On the dichromatic object-colour palette.

The object- and light-colour palettes prove to be different for both trichromats and dichromats. This explains why there is no consensus on what colours dichromats see, since, until now, studies of dichromatic vision have mainly focused on the light-colour palette. By contrast, this study concentrates on the dichromatic object-colour palette, assuming that it is as much determined by optimal reflectances as the trichromatic palette. In this case, the dichromatic object-colour palette is simply part of the trichromatic object-colour palette. This is a consequence of the fact that the dichromatic optimal reflectances bring about identical perceptions in both dichromats and trichromats. Since the optimal reflectances cannot be physically implemented, a set of Munsell chips was selected that was close enough to the dichromatic optimal reflectances. By examining these chips, trichromats can get an idea of what the dichromatic object-colour palette looks like. These chips clearly contain red, green and blue component hues. As to green, it was tinged with such a considerable amount of white that it was hard to judge its presence even for trichromatic observers. By hue scaling, the amount of component hues (Y, B, R, G, W and Bk) that trichromats see in these chips was evaluated. Although the amount of green was found to be low, its presence for some chips was statistically significant. Thus, dichromats should see all six component hues. Also, the opponency of black and white was confirmed, which contradicts the generally accepted view that grey is a mixture of black and white.

On Counting Metamers.

Objects reflecting lights, which invoke the identical responses of the photoreceptors, are called metameric. Metameric objects match each other in color. Assuming all the reflecting objects equiprobable, the probability density distribution on the set of all the classes of mutually metameric objects is evaluated. Metamerism depends on illumination. Objects metameric under one illumination might produce different photoreceptor responses (i.e., mismatch each other) under the other illumination. In particular, objects mapping to a single point in the photoreceptor response space under one illumination map to a volume (referred to as the metamer mismatch volume) in the photoreceptor response space under the second illumination. The probability density distribution in the metamer mismatch volume is also evaluated. Its center is expressed as a function of the photoreceptor spectral sensitivities and the spectral power distributions of the illuminants. Potential application of the obtained results to color management is discussed.

The Achromatic Object-Colour Manifold is Three-Dimensional.

When in shadow, the achromatic object colours appear different from when they are in light. This immediate observation was quantitatively confirmed by Logvinenko and Maloney (2006, Perception & Psychophysics, 68, 76-83) who, using multidimensional scaling (MDS), showed the two-dimensionality of achromatic object colours. As their experiments included only cast shadows, a question arises: is this also the case for attached shadows? Recently, Madigan and Brainard (2014) argued in favour of the negative answer. However, they also failed to confirm the two-dimensionality for cast shadows. To resolve this issue, an experiment was conducted in which observers rated the dissimilarity between achromatic Munsell chips presented in light and in shadows of both types. Specifically, the chips were presented in four conditions: in front in light; at slant in light; in front in shadow; and at slant in shadow. MDS analysis of the obtained dissimilarities confirmed the two-dimensionality of achromatic colours for both types of shadow. Furthermore, the dimension induced by the cast shadow (shadowedness) was found to be different from that induced by the attached shadow (shading). In the three-dimensional MDS output configuration these were represented by clearly different dimensions. This quantitatively supports a fact, well-known to artists, that attached and cast shadows are experienced as different phenomenological entities. It is argued that a shading gradient is perceptually experienced as shape (ie spatial relief).

The color cone.

While the notion of a color cone can be found in writings of Maxwell, Helmholtz, Grassmann, and other scientists of the nineteenth century, it has not been clearly defined as yet. In this paper, the color cone is understood as the set of points in the cone excitation space produced by all possible lights. The spectral curve representing all the monochromatic lights is shown not to entirely belong to the color cone boundary, since its ends turn into the color cone interior. The monochromatic lights represented by the fragment of the spectral curve lying on the color cone boundary make up what is called the effective visible spectrum. The color cone is shown to be a convex hull of the conical surface through the fragment of the spectral curve representing the effective visible spectrum. The effective visible spectrum ends are shown to be determined by the photopigment spectral absorbance being independent of the prereceptor filters (e.g., the spectral transmittance of the lense and macular pigment).

Rethinking Colour Constancy.

