Color difference thresholds in young's theory
Authors: Boynton Robert M. , Knoblauch Kenneth 
Published: June 2007

In the context of widely accepted trichromatic theory, two colors will differ in appearance unless they produce very nearly equal excitations of all three classes of cones. Ordinarily, for colors that do not match, all three will differ. For the special case of chromaticity differences at equal luminance, Yves Le Grand was the first to propose, without wishing to abandon the trichromatic conception, that there might be only two dimensions of chromatic variation. He assumed that S cones make no contribution to luminance, which is hypothesized instead to depend upon the linear sum of L and M cone activity, so that any increase in the excitation of one of these is exactly mirrored by a decrease in the other. Therefore, only one of them need be specified, such as L. In combination with the level of S‐cone excitation, then, that of L will be sufficient to specify any color.

MacAdam's 1941 study, which produced his famous ellipses, had been carefully conducted at equal luminance and provided the data to be explained by Le Grand. the CIE chromaticity diagram, upon which MacAdam plotted his results, is not very useful for the representation of Le Grand's dimensions, which lie along oblique axes whose slopes differ continuously throughout the CIE diagram.

Working almost a decade before Le Grand published his article, MacAdam, a devotee of strictly mathematical interpretation, did not have a physiological model in mind. Therefore, except occasionally by accident, he did not test discriminations along either of the dimensions that interested Le Grand. Consequently, Le Grand's analysis of MacAdam's data had to be rather indirect. Perhaps for this reason, and possibly also because it was written in French, this article didn't attract much attention among English‐speaking researchers until Boynton and colleagues published six articles during the period from 1978 to 1987. the last five of which were heavily based upon it.3‐8

Le Grand's analysis of a MacAdam's ellipse amounted to the following. (1) Determine the slopes of the pair of oblique axes, passing through the center of the ellipse on the CIE diagram, which correspond to the dimensions of pure changes of S‐ and L‐cone excitations, (2) Determine the levels of S‐ and L‐cone excitations represented by a stimulus that plots at the center of the ellipse. (3) Accept the notion that a fitted ellipse constitutes an adequate resentation of MacAdam's data. (4)Determine the magnitude of change along each of the Sand L axes required to achieve a MacAdam discrimination step (the standard deviation of repeated attempts to match the stimulus which plots at the center of the ellipse). (5)Represent that step in terms of changes in S‐ and L‐cone excitations.

Le GrandS approach was actually more algebraic and less geometric than what has just been described. Either way, the analysis cannot be done unless S‐ and L‐cone excitations can somehow be calculated. To do this requires knowledge of the spectral sensitivities of the three classes of cones. Arbitrary transformations (of which CIE tristimulus values are a prime example) will not do. Le Grand considers two sets of coneficndamentals thaf were available in 1949, both based on evidence derived from dichromats, but which differed because of unlike assumptions concerning the cause of their reduced form of normal color vision. One set was based on Fick's hypothesis that signals from L and M cones fuse, perhaps arising from cones of a single mixed type. The other set reflected Konig's conception that one of the two cone classes is entirely missing in dichromacy.

Le Grand carried out his calculations separately for these two sets of sensitivity curves (fundamentals). Figure 2 shows a derived threshold‐vs‐intensity curve (log AB vs. log B) for S‐cone excitation alone, which passes reasonably close to 25 data points‐one for each of MacAdamS ellipses. The data shown are based on Fick's hypothesis, but a change to Konig's would not move the points significantly. The limited scatter ofthe data points around the curve suggests that the level of L‐cone excitation, which varies from point to point, would not significantly afect discriminations uniquely mediated by variations of S‐cone excitation. The data show that the greater the level of prevailing S‐cone excitation, the greater is the increment required for a discriminable difference. This relation is similar to that exhibited in threshold‐vs‐intensity curves for luminance discrimination.

Figure 3 displays calculations of log AR vs. log (R/G); points are plotted that are based on both hypotheses. In each case, the ordinate values are clearly minimal when the L‐ and M‐cone excitations are balanced (R/G = I log R/G = 0), showing that chromatic acuity is best there. Straight lines that intersect at the minimum, one with a negative and the other with a positive slope, describe the Fick data remarkably well. Because such lines (which he does not draw) would apparently not jit the Konig data as accurately, Le Grand takes this as a vindication of the choice of Fick fundamentals, which he accepts in preference to Konig's.

