1 DEFINITIONS USED IN THIS PAPER

There are a number of closely related topics which need to be carefully distinguished in order to understand this paper. These subtle distinctions are not generally made in the literature. To avoid confusion, the following definitions will be used in this paper.

Color constancy—The phenomenon that the perceived color of an object does not appreciably change despite large changes in the magnitude or spectral characteristics of the illumination.

Color inconstancy—The change in the perceived color of an object under change in illuminant. This may be more completely described as real color inconstancy.

Implied color inconstancy—The change in the color coordinates under change in illuminant.

False color inconstancy—The difference between an implied color constancy and a real color inconstancy.

Metamerism—The phenomenon where two or more spectra match under one illuminant but not under another.

Metameric variation—The range in color coordinates under a second illuminant, of a set of metameric spectra under a first illuminant.

Ensemble color inconstancy—The trend in color shift, under a change in illuminant, for a given position in color space. This can be loosely defined as the implied color inconstancy without metameric variation. This is generally taken to mean implied ensemble color inconstancy, since it is very difficult to assess real ensemble color inconstancy. A mathematical definition is given in Appendix D.

The aforementioned definitions are perhaps narrower than those found in the literature. Apologies are made in advance for the inevitable differences that were required to make this paper clear.

2 INTRODUCTION

Figure 1 shows the spectrum of a hypothetical orange–brown printing ink with CIELAB values of (50, 40, 40) when computed under D50/2. This spectrum is achievable—it could hypothetically be obtained through an appropriate combination of the base Pantone inks.

Details are in the caption following the image — Spectrum of a hypothetical ink

The CIELAB values of this hypothetical ink under D65/2 are (48.72, 37.64, 37.74). Figure 2 is an a*b* plot showing the change in a*b* values under this change in illumination. The start of the arrow is the color of the ink under D50/2, and the arrow point is at the color under D65/2. Figure 2 shows the implied color inconstancy of the spectrum from Figure 1.

The change in CIELAB values is 3.51 ΔE_ab, so it would appear that the change in color would be perceptible. But if an observer would view the ink under D50 lighting, and then switch to D65 lighting for a period long enough to adapt to the new lighting condition, would this metric represent the apparent color change? Is this a valid measure of the color inconstancy of that ink formulation?

While on the surface this seems like a reasonable conclusion, it would not be supported by experimental evidence. All of the work that was done to establish the validity of any color difference formula was done by comparing two colors under the same illuminant with the observer fully adapted to that illuminant. A recent paper,¹ Brill proposed that CIELAB under D50/2 and CIELAB under D65/2 are actually different color spaces. Under this assumption, the comparison of CIELAB values under different illuminants is questionable.

Two sets of researchers have investigated the nature of implied color inconstancy using CIELAB.^{2, 3} Based on Brill's assertion, the results of these papers are questionable.

3 COLOR CONSTANCY

3.1 Definition

Color constancy refers to the fact that objects usually do not appreciably change color as illumination changes.^4-7 If an observer looks at a basket of fruit under incandescent lighting, the apple looks red, the orange looks orange, the banana looks yellow, and the lime looks green. If they take the basket outside in bright daylight, they are likely not aware of any change in the colors of the objects. This happens despite the fact that the overall amount of light has changed by multiple orders of magnitude and the balance of light at the red end of the spectrum versus the violet end has changed by a factor of perhaps 10.

This demonstrates that objects generally have color constancy. After the eye has adapted to a drastic change in illumination, the color of an object is reasonably stable.

3.2 Chromatic adaptation

There is an evolutionary advantage to color constancy. This feature helps an animal to recognize objects, and allows it to distinguish between (for example) a ripe berry which is full of nutrients and one which is not ripe. Multiple mechanisms have evolved which facilitate color constancy. The actions of these mechanisms are collectively referred to as chromatic adaptation.

