The Distribution of Color Matchings Around a Color Center*

Color gamut volume and the maximum number of mutually discernible colors based on a Riemannian metric

For the calculation of the color gamut volume and the maximum number of mutually discernible colors, an algorithm based on a Riemannian metric and the densest packing of spheres is proposed. With this algorithm, the color gamut volume was calculated for the conditions of experiments reported in literature. Good agreement was found with the experimental findings of the color gamut volume as a function of the peak luminance. Using the new algorithm, the color gamut volume and the maximum number of mutually discernible colors was calculated for various sets of primary colors corresponding to display standards and various dynamic ranges. Comparisons were made with state-of-the-art methods which are based on the Euclidean metric in approximately uniform color spaces and a simple cubic lattice. It was found that the state-of-the-art methods underestimate the maximum number of mutually discernible colors. However, the relative differences decrease as the primary colors are more saturated. Based on the new algorithm the maximum number of mutually discernible colors was calculated for a range of peak retinal illuminance levels and various sets of primary colors. We found that, for a given set of primary colors, the maximum number of mutually discernible colors is proportional to the logarithm of the ratio of the peak retinal illuminance level and a fitting parameter.

Line element for the perceptual color space

It is generally accepted that the perceptual color space is not Euclidean. A new line element for a 3-dimensional Riemannian color space was developed. This line element is based on the Friele line elements and psychophysical color discrimination models, and comprises both the first and second stage of color vision. The line element is expressed in a contrast space based on the MacLeod-Boynton chromaticities. New equations for the contrast thresholds along the cardinal axes and new metric tensor elements were determined. Visual adaptation effects were incorporated into the model. Color discrimination threshold ellipsoids were calculated with the new line element. Adequate agreement with experimental threshold ellipsoids reported in literature was demonstrated. From a comparison with other color difference metrics a better overall predictability of threshold ellipsoids was found with the new line element.

Metamer mismatching underlies color difference sensitivity

Color difference sensitivity as represented by the size of discrimination ellipsoids is known to depend on where the colors reside within color space. In the past, various color spaces and color difference formulas have been developed as parametric fits to the experimental data with the goal of establishing a color coordinate system in which equally discriminable colors are equal distances apart. These empirical models, however, provide no explanation as to why color discrimination varies in the way it does. This article considers the hypothesis that the variation in color discrimination tolerances reflects the uncertainty created by the degree of metamer mismatching for a given color. Specifically, the greater the degree of metamer mismatching for a color, the wider the range of spectral reflectances that could have led to it and, hence, the more finely a color needs to be discriminated in order to reliably identify materials and objects. To test this hypothesis, the available color discrimination data sets for surface colors are gathered and analyzed. A strong correlation between color discrimination and the degree of metamer mismatching is found. This correlation provides evidence that metamer mismatching provides an explanation as to why color discrimination varies throughout color space as it does.

Neurogeometry of color vision

In neurogeometry, principles of differential geometry and neuron dynamics are used to model the representation of forms in the primary visual cortex, V1. This approach is well-suited for explaining the perception of illusory contours such as Kanizsa's figure (see Petitot (2008) for a review). In its current version, neurogeometry uses achromatic inputs to the visual system as the starting-point for form estimation. Here we ask how neurogeometry operates when the input is chromatic as in color vision. We propose that even when considering only the perception of form, the random nature of the cone mosaic must be taken into account. The main challenge for neurogeometry is to explain how achromatic information could be estimated from the sparse chromatic sampling provided by the cone mosaic. This article also discusses the non-linearity involved in a neural geometry for chromatic processing. We present empirical results on color discrimination to illustrate the geometric complexity for the discrimination contour when the adaptation state of the observer is not conditioned. The underlying non-linear geometry must conciliate both mosaic sampling and regulation of visual information in the visual system.

Métodos utilizados na avaliação psicofísica da visão de cores humana

A cor é um atributo perceptual que nos permite identificar e localizar padrões ambientais de mesmo brilho e constitui uma dimensão adicional na identificação de objetos, além da detecção de inúmeros outros atributos dos objetos em sua relação com a cena visual, como luminância, contraste, forma, movimento, textura, profundidade. Decorre daí a sua importância fundamental nas atividades desempenhadas pelos animais e pelos seres humanos em sua interação com o ambiente. A psicofísica visual preocupa-se com o estudo quantitativo da relação entre eventos físicos de estimulação sensorial e a resposta comportamental resultante desta estimulação, fornecendo dessa maneira meios de avaliar aspectos da visão humana, como a visão de cores. Este artigo tem o objetivo de mostrar diversas técnicas eficientes na avaliação da visão cromática humana através de métodos psicofísicos adaptativos.

