Figure 5.27 A photograph of a plexiglass model of the object color solid.

plane [Figure 5.26(c)]. Some advantages of this color solid over the CIE space in representing psychophysical correlates have been discussed elsewehere (Valberg, 1981).

Color differences

Hue and saturation in the (x, y) diagram

Figure 5.28 displays the CIE chromaticities of the basic chromatic attributes hue and chroma of equal luminance factor of the American Munsell color ordering system. The slightly curved lines radiating from the white point represent the chromaticities of colors of constant hue, varying in chroma. Since a constant hue follows a curved, rather than a straight line, this is an indication that the process determining hue, a perceived attribute, is nonlinearly related to cone excitations. The ellipse-like forms encircling the white point characterize stimuli with the same chroma, or the same color difference from white.

Let us take the hue 5R, represented by a solid curve in Figure 5.29. This hue in the Munsell system comes close to what most people regard as an elementary, or unique red, being ‘neither yellowish nor bluish’. Additive mixtures between a color of maximum chroma and white are in the diagram situated on the straight dashed line connecting these chromaticities, whereas constant perceived hue follows the solid curve. Thus the hue of the mixture of red with white becomes bluer as more white is added, until it reaches a maximum blueness somewhere between the red and white, after which the blueness decreases. This change of perceived hue of an additive

Figure 5.28 Constant hue and chromatic strength (Munsell hue and chroma) for medium reflectance (value 5) plotted in the CIE chromaticity diagram. Color stimuli that have the same hue, but different chroma, are situated on curves radiating from the white point. Equal chromatic differences from white give ellipse-like loci about the white point. The chroma steps are 4, 8, 12, etc.

mixture with white is called the ‘Abney effect’, and it is common for all hues (except maybe for hues close to 5GY and 5P; see also Figure 5.28). For instance, a yellow– green spectral color becomes greener in an additive mixture with white. Later we shall associate such peculiarities with nonlinear responses of the opponent cells beyond the stage of receptor excitations and receptor potentials.

Color scales

When comparing color differences, it is difficult for two persons to reach agreement on what are equal differences. In color science the specification of color differences is a recurrent topic of discussion. Take two clearly different nuances of orange, for example. What is the same color difference for two purple nuances, or between two different greens? Problems of comparison and a simple quantification of color differences (in terms of larger than, equal to or smaller than) occur whenever the color of a product must be reproduced consistently over a period of time in the manufacture of paints, textiles, ceramic tiles, etc. These are examples of cases where the tolerances for deviations from a standard must be quantified. Most of us find it particularly hard to compare color differences if differences in hues are involved, and

Figure 5.29 A section of the chromaticity diagram of Figure 5.28. A solid curve starting at the white point end ending at the spectrum locus represents chromaticities that all have the same hue. The hue curves deviate from the dashed straight lines which represent additive color mixtures of the spectrum colors with white. Let us take the red elementary color 5R of maximum purity as an example: in a mixture with white, the color will become increasingly bluish, depending on how much white is added. Yellow–green colors will become greener when white is added to them. Only the yellow color 10Y and violet 5P seem to retain their hues when mixed with white.

the difficulty increases with increasing difference. Who is able to say what difference between a red and a green is equal to a difference between a certain yellow and a certain blue? Isn’t this a bit like comparing apples and oranges? Nor is it easy to compare even a small step of grays or saturations with a difference in hue. Nevertheless, industry needs objective measures of color differences and quantitative specifications of tolerances across the different color dimensions. This is of particular importance in color reproduction, and it is also required in the quality control of food, in color TV, for color displays in general, lighting engineering, etc.

It has been suggested that psychophysical thresholds, i.e. the just noticeable difference of a repeated color match, can be used as a unit for color difference, and that larger color differences can be characterized by the number of threshold steps they contain. This hypothesis stems from Weber and Fechner’s laws (see the section on ‘Contrast Vision’ in Chapter 4, p. 196), and it is analogous to saying that a distance of 1 cm always equals 10 mm. The problem with this idea is nonlinearity, as with scales of lightness: 10 thresholds do not lead to the same perceived difference everywhere in color space. This is an empirical fact contradicting the

conjecture that it is possible to summate thresholds to get a predictable finite step. However, in some cases, and because of the lack of an alternative, summation can be a useful approximation.

