (CIEDE2000) for 2 to 17 Colors

Contribution of the GM to Predicting Leeds Data Variance beyond Baseline

GM Correlation $R$

GM ${R}^{2}$

Baseline (Std. Tristimulus) Correlation $r$

GM Contribution, Proportion of Variance Beyond Baseline $({R}^{2}-{r}^{2})/(1-{r}^{2})$

GM Contribution if 10% Errors of Measurement $({R}^{2}-{r}^{2})/(1-{r}^{2}-0.1)$

44 deg standard

${L}^{*}$

0.82

0.67

0.71

0.35

0.43

${C}^{*}$

0.90

0.81

0.85

0.30

0.48

19 deg standard

${L}^{*}$

0.74

0.55

0.47

0.42

0.48

${C}^{*}$

0.92

0.85

0.86

0.39

0.65

2 deg standard

${L}^{*}$

0.57

0.33

0.50

0.11

0.12

${L}^{*}$ (boost 44 deg GM LD by 25%)

0.82

0.68

0.50

0.57

0.66

${L}^{*}$ ($\mathrm{GM}+\text{regression on subtense}$)

0.88

0.78

0.85

0.70

0.81

${C}^{*}$

0.88

0.78

0.85

0.23

0.35

Tables (9)

Table 1.

Multivariate Correlations of Cone Primary Attenuations Derived from Experiments of Jacobsen, Highnote, and Zhang–Wandell (for ${120}^{\prime}$, ${60}^{\prime}$, ${30}^{\prime}$, ${15}^{\prime}$ and ${7.5}^{\prime}$; normalized to ${120}^{\prime}\text{\hspace{0.17em}}\mathbf{\text{threshold}}=1$)

Cone Primary Attenuations

CS2010 Highnote

Zhang

dJacobsen

CS2010 Highnote

1.0000

0.9500

0.8885

Zhang

0.9500

1.0000

0.9050

dJacobsen

0.8885

0.9050

1.0000

Table 2.

Proportion of Target Contrast ($|\mathbf{\text{Target Intensity}}-\mathbf{\text{Surround Intensity}}|$) That Is Lost to Scatteringa

Subtense (Arc Min)

${120}^{\prime}$

${60}^{\prime}$

${30}^{\prime}$

${15}^{\prime}$

${7.5}^{\prime}$

Positive contrasts

0.059

0.108

0.175

0.256

0.346

Negative contrasts

0.055

0.103

0.171

0.252

0.341

Out of the target for positive contrasts, in for negative—based upon the van den Berg point spread function convolved with targets having specified subtenses.

Table 3.

Relation between Inverse Search Time ($1/\mathrm{s}$) and CIEDE2000, as Mediated by Various Models of Effects of Subtense

Models of $\mathrm{\Delta}E$

Model ${\mathrm{SS}}_{\mathrm{error}}$

Proportion of Inverse Search Time Variance Explained by the Model, ${R}^{2}$

Statistics Associated with Fig 3, Inverse Search Time versus CIEDE2000 Calculated from Tristimulus Values Preprocessed with the GM Using Zhang–Wandell Kernels and the van den Berg PSF

SSE

DFE

MSE

RMSE

12.357187449

117

0.105617

0.3249877

Parameter

Estimate

Approx. Std. Err.

Lower CL

Upper CL

Intercept

0.2838482652

0.10077022

0.0670307

0.49100222

RT

1.7924461538

0.04030976

1.71141942

1.87799599

Knee

14.028720798

0.86922935

12.1972424

16.6558905

Table 5.

Display Primary Tristimulus Values

$X$

$Y$

$Z$

R

84.65455

43.65

3.968182

G

71.725

143.45

23.90833

B

32.25

12.9

169.85

Table 6.

Example of Optimal Selection of Four Colors Corresponding to the 4-Color 2 Deg Result in Table 7

Color Number & Display Primary

Display Primary Intensity, Proportion of Maximum ($=1$)

Color 1 R

0

Color 1 G

1

Color 1 B

0

Color 2 R

0

Color 2 G

0.01052

Color 2 B

0

Color 3 R

1

Color 3 G

0.42987

Color 3 B

1

Color 4 R

0.00799

Color 4 G

0.02360

Color 4 B

0.57355

Table 7.

Maximized Minimum Color Difference (CIEDE2000) for 2 to 17 Colors