Upgrading color-difference formulas

We propose a method to improve the prediction performance of existing color-difference formulas with additional visual data. The formula is treated as the mean function of a Gaussian process, which is trained with experimentally determined color-discrimination data. Color-difference predictions are calculated using Gaussian process regression (GPR) considering the uncertainty of the visual data. The prediction accuracy of the CIE94 formula is significantly improved with the GPR approach for the Leeds and the Witt datasets. By upgrading CIE94 with GPR we achieve a significantly lower STRESS value of 26.58 compared with that for CIEDE2000 (27.49) on a combined dataset. The method could serve to improve the prediction performance of existing color-difference equations around particular color centers without changing the equations themselves.

Estimativa da sensibilidade ao contraste espacial de luminância e discriminação de cores por meio do potencial provocado visual transiente

O potencial provocado visual (VEP) é uma resposta cortical registrável na superfície do couro cabeludo, que reflete a atividade dos neurônios de V1. é classificado, a partir da freqüência temporal de estimulação, em transiente ou de estado estacionário. Outras propriedades do estímulo parecem provocar uma atividade seletiva dos diversos grupos de neurônios existentes em V1. Desse modo, o VEP vem sendo usado para estudar a visão humana acromática e cromática. Diversos trabalhos usaram o VEP para estimar a sensibilidade ao contraste de luminância no domínio das freqüências espaciais. Mais recentemente, há estudos que empregaram o VEP para medir os limiares de discriminação de cores. O VEP transiente pode complementar as medidas psicofísicas de sensibilidade ao contraste espacial de luminância e de discriminação cromática, e constitui um método não invasivo para estudar a visão de indivíduos com dificuldades de realizar testes psicofísicos.

Perception of spatiotemporal random fractals: an extension of colorimetric methods to the study of dynamic texture

Recent work establishes that static and dynamic natural images have fractal-like l/falpha spatiotemporal spectra. Artifical textures, with randomized phase spectra, and 1/falpha amplitude spectra are also used in studies of texture and noise perception. Influenced by colorimetric principles and motivated by the ubiquity of 1/falpha spatial and temporal image spectra, we treat the spatial and temporal frequency exponents as the dimensions characterizing a dynamic texture space, and we characterize two key attributes of this space, the spatiotemporal appearance map and the spatiotemporal discrimination function (a map of MacAdam-like just-noticeable-difference contours).

Evidence for the contribution of S cones to the detection of flicker brightness and red-green

We were interested in the question of how cones contribute to the detection of brightness, red-green, and blue-yellow. The linear combination of cone signals contributing to flicker detection was determined by fitting a plane to 64 points (colors) of equal heterochromatic flicker brightness. A small S-cone contribution to flicker brightness of similar amplitude in all five subjects was identified. The ratio of L- to M-cone contribution was found to vary considerably among subjects (1.7-4.1). Chromatic detection thresholds were determined for small patches in the isoluminant plane defined by flicker brightness. These stimuli were presented at an eccentricity of 40 arc min. By using color naming at the detection threshold, one can attribute different segments of the resulting detection ellipses to different chromatic mechanisms. Linear approximation of these segments provided an estimate for the contribution of the different cone types to the detection of red-green and blue-yellow. The results are consistent with the hypothesis that S cones contribute to the red-green mechanism with the same sign as that of the contribution from L cones. The blue-yellow mechanism very probably subtracts S-cone contrast from luminance contrast. The detection ellipse can be mapped into a circle in cone difference space. The base of this canonical transformation is a set of three cone fundamentals that differs from previously published estimates. Projecting the circle onto the three cone difference axes produces sinusoidal changes within the respective excitations. We propose that simultaneous sinusoidal changes of equal increment in the three cone difference excitations generate stimuli differing by equal saliency.

Statistics of color-matching experimental data

By assuming that color matches are normally distributed in XYZ space, we present a rigorous statistical technique to obtain regions of equally noticeable chromaticity differences. The probability density function of color matches in (x, y, Y) space is calculated according to standard techniques in probability theory. The geometry of the chromaticity thresholds is computed for various confidence levels alpha. Because of the asymmetry of the probability density function in (x, y, Y) space, the chromaticity thresholds are not symmetrical around the color center. The asymmetries depend on the color center, and they increase when high confidence levels (alpha < 0.32) are considered. It is our opinion that the technique proposed here can provide a useful tool for checking and evaluating deviations from the elliptic geometry of the chromaticity thresholds. It is formally demonstrated that regions of equally noticeable chromaticity differences are not ellipses when the normality hypothesis is assumed in XYZ space.