Based on empirical judgements of color differences by many individuals, atlases have been constructed that visualize a color scaling system for the entire color space. The Munsell system (USA) is probably the best known. Although far from perfect, this system is based on extensive scaling experiments with many observers. In the past it has often been used as a ‘yard-stick’ in quality control. It has also served as a basis for the development of quantitative formulae of color differences. Such formulae are the CIE-Lab and CIE-Luv formulae (Wyszecki and Stiles, 1982; CIE, 1970). A review of old and new formulae for color scaling can be found in Richter (1996).

The need for fast and reproducible control procedures has led to the development of standardized instruments to specify color differences. However, such instruments often lack the desired precision. This has two main causes: (a) the failure of the three light sensors to match the standardized CIE spectral sensitivity curves Xð Þ, Yð Þ, and Zð Þ, and (b) the use of questionable mathematical model approximations to convert differences in tristimulus values to perceived color differences.

Color difference discrimination

MacAdam (1942) made a study of the precision of color matches, and showed that, for a number of repeated matches, the chromaticities of the matching stimuli were distributed on an ellipse around the targeted chromaticity in the ðx; yÞ diagram, These so-called MacAdam ellipses varied in size across the diagram, as shown in Figure 5.30 (here magnified 10 times). This means that equally large color differences are reproduced in the diagram as different lengths, depending on the target chromaticity and on the direction from the target. The ellipses are rather small in the blue and violet corners of the diagram and large in the green region. Later works demonstrated that these ellipses were cross-sections through three-dimensional ellipsoids in XYZspace.

Because of its importance in practical applications, many attempts have been made over the years to develop good empirical formulae for color differences. They have, however, met with little success. One requirement for a good formula has been that it describes the scaling of color differences in the Munsell system. Another system against which to test color difference formulations is the American Optical Society Uniform Color Scale (OSA-UCS; see Wyszecki and Stiles, 1982).

Most descriptions of small and large color differences have been based directly on mathematical manipulations of cone excitations, or of the XYZ tristimulus values. Relatively few attempts have been made to utilize knowledge about the physiological processing of color information later in the nervous system. One of the earliest attempts in this direction was that of Adams (1942). In want of something better, he

Figure 5.30 MacAdam’s discrimination ellipses in the CIE ðx; yÞ diagram, magnified 10 . The ellipses represent the uncertainty (standard deviation) of repeated matches of the center color. They are therefore taken to represent the same sensory difference from the center color in all directions.

used nonlinear transformations of X, Y and Z in an opponent formulation of color processing. A more recent attempt in this direction was the SVF formula developed by Seim and Valberg (1986). The magnitudes S, V and F represent the three most significant steps in the process; the linear excitations S of the three cone types, the nonlinear receptor potentials V of the same cones, and the responses F of opponent cells and their linear combinations. The introduction of nonlinear intensity–response curves for each of the three cone types was the most important feature of the SVF model. All later steps in the model’s signal transmission were assumed to be linear. These assumptions were later confirmed by a mathematical modeling of the responses of opponent ganglion cells in the retina and in the lateral geniculate nucleus (LGN) of the macaque monkey (Valberg et al., 1986a; see Chapter 6).

These responses were obtained from recordings where color stimuli were exchanged with a neutral, white background. This viewing situation resembles the way in which object colors are seen in nature, related as they are to near and far surrounds that determine the adaptation level of the eye, and with eye movements introducing a temporal component. These cell recordings also showed that the opponent ONand OFF-cells (here called Increment and Decrement cells; I- and D-cells) divided the luminance range between them, with the opponent D-cells responding to darker colors than the I-cells. In fact, opponent D-cells are so strongly inhibited that they are silenced

for a luminance just slightly higher than the adaptation luminance of a white stimulus, and therefore they cannot code for color of such stimuli. The obvious next step was to develop a model adding the responses of I- and D-cells with the same opponent cone input (in other parallel channels, however, these cells should maintain their independence). As we shall see later, a vector treatment of cells with different opponency, for instance of ‘M–L’ and ‘S–L’ cells, provides reasonably good correlates to such properties of colors as the Abney effect, the Bezold–Bru¨cke phenomenon, and equidistant color scales.