Colour constancy needs to be reconsidered in light of the limits imposed by metamer mismatching. Metamer mismatching refers to the fact that two objects reflecting metameric light under one illumination may reflect non-metameric light under a second; so two objects appearing as having the same colour under one illuminant can appear as having different colours under a second. Yet since Helmholtz, object colour has generally been believed to remain relatively constant. The deviations from colour constancy registered in experiments are usually thought to be small enough that they do not contradict the notion of colour constancy. However, it is important to determine how the deviations from colour constancy relate to the limits metamer mismatching imposes on constancy. Hence, we calculated metamer mismatching's effect for the 20 Munsell papers and 8 pairs of illuminants employed in the colour constancy study by Logvinenko and Tokunaga and found it to be so extensive that the two notions-metamer mismatching and colour constancy-must be mutually exclusive. In particular, the notion of colour constancy leads to some paradoxical phenomena such as the possibility of 20 objects having the same colour under chromatic light dispersing into a hue circle of colours under neutral light. Thus, colour constancy refers to a phenomenon, which because of metamer mismatching, simply cannot exist. Moreover, it obscures the really important visual phenomenon; namely, the alteration of object colours induced by illumination change. We show that colour is not an independent, intrinsic attribute of an object, but rather an attribute of an object/light pair, and then define a concept of material colour in terms of equivalence classes of such object/light pairs. We suggest that studying the shift in material colour under a change in illuminant will be more fruitful than pursuing colour constancy's false premise that colour is an intrinsic attribute of an object.

How metamer mismatching decreases as the number of colour mechanisms increases with implications for colour and lightness constancy.

Metamer mismatching has been previously found to impose serious limitations on colour constancy. The extent of metamer mismatching is shown here to be considerably smaller for trichromats than for dichromats, and maximal for monochromats. The implications for achromatic colour perception are discussed.

The geometric structure of color.

Color is commonly described in terms of the three perceptual attributes-hue, saturation, and brightness-of which only hue has a qualitative nature, saturation and brightness being of a quantitative nature. A possible reason for such a phenomenological structure of the color manifold, and its geometric representation, are discussed.

Unique hues as revealed by unique-hue selecting versus partial hue-matching.

Unique hues are usually defined as those that cannot be introspectively reduced to any other hue. According to a major dogma of color science, there are four unique hues: yellow, blue, red, and green. Yet only 55 of the 173 inexperienced observers who participated in our experiment selected exactly four Munsell papers that, according to their judgment, had a unique hue. The number of papers selected by the rest of the observers varied from zero to nine. We believe that such variability of unique hue selection is due to the ambiguity of the introspective criteria for hue uniqueness. Along with the traditional technique of unique hue selection, an alternative method based on partial hue-matching has also been used to establish the nomenclature of unique hues. The partial hue-matching method is based on observer judgments concerning the presence of a common hue in a pair of colors. Observers are not supposed to name (or make any other judgments of) this common hue. Without presupposing their number, the unique hues are derived from the observer's responses to a sample of color pairs. The results obtained by this new method generally support the classical notion of four unique hues.

Metamer mismatching.

A new algorithm for calculating the metamer mismatch volumes that arise in colour vision and colour imaging is introduced. Unlike previous methods, the proposed method places no restrictions on the set of possible object reflectance spectra. As a result of such restrictions, previous methods have only been able to provide approximate solutions to the mismatch volume. The proposed new method is the first to characterize precisely the metamer mismatch volume for any possible reflectance.

Color constancy investigated via partial hue-matching.

Each hue is believed to be made up of the four component hues (yellow, blue, red, and green). A hue consisting of just one component hue is called unitary (or unique). A new technique--partial hue-matching--has been used to reveal the component and unitary hues for a sample of 32 Munsell papers, which were illuminated by neutral, yellow, blue, green, and red lights and assessed by four normal trichromatic observers. The same set of four component hues has been found under both the neutral and the chromatic illuminations for all of the observers. On average, more than 87% of the papers containing a particular component hue under the neutral illumination also have this component hue when lit by the chromatic lights. However, only a quarter of the papers perceived as unitary under the neutral illumination continues being perceived as unitary under all of the chromatic illuminations. In other words, most unitary colors shift along the hue circle due to change in an illuminant's chromaticity. Still, this shift of unitary colors is relatively small: On average, it does not exceed one Munsell hue step.

A test for psychometric function shift.

A nonparametric, small-sample-size test for the homogeneity of two psychometric functions against the left- and right-shift alternatives has been developed. The test is designed to determine whether it is safe to amalgamate psychometric functions obtained in different experimental sessions. The sum of the lower and upper p-values of the exact (conditional) Fisher test for several 2 × 2 contingency tables (one for each point of the psychometric function) is employed as the test statistic. The probability distribution of the statistic under the null (homogeneity) hypothesis is evaluated to obtain corresponding p-values. Power functions of the test have been computed by randomly generating samples from Weibull psychometric functions. The test is free of any assumptions about the shape of the psychometric function; it requires only that all observations are statistically independent.