The first mechanism is the dilation and contraction of the pupil, which occurs within a few tenths of a second of the change in illumination.⁸This is certainly not the only mechanism, since it only accounts for only a factor of 10 change in light intensity, and it is not spectrally selective. It cannot account for our adaptation to different color temperatures.

The second mechanism is in the cones.^9-12 There are processes of chromatic adaptation in the cones which effectively serve as an automatic gain control to better adapt to changes in illumination. As light intensity increases, the gain decreases. This AGC happens independently in the L, M, and S cones, so the cones will individually adjust their gain according to the intensity of light in their respective wavelength ranges.

There is also evidence of a third mechanism for chromatic adaptation that occurs after the cones, in the neural pathways to the brain or in the lower parts of the brain.^{10, 13} Experiments have performed^{14, 15} which suggest that there are at least two mechanisms; one which adapts in tenths of a second, and the other which requires seconds for adaption. Presumably, the former is a neural or cognitive effect and the latter is a slower chemical effect in the cones.

All three of these mechanisms are no doubt at play in the human visual system. A more thorough summary of the literature can be found in Appendix A.

Still, for a variety of reasons outlined in Appendix B, perfect color constancy is not always possible. The presence of metamerism demonstrates that color inconstancy can occur under normal lighting conditions. Two objects are metameric if they have dissimilar spectra and match under one lighting, but do not match under another. It stands to reason that if they no longer match under the second illuminant, at least one of them has changed color.

3.3 The wrong von Kries transform

Since a significant amount of chromatic adaptation occurs in the cones or on the signals from the cones before L, M, and S are combined, an ideal simulation of that chromatic adaptation should occur on LMS values. However, CIELAB computations are based on XYZ values, which are computed with the aid of the $x ¯$ , $y ¯$ , and $z ¯$ tristimulus functions. These are an estimate of one particular linear combination of the cone sensitivities, but the linear combination was chosen so as to minimize the amount of hand calculation that went into computing color coordinates, and not to closely emulate anything that might be in the actual visual pathway.^{16, 17}

Normalization of XYZ values is one step in the computation of CIELAB. This step serves two purposes: (1) it forces a pure white (100% reflectance at each wavelength) to have an L* of 100 and a* and b* of 0, and (2) it provides a rough simulation of the chromatic adaptation that occurs in the human visual system. The first purpose is not in dispute. As for the second purpose, the inaccuracy of the simulation has been noted previously. This has been called a wrong von Kries transform.^18-20 The improper normalization will introduce an artifact to CIELAB values, specifically when one compares values computed under different illuminants.

While there is some debate on the extent that chromatic adaptation occurs in the cones, there is no evidence that it occurs in the XYZ values. Virtually all successful attempts to predict chromatic adaptation convert from XYZ values to a cone space for the normalization. Several color spaces (the models from Nayatani, Hunt, RLAB, LLAB, CIECAM97s, and CIECAM02) perform normalization on cone response values (LMS) rather than XYZ.

The CIELAB calculations also apply a nonlinearity to XYZ values. Again, while the exactly location(s) where the majority of the nonlinearity occurs in the HVS is up for debate, there is no evidence that applying the nonlinearity to the XYZ values is a better emulation.

The assumption is made in this paper that any difference in the implied color inconstancy for CIELAB and the implied inconstancy of an LMS-based color space is likely due to the wrong von Kries transform, or possibly the misapplication of the nonlinearity function.

4 LOOK AT ENSEMBLE COLOR INCONSTANCY WITH A CHROMATIC ADAPTATION TRANSFORM

The Introduction section looked at implied color inconstancy for one spectrum. But one spectrum does not tell us the overall size of the implied color inconstancy in L*a*b*. Is there a trend to the color change?

Ideally, this question would be addressed by using a set of representative spectra covering a large amount of color space. Such a set of spectra is, however, not widely available. But a chromatic adaptation transform (CAT) can simplify the process. CATs work directly on XYZ values, so they do not require the generation of spectra for each CIELAB value.