Measurements of colour constancy by using a forced-choice matching technique

Colour constancy is typically measured with techniques involving asymmetric matching by adjustment, in which the observer views two scenes under different illuminants and adjusts the colour of a reference patch in one to match a test patch in the other. This technique involves an unnatural task, requiring the observer to predict and adjust colour appearance under an illumination shift. Natural colour constancy is more a simple matter of determining whether a colour is the same as or different from that seen under different illumination conditions. There are also technical disadvantages to the method of matching by adjustment, particularly when used to measure colour constancy in complex scenes. Therefore, we have developed and tested a two-dimensional method of constant-stimuli, forced-choice matching paradigm for measuring colour constancy. Observers view test and reference scenes haploscopically and simultaneously, each eye maintaining separate adaptation throughout a session. On each trial, a pair of test and reference patches against multicoloured backgrounds are presented, the reference patch colours being selected from a two-dimensional grid of displayable colours around the point of perfect colour constancy. The observer's task is to respond "same" or "different". Fitting a two-dimensional Gaussian to the percentage of "different" responses yields (1) the subjective colour-constancy point, (2) the discrimination ellipse centred on this point, and (3) a map of changes in sensitivity to chromatic differences induced by the illuminant shift. The subjective colour-constancy point measured in this way shows smaller deviations from perfect colour constancy-under conditions of monocular adaptation-than previously reported; discrimination ellipses are several times larger than standard MacAdam ellipses; and chromatic sensitivity is independent of the direction of the illuminant shift, for broad distributions of background colours.

Testing the indeterminacy of linear color mechanisms from color discrimination data

It have previously been reported that, for some choices of the fixed spatial and temporal characteristics of test stimuli, it was possible to estimate the spectral sensitivities of chromatic mechanisms from chromatic discrimination data alone. If mechanism sensitivities could be reliably estimated for any choice of test stimuli characteristics, the influence of spatial and temporal factors on chromatic discrimination performance could be directly measured. Previous studies, using test stimuli with other spatio-temporal characteristics, have found equi-discrimination contours whose ellipsoidal shapes seem to preclude estimation of mechanisms. Since there is no commonly-accepted method for testing the adequacy of ellipsoidal fits of chromatic equi-discrimination contours, it is possible that alternative psychophysical procedures combined with more powerful statistical tests could detect the pattern of deviations from ellipticality reported previously. In this paper, we described psychophysical tests and statistical analyses that, taken together, provide a more powerful test of the indeterminacy of mechanisms than previous methods. We develop a method based on analysis of residuals for detecting the pattern of deviations from ellipticality. We apply these tests under fixed experimental conditions similar to those in which other researchers have found ellipsoidal equi-discrimination contours. For these conditions, for any of the tests performed, we do not reject the hypothesis that equi-discrimination surfaces are ellipsoidal.

Estimation of linear detection mechanisms for stimuli of medium spatial frequency

Detection thresholds were obtained for a circularly-symmetric Gabor profile and Craik-Cornsweet profiles presented on a large white adapting field. These stimuli possessed peak spatial power between 1 and 6 c/deg. Their contrast was represented in an L, M and S cone contrast space. Detection thresholds were obtained for many vectors close to specific but theoretically important planes within this space. These data were fitted with a model comprising independent mechanisms, each a weighted sum of cone contrasts. The fit revealed a chromatic mechanism driven by delta L/L-delta M/M with no S cone input. Within cone contrast space, this mechanism was more sensitive than both a luminance mechanism with little S cone input but considerable variation in relative L to M cone input, and a blue-yellow chromatic mechanism.

Detection mechanisms in L-, M-, and S-cone contrast space

Detection thresholds were obtained for a 2 degrees Gaussian-blurred spot flashed for 200 ms on an 8.9 degrees white adapting field of 1070 trolands. The spot's contrast was represented in an L-, M-, and S-cone contrast space. Detection thresholds were obtained for many vectors close to specific but theoretically important planes within this space. A three-dimensional surface was fitted to the data generated by the probability summation of three mechanisms, each a weighted sum of cone contrasts. The fit revealed a red-green chromatic mechanism driven by delta L/L--delta M/M with no S-cone input that was 1 order of magnitude more sensitive than the two other mechanisms. The latter consisted of a luminance mechanism with little S-cone input and a blue-yellow chromatic mechanism with the S cone opposed to L and M cones.