Colour constancy as measured by least dissimilar matching.

Although asymmetric colour matching has been widely used in experiments on colour constancy, an exact colour match between objects lit by different chromatic lights is impossible to achieve. We used a modification of this technique, instructing our observers to establish the least dissimilar pair of differently illuminated coloured papers. The stimulus display consisted of two identical sets of 22 Munsell papers illuminated independently by neutral, yellow, blue, green and red lights. The lights produced approximately the same illuminance. Four trichromatic observers participated in the experiment. The proportion of exact matches was evaluated. When both sets of papers were lit by the same light, the exact match rate was 0.92, 0.93, 0.84, 0.78 and 0.76 for the neutral, yellow, blue, green and red lights, respectively. When one illumination was neutral and the other chromatic, the exact match rate was 0.80, 0.40, 0.56 and 0.32 for the yellow, blue, green and red lights, respectively. When both lights were chromatic, the exact match rate was found to be even poorer (0.30 on average). Yet, least dissimilar matching was found to be rather systematic. Particularly, a statistical test showed it was symmetric and transitive. The exact match rate was found to be different for different papers, varying from 0.99 (black paper) to 0.12 (purple paper). Such a variation can hardly be expected if observers' judgements were based on an illuminant estimate. We argue that colour constancy cannot be achieved for all the reflecting objects because of mismatching of metamers. We conjecture that the visual system might have evolved to have colour constant perception for some ecologically valid objects at a cost of colour inconstancy for other types of objects.

Partial hue-matching.

It is widely believed that color can be decomposed into a small number of component colors. Particularly, each hue can be described as a combination of a restricted set of component hues. Methods, such as color naming and hue scaling, aim at describing color in terms of the relative amount of the component hues. However, there is no consensus on the nomenclature of component hues. Moreover, the very notion of hue (not to mention component hue) is usually defined verbally rather than perceptually. In this paper, we make an attempt to operationalize such a fundamental attribute of color as hue without the use of verbal terms. Specifically, we put forth a new method--partial hue-matching--that is based on judgments of whether two colors have some hue in common. It allows a set of component hues to be established objectively, without resorting to verbal definitions. Specifically, the largest sets of color stimuli, all of which partially match each other (referred to as chromaticity classes), can be derived from the observer's partial hue-matches. A chromaticity class proves to consist of all color stimuli that contain a particular component hue. Thus, the chromaticity classes fully define the set of component hues. Using samples of Munsell papers, a few experiments on partial hue-matching were carried out with twelve inexperienced normal trichromatic observers. The results reinforce the classical notion of four component hues (yellow, blue, red, and green). Black and white (but not gray) were also found to be component colors.

Lightness constancy and illumination discounting.

Contrary to the implication of the term "lightness constancy", asymmetric lightness matching has never been found to be perfect unless the scene is highly articulated (i.e., contains a number of different reflectances). Also, lightness constancy has been found to vary for different observers, and an effect of instruction (lightness vs. brightness) has been reported. The elusiveness of lightness constancy presents a great challenge to visual science; we revisit these issues in the following experiment, which involved 44 observers in total. The stimuli consisted of a large sheet of black paper with a rectangular spotlight projected onto the lower half and 40 squares of various shades of grey printed on the upper half. The luminance ratio at the edge of the spotlight was 25, while that of the squares varied from 2 to 16. Three different instructions were given to observers: They were asked to find a square in the upper half that (i) looked as if it was made of the same paper as that on which the spotlight fell (lightness match), (ii) had the same luminance contrast as the spotlight edge (contrast match), or (iii) had the same brightness as the spotlight (brightness match). Observers made 10 matches of each of the three types. Great interindividual variability was found for all three types of matches. In particular, the individual Brunswik ratios were found to vary over a broad range (from .47 to .85). That is, lightness matches were found to be far from veridical. Contrast matches were also found to be inaccurate, being on average, underestimated by a factor of 3.4. Articulation was found to essentially affect not only lightness, but contrast and brightness matches as well. No difference was found between the lightness and luminance contrast matches. While the brightness matches significantly differed from the other matches, the difference was small. Furthermore, the brightness matches were found to be subject to the same interindividual variability and the same effect of articulation. This leads to the conclusion that inexperienced observers are unable to estimate both the brightness and the luminance contrast of the light reflected from real objects lit by real lights. None of our observers perceived illumination edges purely as illumination edges: A partial Gelb effect ("partial illumination discounting") always took place. The lightness inconstancy in our experiment resulted from this partial illumination discounting. We propose an account of our results based on the two-dimensionality of achromatic colour. We argue that large interindividual variations and the effect of articulation are caused by the large ambiguity of luminance ratios in the stimulus displays used in laboratory conditions.