Thus, a CAT can be used to estimate implied color inconstancy for a sampling of positions in CIELAB space.

There are many CATs that could be used, which differ in the method to transform from XYZ values to LMS values. They all convert from XYZ to some estimate of LMS, perform white normalization in LMS space, and then convert back to XYZ. The Bradford transform was chosen for this paper simply because it is in common use. We shall see in a later section that this was a propitious choice.

Figure 3 shows the predicted changes in CIEALB values when one goes from D50 to D65. The points are all at L* = 50, and are grid points, with a* = −40, −35,… +40, and the same range for b*.

The change in a*b* values appears to be a relatively well-behaved warping of one grid to another. Based on this, it appears (for example) that the change from D50 to D65 will cause a small change in chroma for orange and fairly large reduction in a* for cyan.

It would appear that the size of the (real) color inconstancy for a relatively small change in lighting (from D50/2 to D65/2) is fairly large. In rare or contrived situations where the two eyes see the same object illuminated by different illuminants, the difference may be perceptible based on this analysis.

There are two caveats however. The first is that the comparison was done in CIELAB. This is an implied color inconstancy based on a poor approximation of chromatic adaptation.

The second caveat is that the results shown in Figure 3 depend upon a chromatic adaptation algorithm which is also an approximation to what happens in the human visual system. The Bradford transform is an improvement to direct use of CIELAB, but it does not take spectral information into account. The fact of metamerism shows that the generalization “orange has a slight reduction in chroma” may not be true for all spectra of orange with L*a*b* of (50, 40, 40).

In order to sift one caveat from the other, we need to investigate the effects of metamerism a bit deeper.

5 A SECOND LOOK AT CIELAB (50, 40, 40)

A database of printing metamers was created by this author.²¹ This database was created by searching through spectra of actual print—characterization data sets from seven types of printing and ink sets. For each grid point in CIELAB (steps of five in L*, a*, and b*), each of the databases were searched for a close match. If a close match was found, then the spectrum of the real data was mathematically adjusted to create a spectrum that is likely to be produced by a printing process and that has the exact CIELAB value under D50/2. The resulting database contains sets of metamers at in-gamut grid points. Some grid points (those in the intersection of all the print gamuts) have metameric septuplets. Others, near the edge of the gamuts, have only metameric pairs. All told, there are 38 322 spectra in this database.

More details on the database can be found in Appendix C.

A set of metameric septuplet spectra, all with CIELAB value of (50, 40, 40) under D50/2 were culled from the database. The spectra of the seven metamers are shown in Figure 4.

These seven spectra were converted to CIELAB values under D65/2. Figures 5A and 6 show the direction of change in CIELAB values, with Figure 5A being the traditional view from the top of CIELAB space and Figure 6 being a side view.

Figure 5B illustrates two of the definitions. All of the spectra produced the same L*a*b* values under D50, but there is considerable variation under D65. This metameric variation is represented by the red double-sided arrow.

The length of each individual line segment is the implied color inconstancy for one particular spectrum (as implied by CIELAB). The blue arrow represents an average of these implied color inconstances. This is an estimate of the ensemble color inconstancy. To the extent that the spectra that went into the database is uncorrelated, this is an unbiased estimate. On the other hand, the fact that the spectra are all close to real-world samples means that the estimate of the ensemble color inconstancy is biased toward spectra that occur in real life.

The changes seen in Figures 5 and 6 cannot be ascribed to any issues with the Bradford CAT, since the CIELAB values were computed directly from spectra. The changes in CIELAB values are due to some combination of the wrong von Kries transform enshrined in CIELAB (false color inconstancy) and real changes in the color as seen by the human visual system (real color inconstancy).

Note: Under the premise from Brill,¹ the plots in Figures 5 and 6 are mislabeled. The starting points are in one color space and the endpoints are in another.