Although Le Grand noted the heightened acuity of redgreen chromatic discrimination at the balance point of L‐and M‐cone excitation, he failed to speculate about why it happens. Opponent‐color theory can easily make sense of it. Here, as with S‐cone and luminance discriminations, the greater the prevailing level of |L‐M| activity in the r‐g opponent channel, the greater is the increment required for a discriminable change. By 1957, in his text book Light, Colour and Vision, (without, curiously, any specific text reference to this article) Le Grand reached essentially this position on page 447. There he rewrites Eq. (7) by taking the antilogarithms of both sides. Had he kept the same symbols for R and G (which he did not), the revised formula would read ΔR = .0022(1.26R/G)n, (where n is either 2/3 or − 2/3 depending upon whether 1.26RIG is greater or less than unity). Then he writes: … it is just as ifthe fundamentals [R] and [G] acted in close association and neutralized each other's responses so that the difference threshold corresponds to that of much weaker stimuli. As a result of this association, the red and green fundamentals acting together have (especially near the points on the chromaticity diagram where their responses are balanced) a difference sensitivity which is much greater than that of the blue fundamental…

A classic approach to the understanding of color differences, which began with Helmholtz, and was later embellished and improved by Stiles, lo involves the notion of the “line element.” It is a receptor‐based theory, according to which thresholds depend on the distance between vectors in a three‐space in which the L, M, and S receptor outputs dejne three orthogonal dimensions. Le Grand points out that such an analysis by Stiles predicted the orientation of the long axes of MacAdam's ellipses fairly well, but failed completely to account for the short ones. From our point of view, such an approach is bound to fail because of the assumption that signals generated by the L and M cones remain separated at a level of processing where they each can be compared to signals generated by S cones, whereas the weight of the evidence indicates that the L‐M difference is computed well before any such comparison can take place. Le Grand makes a similar argument, stating that “…R and G are associated and cannot in any way be considered as independent receptors, and definitely not as Fechnerian receptors.” Indeed, Wyszecki and Stiles show (page 687‐689) that a line‐element theory, even one that includes a chromatic difference term arising.fiom independent receptors each subject to Weber's Law, falsely predicts an elevation of threshold, rather than the observed reduction, when L‐ and M‐cone excitations are balanced.

Most of the time, when two stimuli are compared at equal luminance, dierences bet ween them exist along the L and S dimensions simultaneously. MacAdam's ellipses provide a contour that describes his experimental results for such combined differences, and Le Grand raises the question of why an ellipse should result. He considers three possibilities. Thejirst is that the two components of difference act independently and are then combined in a root‐mean‐squarefashion [Eq. (8)], as would be expected mathematically for the distances between the each of vectors represented on three orthogonal axes. But he rejects this idea, as well as a suggestion made by Silherstein and MacAdam, who had posited a “two‐dimensional normal probability law.” Near the end of his article, Le Grand makes the excellent point that be should definitely not forget that color vision is based on biology and not on geometric analogies. Instead, heproposes (without using the expression) that probability summation ofthresholds mediated by independent receptors could also predict an ellipse, provided that Crozier's law (which states that the variability of threshold estimates is proportional to their mean values) is valid. To do this exactly requires knowledge oflfrequency‐of seeing curves. However, these were not obtained with MacAdam's method, so additional assumptions are needed to yield the prediction of an ellipse. It is not clear why this hypothesis is not subject to the same kind of objection as thefirst two.

On the basis ofwork in San Diego in which thresholds were tested directly on the L and S axes, as well as those that produced combined changes, the heretical notion was proposed that a hexagon might be the “true” form with which to Jit the data. A tendency toward perfect summation was consistently found when L and S were simultaneously increased or decreased, and something less than probability summation resulted when their signs diflered, suggesting that the combined threshold in this instance was determined exclusively by whichever component ofthe combined change reached its threshold valuejrst. An appeal to color appearance was also made. A pure increase in Sfvom the white point, which could be produced in a very direct fashion with the La Jolla Analytic Calorimeter, produces not a pure blue, but a sensation of increasing saturation that is as much red as blue, suggesting that when more redness is also introduced by simultaneously increasing L‐cone excitation, the two rednesses will add. It was also observed that individual diferences are accounted for mostly by the size of threshold steps along the critical dimensions, but when these are normalized the rules just described apply similarly to all subjects‐not only those ofthe La Jolla experimen Is, but those ofprevious studies re‐analyzed from this point of view.8 A hexagon appears to fit the data about as well as an ellipse. However, given a set of discrimination data obtained along many axes, the diflerences between a best‐fitting hexagon and a best‐fitting ellipse are far too small, relative to the experimental variability, to allow a choice based on curve‐fitting alone.

Following the publication of the Jirst La Jolla experiment, Krauskopfand his colleagues l 5 independently discovered, based upon their experimental data alone, the special nature of the L and S dimensions. Many of their experiments were electrophysiological, with nerve impulses recordedfrom cells in the lateral geniculate nuclei of Macaque monkeys, an ideal human surrogate. Stimulation started at a white point, and movedgingerly in various directions in color space. Whereas Boynton termed the L and S axes “critical,” Krauskopfet al. referred to them as “cardinal” instead (which has a nice ring to it). Over the years, there has been considerable argument concerning whether S cones contribute nothing to luminance, as Le Grand assumed (see also the translation qf Schrodinger and Zaidi's commentary), or some very small amount (no one has proposed a contribution any‐ where near that ofthe other cone types). This is not the place to review the controversy, but it is safe to say that the S‐cone contribution (ifany) is so small, and its inclusion would be so destructive to the elegance ofLe Grand's proposed analysis and others that have followedfrom it, that the assumption of zero S‐cone contribution to luminance is the one that should befavoredfor an analytic view of chromatic discrimination.