Hue manifold.

It is generally accepted that hues can be arranged so as to make a circle. The circular representation of hue has been supported by multidimensional scaling, which allows for the representation of a set of colored papers as a configuration in a Euclidean space where the distances between the papers correspond to the perceptual dissimilarities between them. In particular, when papers of various hues are evenly illuminated, they are arranged in a one-dimensional circular configuration. However, under variegated illumination we show that the same papers in fact make a two-dimensional configuration that resembles a torus.

Material and lighting hues of object colour.

Observers can easily differentiate between a pigmented stain and the white surface that it lies on. The same applies for a colour shadow cast upon the same surface. Although the difference between these two kinds of colour appearance (referred to as material and lighting hues) is self-evident even for inexperienced observers, it is not one that has been captured by any colour appearance model thus far. We report here on an experiment supplying evidence for the dissociation of these two types of hue in the perceptual space. The stimulus display consisted of two identical sets of Munsell papers illuminated independently by yellow, neutral, and blue lights. Dissimilarities between all the paper/light pairs were ranked by five trichromatic observers, and then analysed by using non-metric multidimensional scaling (MDS). In the MDS output configuration, the Munsell papers lit by the same light made a closed configuration retaining the same order as in the Munsell book. The paper configurations for the yellow and blue lights were displaced transversally and in parallel to each other, with that of the neutral light located in between. The direction of the shift is interpreted as the yellow-blue lighting dimension. We show that the yellow-blue lighting dimension cannot be reduced to that of the reflected light.

Material and lighting dimensions of object colour.

The dimensionality of the object colour manifold was studied using a multidimensional scaling technique, which allows for the representation of a set of coloured papers as a configuration in a Euclidean space where the distance between papers corresponds to the perceptual dissimilarities between them. When the papers are evenly illuminated they can be arranged as a three-dimensional configuration. This is in line with the generally accepted view that the object colour space is three-dimensional. Yet, we show that under variegated illumination another three dimensions emerge. We call them lighting dimensions of object colour in order to distinguish from the traditional three referred to as material dimensions of object colour.

An object-color space.

Putting aside metaphorical meanings of the term, color space is understood as a vector space, where lights having the same color (i.e., subjectively indistinguishable) are represented as a point. The CIE 1931 color space, empirically based on trichromatic color measurements, is a classical example. Its derivatives, such as CIELAB and sRGB, have been successfully used in many applications (e.g., in color management). However, having been designed for presenting the color of self-luminous objects, these spaces are less suitable for presenting color of reflecting objects. Specifically, they can be used to represent color of objects only for a fixed illumination. Here I put forward a color space to represent the color of objects independently of illumination. It is based on an ideal color atlas comprising the reflectance spectra taking two values: k or 1 - k (0 < or = k < or = 1), with two transitions (at wavelengths lambda(1) and lambda(2)) across the spectrum. This color atlas is complete; that is, every reflecting object is metameric to some element of the atlas. When illumination alters, the classes of metameric reflectance spectra are reshuffled but in each class there is exactly one element of the atlas. Hence, the atlas can uniquely represent the metameric classes irrespective of illumination. Each element of the atlas (thus, object color) is specified by three numbers: (i) lambda = (lambda(1) + lambda(2))/2, which correlates well with hue of object color (as dominant wavelength correlates with hue of light color); (ii) delta =/lambda(1) - lambda/, which correlates with whiteness/blackness; and (iii) alpha =/1 - 2k/, which correlates with chroma of object color (as colorimetric purity correlates with saturation of light color). Using a geographical coordinate system, each element of the atlas (thus, each object color) is geometrically represented as a radius vector so that its length equals alpha, the latitude and longitude being proportional to delta and lambda, respectively.

Dissimilarity of yellow-blue surfaces under neutral light sources differing in intensity: separate contributions of light intensity and chroma.

Observers viewed two side-by-side arrays each of which contained three yellow Munsell papers, three blue, and one neutral Munsell. Each array was illuminated uniformly and independently of the other. The neutral light source intensities were 1380, 125, or 20 lux. All six possible combinations of light intensities were set as illumination conditions. On each trial, observers were asked to rate the dissimilarity between each chip in one array and each chip in the other by using a 30-point scale. Each pair of surfaces in each illumination condition was judged five times. We analyzed this data using non-metric multi-dimensional scaling to determine how light intensity and surface chroma contributed to dissimilarity and how they interacted. Dissimilarities were captured by a three-dimensional configuration in which one dimension corresponded to differences in light intensity.