A comparison of the early SVF formula with the CIE-Lab formula showed a clear advantage of the former in describing the data of the Munsell system. In Figure 5.31

Figure 5.31 A plane defined by the achromatic axis of the Munsell system and the line through the opposing hues 10PB and 10Y. The parallel vertical lines represent an ideal scaling of chroma in steps of two chroma units up to chroma 12. The SVF formula, using nonlinear intensity–response functions for the cone signals, comes closer to modeling an ideal spacing than does the empirical CIE formula. Solid symbols are for the experimental data and open circles for interpolated and extrapolated data.

one sees the chroma scaling (abscissa) of the two representations for two opposite hues 10Y and 10PB and value V 1–9. The chroma scales in these two opposite hue directions are more equidistant for SVF than when using CIE-Lab (the vertical bars indicate the targeted Munsell scaling). In Figures 5.32(a) and (b) the coordinates of the color chips of the Munsell system (for value 5) are compared in CIE-Lab space

b

a

f2

f1

(c)

F2

F1

Figure 5.32 Munsell hue and chroma for value 5, as modeled by (a) the CIE formula, (b) the SVF formula, and (c) a neural model of color vision to be presented later. The chroma steps are 2, 4, 6, . . ., 12. Solid symbols represent the original data of Munsell scaling, whereas the open circles are for inerpolated and extrapolated data. Compared with (a), the scaling of hue and chroma is more uniform in (b) and (c). In the neural model used to obtain (c), the scaling of Munsell hue and chroma is based upon a mathematical transformation ðF1; F2Þ of a combination of six opponent cell types found in recordings from the LGN of the macaque monkey. In this model, the hue steps around the hue circle are more evenly spaced and the contours for equal chroma are more radial symmetric than for the empirical CIE formula in (a).

and in SVF-space. Figure 5.32(c) shows the result of the physiological model mentioned above, which was based on our recordings from macaque opponent cells. In this model, to be described later, representative opponent cells are the actual response units.

A calculation of deviations between an ‘ideal’ Munsell representation and the coordinates of the CIE-Lab, the SVF-system and the cell response coordinates (F1, F2) of the physiological model, demonstrates that the two latter representations of Figures 5.32(b) and (c) are both better than the empirical CIE-Lab representation in Figure 5.32(a). Thus, major improvements of the quantitative formulation of color differences can be achieved by taking into account the nonlinear response curves of cone receptors and their subtraction in the cone opponent cells. The result of Figure 5.33,

Figure 5.33 A representation of the chromaticities of the Uniform Color Scale of the Optical Society of America (OSA-UCS) in terms of the coordinates ðF1; F2Þ of the neural color vision model also used in Figure 5.32(c). The original data of OSA-UCS have equal perceptive steps sizes horizontally and vertically, and this is very well rendered by the neural model. The reflection factor for these colors were near Y ¼ 30% (from Valberg et al., 1986a).

showing how the physiological model yields a close to perfect scaling of a plane in the OSA-UCS system, proves the usefulness of this approach. These results lend strong support to the idea that cone nonlinearity and a simple subtractive opponent processing is a better model for color scaling than using Weber ratios of cone receptors and combining them to form line elements (see below).

However, the theory of line elements has played an important role in developing a physical measure of color difference and contrast, and in the next section we shall give a brief description of this approach.

Line elements

Although it has not been very successful, the theory of line elements has played an important historical role in attempts to describe small color differences. Von Helmholtz was the first to refer the Weber law to the relative excitation of the three cone receptors, stating that the Weber fraction was constant at threshold for each of them and independent of cone excitation.

In a three-dimensional vector space (u1, u2, u3), a small distance, ds, the length of a line element, is found by (see Figure 5.34):

ðdsÞ2 ¼ ðdu1Þ2 þ ðdu2Þ2 þ ðdu3Þ2

u2

du2

ds

du1 u1

du3

u3

Figure 5.34 The definition of a line element as the distance ‘ds’ between two vectors in the three-dimensional vector space with the basic vectors u1, u2 and u3.