5.1 The trend

Although there is some scatter to the endpoints in Figures 5 and 6, they all seem to tend in a similar direction. Is the trend statistically significant, or is this a case of seeing a pattern that does not exist? In all cases, the L*, the a*, and the b* values are smaller under D65/2. One simple way to consider the likeliness of this pattern is to assign the change of each of the metamers into eight categories depending on whether L* increased or decreased, whether a* increased or decreased, and whether b* increased or decreased. All seven of the changes wound up in the same bucket. This has a chance of about one in a quarter million of happening at random. Thus, there is a statistically significant trend.

The seven spectra came from seven physically different patches, each with its own unique combination of pigments. It seems unlikely that there is any hidden dependency in the data. The most likely explanation is that, at least for this particular L*a*b* value, there is a common trend for the change in CIELAB values when the illuminant changes from D50/2 to D65/2. There is an implied ensembled color inconstancy.

Figure 7 shows the results of D65/2 conversions on 81 sets of metamers from the printing metamer database. The D50/2 CIELAB values all have L* = 50, and have a* and b* values of −40, −30, −20,… 30, 40. The 81 sets of metamers shown here bear out the conclusions drawn from Figures 5 and 6 about one set of metamers. There is metameric variation, but there is also ensemble color inconstancy. Furthermore, this diagram shows a coherency among all the trends. Neighboring L*a*b* values show similar trends.

Figure 8 is a simplification of Figure 7. The CIELAB values for each set of D65-converted metamers have been averaged to create one set of D65/2 color coordinates for each grid point. The points were then connected by lines horizontally and vertically to create a grid. If one assumes that the seven metamers are a representative sampling of all real-world metamers for each grid point, then these averages are a fair approximation of how CIELAB values of real objects change when going from D50/2 to D65/2. Figure 8 is an implied ensemble color inconstancy graph for D65/2 to D50/2.

One notable feature of Figure 8 is that, with a few exceptions, the warping on the gird is consistent among neighbors. This further reinforces the claim that this is an fair estimate of the implied ensemble color inconstancy from D50/2 to D65/2.

Finally, Figure 9 combines the results from the experiments shown in Figures 4 and 8. The averages of metameric data is again shown in white, and the color shifts predicted by the Bradford CAT are shown in dashed black lines.

The two methods of determining the warp each have weaknesses: the Bradford CAT does not look at spectra and as a result is blind to the effect of metamers, and the metameric database approach only looks at a subset of all possible metamers. For much of the plot in Figure 9, the distortions of color are very close. This moderate agreement between two completely different ways of determining the shift in CIELAB values bolsters the claim that these shifts in CIELAB are a reasonable approximation for implied ensemble color inconstancy from D50/2 to D65/2.

The conclusion from this analysis is that there is a systematic warping of CIELAB values when going from D50/2 to D65/2. It remains to be demonstrated whether this warping is an artifact of the wrong von Kries transform built into CIELAB or if it is a potentially real prediction of the response of the human visual system. Is Figure 9 a demonstration of the failings of CIELAB, or is it a depiction of the systematic effect of color inconstancy?

6 AN LMS VARIATION OF CIELAB

An adaptation of CIELAB which normalizes LMS values rather than XYZ values was proposed earlier²² (adapted from an earlier paper by Berns and Billmeyer). In this unnamed color space, the XYZ values are converted to LMS, normalized against the illuminant in LMS, and then converted back to XYZ. The resulting values are then converted to L*a*b* using the existing formula, but using values of $X n$ , $Y n$ , and $Z n$ that are fixed. In the B&B version of CIELAB, the nonlinearity is applied to the XYZ values. Since the actual X and Z values do not exist in the human visual system, the B&B version of CIELAB emulation of the nonlinearity in the HVS.

CIELAB also applies its nonlinearity correction to XYZ values, which is another poor emulation of the HVS. It is expected that the misplacement of the nonlinearity correction does not contribute appreciably to false color inconstancy, but this analysis will include this effect as well.

