Statistical explanation of the Bezold–Brücke effect. (a) Psychophysical data showing the changes in wavelength needed to match the hue of a monochromatic light presented at a higher intensity (after ref. 6). (b) Joint probability distribution of the physical correlates of hue and brightness, P(Hp,Bp), obtained from the database. (c) The cumulative density functions of P(Hp|Bp) calculated within a ±0.1 window centered at 0.3 as the value of the physical correlate of hue, given a lower and a higher levels of the physical correlate of brightness (indicated by the dashed lines in b). (d) The predicted hue shifts for the indicated points; the solid line is a polynomial fit.

Statistical explanation of the Hunt effect. (a) Psychophysical data showing that increasing luminance elicits an increasing sense of saturation; the curves are examples of stimuli that elicit yellow (upper curve) and green (lower curve) percepts (after ref. 3). (b) Joint probability distribution of the physical correlates of saturation and brightness, P(Sp,Bp), obtained from the database when the physical correlate of hue Hp was 0.1. (c) Cumulative density functions of P(Sp|Bp) at brightness values of 0.4 (blue curve) and 0.7 (red curve), corresponding to the dashed lines in b. (d) Saturations predicted from cumulative density functions under different levels of the physical correlate of brightness (Bp = 0.20, 0.31, 0.43, 0.55, 0.67, 0.78, 0.90) when the physical correlate of saturation (Sp) was 0.3 and the physical correlates of hue (Hp) were 0.1 and 0.3, thus corresponding to the examples in a.

Statistical explanation of the saturation discrimination function. (a) The amount of monochromatic light needed to generate the same perceived saturation difference with respect to neutral gray at different wavelengths (after ref. 2). (b) Joint probability distribution of the physical correlates of hue and saturation, P(Hp|Sp), obtained from the image database. (c) The cumulative density functions of P(Sp|Hp) when the physical correlate of hue (Hp) was 0.1 (blue curve; ≈575 nm) or 0.4 (red curve; ≈515 nm). (d) The predicted changes in the physical correlates of saturation needed to maintain a constant difference in perceived saturation as a function of the wavelength (the value illustrated is 0.3 in the cumulative density function, as shown in c). The points were derived from the cumulative density functions at the indicated wavelengths; the solid curve is a polynomial fit.

Statistical explanation of the Helmholtz–Kohlrausch effect. (a) Psychophysical changes in brightness as a function of both hue and saturation, plotted in the 1931 CIE chromaticity diagram (after ref. 8; the red dot indicates the white point). (b) Joint probability distribution of the physical correlates of saturation and brightness, P(Sp,Bp), at a particular value of the physical correlate of hue (Hp = 0.1). (c) The cumulative density functions of the physical correlate of brightness when the physical correlate of saturation is 0.3 and the physical correlate of hue 0.4 (blue curve) and when the physical correlate of saturation is 0.39 and the physical correlate of hue 0 (red curve). The cumulative probability values in these functions at a physical correlate of brightness value of 0.4 are the same. (d) The equal-brightness curves predicted by these statistics. Each of the points on a given curve has the same physical correlate of brightness value (0.2) and the same predicted brightness. The relative closeness of these curves reflects the limited number of highly saturated samples in the database and the limited gamut of the xy values in the CIE chromaticity diagram that can be represented by RGB values.

Statistical explanation of the Abney effect. (a) Psychophysical data showing variations in apparent hue as a function of saturation (after ref. 5). Lines of constant hue radiating from the white point (red dot; D65 daylight) curve systematically as saturation varies (after ref. 5). (b) Joint probability distribution of the physical correlates of hue and saturation, P(Hp,Sp), derived from the database. The cumulative density functions of P(Hp|Sp) were calculated within a local window centered on the physical correlate of hue in question (black bars). (c) Cumulative density functions when the physical correlate of saturation was 0.2 (blue curve) or 0.45 (red curve) and the physical correlate of hue 0.3. The abscissa shows the relative value of the physical correlate of hue, defined as in Fig. 2. (d) The constant hue curves predicted by the cumulative density functions, plotted as in a. The relatively short lengths of curves D and E reflect the limited number of highly saturated samples in the database and the limited gamut of xy values in the CIE chromaticity diagram that can be represented by RGB values.

Statistical explanation of the hue discrimination function. (a) The wavelength change in a monochromatic stimulus needed to elicit a just noticeable difference in hue (after ref. 1). (b) Joint probability distribution of the physical correlates of hue and brightness, P(Hp,Bp), obtained from the image database (probability values in this and subsequent figures are indicated by color coding; see bar on right). (c) Cumulative density functions of P(Hp|Bp) at two different values of the physical correlate of hue (Hp₁ = 0.6, Hp₂ = 0.1) for the same physical correlate of brightness (Bp = 0.6), obtained by normalizing and accumulating the probability values of P(Hp|Bp) within a local window. The abscissa shows the relative value of the physical correlate of hue with respect to the center of the local window. (d) The predicted changes in wavelength needed to maintain constant perceived differences in hue. The points are the constant differences in cumulative density functions at the indicated wavelengths, using 0.02 as a value for d (the scale of d in c has been enlarged for clarity); the solid curve is a polynomial fit.