The three vector segments du1, du2 and du3 may be regarded as increments or decrements of the cone excitations L, M and S, but this is not the only possibility. One might also imagine them as the response differences of three independent channels at a later stage of visual processing, for example of one luminance channel and two opponent chromatic channels. If we imagine the three vectors (u1, u2, u3) in Figure 5.34 as those of cone fundamentals L, M and S, we can write

ðdsÞ2 ¼ ðdLÞ2 þ ðdMÞ2 þ ðdSÞ2

However, according to Weber’s law, it is the differential excitation, dL divided by excitation, L, and not dL alone, that is constant at threshold. If the excitation is increased from L to 2L, the increment dL must be increased to 2dL to maintain threshold visibility (but as we have seen, this is restricted to low spatial and temporal frequencies of the stimulus; see Figures 4.24 and 4.28). Such considerations led von

(L þ M) and the chrominance direction (L M). Noorlander and Koenderink (1983) determined the difference thresholds for combinations of red–green contrast and luminance contrast when the spatial and temporal frequencies were changed. Figure 5.35 shows a schematic example for a large field of how the discrimination ellipses depend on temporal frequency. For a large stationary stimulus and for slowly varying stimuli, the discrimination ellipse is oriented with the long axis along the luminance

M

>10 Hz

Y (L+M )

1 Hz

L

4 Hz

8 Hz

F1 (L − M )

Figure 5.35 A schematic drawing of the contours for discrimination thresholds in different directions of color space with respect to white and for different temporal frequencies. The major axes for the threshold ellipses represent directions of pure luminance change ðYÞ or isoluminant red–green changes ðF1Þ. Stationary and slowly changing stimuli ( 1 Hz) give best discrimination (lowest threshold) for color (chrominance) and least discrimination (highest threshold) for luminance. This causes the long axis of the ellipse to be orientated along the luminance direction Y ¼ L þ M. As frequency increases, the luminance threshold decreases, whereas the chromatic threshold for isoluminant stimuli increases. This results in a circular threshold contour around 4 Hz. From 5 Hz and up to about 7–8 Hz, the luminance threshold changes little, whereas chrominance threshold increases significantly, resulting in a major axis (low sensitivity) oriented along the chrominance direction L–M. For still higher frequencies (>10 Hz), both luminance and chrominance thresholds increase (after Noorlander and Koendrink, 1983).

direction Y ¼ L þ M and the short axis along the chrominance direction F1 ¼ L M . Thus, discriminability is less for luminance than for chrominance. As the temporal frequency increases, the long axis becomes shorter, and around 4–5 Hz the ellipse resembles a circle. If the temporal frequency is increased further, chrominance discrimination becomes poorer and the axis increases in the red–green chrominance direction F1 ¼ L M . The transition from a circle to having a long axis in the chrominance direction takes place between 5 and 8 Hz for large fields. Above 8 Hz, discrimination of both luminance and chrominance becomes increasingly poorer.

If the same holds true for the yellow–blue opponent direction, one would expect to obtain a better line element by introducing orthogonal luminance and chrominance channels:

ðdsÞ2 ¼ aðdY = Y Þ2 þ bðdF1= F 1Þ2 þ c ðdF2= F 2Þ2

where F1 is a red–green opponent process and a function of L M , and F2 is a yellow– blue process that may be a function of the cone combination ðL þ M Þ S. is the uncertainty in Y, F1 and F2. For high luminances, is proportional to Y, F1 and F2 as in the Weber ratio. a, b and c are weighting factors that depend on experimental parameters.

In the experiment described in Figure 5.35, the discrimination was made along a line where F2 (yellow–blue) was constant. We will come back to Figure 5.35 later when we compare the responses of opponent cells with psychophysical results.

Combined luminance and chrominance contrast

Line elements were early attempts to find a common, combined measure for contrasts that included differences in luminance as well as in chrominance. However, experiments have shown that there is no simple way to design a psychophysical ‘yard stick’ for contrast that can be used anywhere in color space. No known formula is able to give a satisfactory account of equivalent luminance and chromatic differences. For a given stimulus size, perceptive color differences can be empirically characterized in the CIE space, as MacAdam did by plotting discrimination ellipses and ellipsoids. However, there is no obvious and direct way to model these differences mathematically, and there is no measure that can directly translate a given luminance contrast into an equivalent chromatic difference, or the other way around. There is no simple way to combine photometric and colorimetric magnitudes to get something that resembles perceptual scaling. Moreover, color and luminance discriminations are differently affected by spatial and temporal parameters, making the task even more complex.