Changes in pattern induced flicker colors are mediated by the blue-yellow opponent process

The colors of Benham's Top [pattern induced flicker colors (PIFCs)] were matched with color stimuli provided by a computer aided color mixer. Subjects viewed a series of specifically modified black and white disks and matched the resulting subjective color with a comparison field containing the color generated by additive mixing. Different phase relations between the apparently colored ring and the surround were tested. The color loci of all PIFCs were found to lie on a plane in receptor three-space which is given by the axis of the shortwave receptor excitation and a vector given by combining the middle and long wave receptor excitation directions in a fixed ratio of nearly 1:1. From the orientation of this plane it can be deduced that the blue-yellow opponent process (the blue-on-center cells) alone accounts for the different PIFCs.

Color induction via non-opponent lateral interactions in the human retina

Retinal connections causing colors in Benham's top (pattern induced flicker colors, PIFCs) are investigated by psychophysical experiments. PIFCs are still seen when stimuli to different cones are demodulated selectively, indicating the involvement of non-opponent channels. PIFCs also occur on retinal areas next to those affected by modulated stimuli; further, both monochromat and dark-adapted trichromats perceive PIFCs which are achromatic. These additional findings point to horizontal cells as neuronal mediators of modulated excitation leading to PIFCs. The unspecifity of the postulated connection with respect to cone types agrees with anatomic findings of Boycott, B. B., Hopkins, J. M. and Sperling, H. G. (1987, Proceedings of the Royal Society of London, B, 229, 345-379) on horizontal cells.

A survey of transforms of the CIE 1931 chromaticity diagram with some new non-linear transforms and histological illustrations of their utility

Surveys are presented of the published data on colour matching, and of the published transforms of the CIE 1931 chromaticity diagram. All transforms are given in the form of standardized equations and tables of coefficients. Both linear and non-linear transforms are considered, although the latter are given special attention. The concept of a uniform chromaticity space (UCS) is discussed, and the published transforms are assessed for uniformity using the colour matching data. 2 series of new, non-linear transforms are presented, both based on 2-dimensional polynomial equations, and both derived using iterative optimization techniques. The 1st series attempts to approximate Farnsworth's (1961) diagram. The 2nd uses Nutting's colour matching points directly. The coefficients of these transforms are tabulated. It was found that increasing the number of coefficients in the 2-dimensional polynomial, beyond 15 each for x and y, does little to increase the uniformity of the resulting transform. The usefulness of UCS's in assessing stain performance is illustrated by examples drawn from the areas of azure B/eosin staining of blood cells and Papanicolaou staining of epithelial cells from the uterine cervix.

Geodesic chromaticity diagram based on variances of color matching by 14 normal observers

A nonlinear transformation of the CIE x,y chromaticity coordinates has been derived from the combined color-matching-variance data of 14 normal observers. In the resulting diagram, the series of equiluminous chromaticities entailing the least number of standard deviations of color matching (sigma-units) between any two-terminal, equiluminous chromaticities is the straight line drawn between the points that represent those terminal colors. The total number of sigma-unit differences between those terminal colors is the euclidean distance between those two points. According to Schrödinger's hypothesis, the loci of constant hue are the straight lines (geodesics) radiating from the point that represents hueless colors in this diagram. The horizontal coordinate in the geodesic chromaticity diagram is xi = 3751a(2) - 10a(4) - 520b(2) + 13295b(3) + 32327ab - 25491a(2)b - 41672ab(2) + 10a(3)b - 5227a((1/2)) + 2952(4)a((1/4)), where a = 10x/(2.4x + 34y + 1) and b = 10y/(2.4x + 34y + 1). The vertical coordinate in the geodesic chromaticity diagram is eta = 404b - 185b(2) + 52b(3) + 69a(1 - b(2)) - 3a(2)b + 30ab(2), where a = 10x/(4.2y - x + 1) and b = 10y/(4.2y - x + 1). These formulas were obtained by use of averages of data for two observers whose individual data were published in 1949 and the weighted averages for 12 young observers, which were published in 1957, together with the data for the single observer, PGN, whose data were published in 1942-45. On the basis of extensive studies of these data, the PGN data were assigned 30% weight in the derivation of the new xi,eta diagram. The 1949 data were assigned 44% weight, or 22% per observer, and the 1957 data were assigned 26%, or about 2.2% per observer. The best fit was found by assuming that the over-all mean of the standard deviation of color matching according to the 1949 data was 1.2 times as much as the standard deviation for PGN, and that the weighted-mean standard deviation for the 12 observers was 1.04 times the standard deviation for PGN. When adjusted to this basis, the radii of the variance ellipses for the three sets of observations fit unit distance on the xi,eta diagram with a mean-square error of 0.056. The mean-square error for the PGN data is 0.052, which may be compared with 0.02 for a version of the xi,eta diagram that was derived from the PGN data alone. The mean-square deviation from unit distance of the 1949 average data is 0.053, and for the 1955 weighted-average ellipses it is 0.076.