There are some remaining puzzles concerning Le Grand's personal reaction to his very clever article. It seems odd that his analysis did not suggest to him (note his title) that behavior along the R‐G dimension might somehow relate to opponent‐color processing (even though he actually uses the word inhibition at one point in the Conclusions section of his article). Nor was this idea apparently suggested by any ofhisjve distinguished referees, whose comments he discusses in the addendum, except in reverse, fashion by De Vries who wonders “whether the physiological linkage that we find between R and G receptors would not arise from the equal luminance condition … that imposes a mathematical linkage between the R and G responses.” Le Grand replies by stating that “we would only be able to decide this by applying our analysis to the cases where luminance and chromaticity vary simultaneously.” This is a curious exchange, because in the body of the article Le Grand proposes only one kind ofphysiological linkage, namely that luminance is based on the sum ofL‐ and M‐cone excitations. This assumption obviously does produce a mathematical linkage between them, and this is a principal reason why the system works, allowing him to reduce three dimensions to two. If DeVries (who unjortunately died young) were alive today, it seems very likely that he would agree that the physiological opponent‐color linkage is very real.

Judd felt that Le Grand's analysis of Figure 3 would not allow a choice ofFick's fundamentals in preference to Konig's. From the perspective gained more than 40 years later, one must agree with Judd. Boynton and colleagues have carried out a similar analysis ofMacAdam 's data, as well as those from some other studies, using the Smith‐Pokorny fundamentals, and as already noted they have experimentally examined discriminations along the S and L dimensions of variation.3‐7 All of the experimental data make sense in the context of Le Grand's system when the Smith‐Pokorny fundamentals are used. The fact is that discrimination data are much too variable to serve as a critical test among fundamentals that difer little from one another; there is so little diference anyway between the Smith‐Pokorny fundamentals and other recent candidates that the issue should be considered moot.

Of the four other referees (MacAdam, Judd, Wright, and Stiles), all but Judd were strict trichromaticists. (MacAdam, the sole survivor of this quartet, still is.) Judd, on the other hand, was fully aware and very sympathetic to opponent‐color concepts, and it is possible that he made mention of them in the critique he sent to Le Grand, but that Le Grand failed to note it. This seems plausible because, in Light, Colour and Vision, Le Grand gives very short shrift to opponent color notions. In his 32‐page chapter on “Theories of Color Vision,” there is no such mention until the last two pages where he introduces the subject by saying that (italics ours) “Young's system must now be left (although with reluctance) for a consideration of that proposed by Hering in 1872…”

It is also curious that, although Le Grand lists this article among the references in the end matter of his textbook, he doesn't specifically cite it anywhere in the body of the text—almost as if he wanted to have it overlooked. It is, therefore, not surprising that Robert Rodieck, in his textbook, developed similar ideas without apparent knowledge of Le Grand's work, or that Boynton did not know of it either when the first article of the La Jolla series was published in 1978. Nor does it seem likely that Krauskopf and his colleagues were influenced by it.

Almost all of the ideas outlined in 1986 in “A system of photometry and colorimetry based on cone excitations,” which had originally been proposed to the CIE in 1979, are evident in Le Grand's article, and the 1979 Mac‐Leod‐Boynton chromaticity diagram is essentially a representation of Le Grand's two dimensions using orthogonal coordinates and the Smith‐Pokorny fundamentals. to the extent that the use of this diagram is gaining favor among basic vision scientists for representing pure chromaticity variations, it is because it clearly represents exactly what Le Grand was writing about, namely the free‐floating excitation level of S cones versus the linked tradeoff between the activity levels of the L and M receptors.

It has taken the better part of a half‐century, but Le Grand's ideas as put forth in this article (despite his tendency to abandon them) are beginning to catch on, Maybe, before another 45 years go by, the CIE will see the light as well. Actually, the situation looks more hopeful at present than in the past. A CIE committee established in 1979 to promote an alternate system of photometry and colorimetry, one heavily based upon Le Grand's trail‐blazing insights, was not very productive untilfairly recently—partly because it got bogged down in controversy about choice of fundamentals, and also because of strong differences of opinion concerning whether it is safe to assume no contribution of the S‐cones to luminance. Now, under the competent and vigorous leadership of Françoise Viénot, who recently assumed the chairmanship of a reconstituted committee, it appears likely that an alternative system of photometry and colorimetry, one consistent with Le Grand's insights, will be forthcoming without too much further delay beyond the 15 years that have already elapsed since the CIE met in Kyoto.

The set of ellipses that represent dierence thresholds at constant luminance in the chromaticity diagram can be readily explained by the Young‐Helmholtz theory if one assumes Fick's hypothesis (that dichromacy arises from trichromacy byfusion oftwo,fundamentals) . Wefind that the blue.fundamenta1 functions as an independent receptor of a generalized Fechnerian type, while the red and green fundamentals remain on the contrary closely linked. The existence ofthe ellipses is interpreted, on the other hand, by means ofa simple statistical theory, taking into account Crozier's Law.