It is, however, possible to use a simple measure of contrast, for example one that is linearly related to the physical processes of excitation. The nonlinear CIE Lab formula and the other empirical scaling formulae are not what we are thinking of, developed as they were from a series of ad hoc concepts and hypotheses. It appears that the closest one can come to a physical measure is to reconsider the excitations of the three types of cone. Threshold values of relative cone excitations are expressions of the effectiveness of a stimulus in evoking physiological and psychophysical response differences.

One simple possibility is the Helmholtzian approach – to combine the Weber ratios or the Michelson contrasts of the three cone excitations, with equal weights on all three cone types. This combination of cone contrasts should not be attached to a particular color vision model; it simply represents the last linear stimulus stage

before the generation of non linear signals in the receptors and in the neural pathways. A common measure for luminance and chrominance contrast would thus be to sum three quadratic contrast expressions, one for each cone type, divide by 3, and calculate the square root:

CWeber2 ¼ 1=3½ð L=LÞ2 þ ð M=MÞ2 þ ð S=SÞ2&

If the stimulus is periodic, as is the case for a sinusoidal grating, it is natural to replace the Weber ratio with the Michelson contrast for each cone type; for example CM ¼ ðMmax MminÞ=ðMmax þ MminÞ for the M-cone and similarly for the L- and S-cone:

CMich2 ¼ 1=3½CL2 þ CM2 þ CS2&

For a pure luminance increment or decrement, the contrast magnitude is the same in each of the three cone types and, since we divide by 3, the resulting combined contrast will be the same as if we used either the Weber or the Michelson contrast definitions directly on photometric luminance. Thus we have here a formula that leaves the luminance contrast definition unchanged, while allowing us to compute a combined contrast based on linear cone excitations. These two formulae treat the contrast, if it is in luminance or in chrominance, or a combination of both, as a distance in a three-dimensional cone vector space. However, nothing is said about the relationship between the contrast measures and perceived differences. We know already that identical contrast values, calculated by these equations for different cone excitations (and hence for different locations in LMS color space), do not correspond to a judgement of equal differences, even if spatial and temporal parameters are left unchanged. Therefore, as for the relationship between luminance and lightness, the relationships between cone excitations and the magnitude of a perceptual color difference must be worked out separately for different experimental conditions. Examples are shown in Figure 5.35 for temporal modulations, and in Figures 4.29 and 5.36 for changes of spatial frequency (size). Only for stationary sinusoidal gratings with a spatial frequency of about 2.5 c/deg, where the curves of Figure 5.36 cross, is the threshold, in terms of combined cone contrasts, the same for luminance as for red–green chrominance. For yellow–blue color differences, the chrominance and luminance thresholds are equal for a stimulus of about 0.6 c/deg (see Figure 4.29).

Color is what the eye sees best

For stationary sinusoidal gratings, Figure 5.36 displays human cone contrast sensitivities (1/CMich) as a function of spatial frequency for luminance and red– green chrominance gratings. The curves are the mean values of 11 and 10 subjects, respectively. Compared with a similar result for yellow–blue discrimination in

Spatial frequency (cycles/deg)

Figure 5.36 Spatial contrast sensitivity for the detection of a static sinusoidal grating modulated either in luminance or in red–green chromaticity at isoluminance. Contrast sensitivity is given as the inverse of combined cone contrasts at detection threshold. At low spatial frequencies, chromatic red–green contrast sensitivity is more than 10 times higher than pure luminance sensitivity (see text).

Figure 4.29, we see that cone contrast sensitivity is higher for red–green discrimination than for yellow–blue. At 0.2 c/deg, threshold sensitivity along the red–green dimension is about 1500. This is about five times higher than the maximum cone contrast sensitivity for pure luminance contrast (which has a maximum at 3.5 c/deg), and about 50 times higher than the luminance contrast sensitivity at 0.2 c/deg. The red–green sensitivity is about six times higher than for yellow–blue at 0.2 c/deg. However, since the curves do not seem to have yet reached a plateau at low spatial frequencies, we cannot exclude the possibility that the maximum sensitivities are even larger for still lower spatial frequencies.

These results confirm the contention that one needs less cone modulation to see pure chromatic differences than to see pure luminance differences (Mullen, 1985; Chaparro et al., 1993). At the cone level, the sensitivity for chrominance contrast is significantly greater than for luminance, and thus isoluminant chromatic stimuli are the more effective in creating a perceived difference. The dependence of discriminability on spatial and temporal parameters is related to the processing of cone signals in nerve cells at later stages.