To investigate the artifact created by the use of XYZ for normalization and nonlinearity correction in CIELAB, a color space called ConeLab2 was created that mimics CIELAB, but which performs the normalization and the nonlinearity calculation in LMS rather than XYZ. If the mimicry of output values can be made sufficiently close, then any differences in behavior between the two color spaces are likely to be an artifact of use of XYZ in CIELAB.

ConeLab2 is not being suggested as an alternative to any of the recent color spaces. In fact, one of the key design criteria for ConeLab2 is that it approximates the corresponding CIELAB values. Since it is well established that CIELAB does not do a good job of emulating visual color differences, ConeLab2 is explicitly not recommended as a color space for general use! Because of this, the only evaluation of ConeLab2 will be of how well it emulates CIELAB ΔE_ab color differences.

On the other hand, the results of this paper are applicable to any of the recent color spaces which use LMS normalization and nonlinearity correction. If XYZ normalization is shown to provide a significant artifact in illumination transforms, then any color space which uses LMS normalization can be expected to provide a better emulation of color change under different illuminants.

Note:

6.1 CIELAB equations

For reference, the formulas for computation of L*, a*, and b* are as follows:

$L * = 116 f Y / Y n − 16$ (1)

$a * = 500 f X / X n − f Y / Y n$ (2)

$b * = 200 f Y / Y n − f Z / Z n$ (3)

where f is

$f q = q 1 / 3 if q > 24 / 116 3$ (4)

$f q = 841 / 108 q + 16 / 116 if q ≤ 24 / 116 3$ (5)

6.2 Description of ConeLab2

The LMS values are computed from spectra in the same way as the XYZ tristimulus values, except with estimations of the L, M, and S cone functions. But which estimates of the cone functions should be used? A valid argument could be made for any of the variety that are available, but one consideration puts the decision in perspective: the differences between the various LMS approximations are insignificant compared to the difference between any of them and the Standard Observer.

A second consideration is that it would be useful for metamers under CIELAB to still be metamers in the new color space. This allows for the metameric data base to be used again. Ultimately, it was decided to go with an estimate of the LMS functions which is based on the Bradford CAT. The fact that this transform is a linear combination of the XYZ values allows for CIELAB metamers to continue to be metamers.

The cone functions

l λ

m λ

, and

s λ

are defined in Equations (6)–(8-6)–(8), where

λ

is the wavelength.

$l λ = 0.8951 x ¯ λ + 0.2664 y ¯ λ − 0.1614 z ¯ λ$ (6)

$m λ = − 0.7502 x ¯ λ + 1.7135 y ¯ λ + 0.0367 z ¯ λ$ (7)

$s λ = 0.0389 x ¯ λ − 0.0685 y ¯ λ + 1.0296 z ¯ λ$ (8)

The LMS values are computed analogous to XYZ from the reflectance spectrum of the sample,

r λ

under the designated illuminant,

I λ

$L = ∑ λ r λ I λ l λ$ (9)

$M = ∑ λ r λ I λ m λ$ (10)

$S = ∑ λ r λ I λ s λ$ (11)

Next, we need to replicate the functionality of Equations (1)–(5-1)–(5) that define CIELAB.

The analog of L*, which we shall call L_c, is worth a little discussion. The Y tristimulus function corresponds well with our perception of lightness, so this functionality needs to be preserved. L_c must be based on something like Y. To be consistent with ConeLab2, L_c should be computed from LMS. But Y is not one of the cone functions, but rather a weighted sum of the LMS functions.

The human visual system incorporates a nonlinearity which must be reflected in L_c. The question comes, do we compute a weighted sum of LMS and then apply to nonlinearity to that? This would mean that $L c = L *$ . Or do we perform the weighting function separately on each of L, M, and S, and then compute a weighted sum of these values?

If the nonlinearity occurs in the cones or the encoding of the signal from the cones, then individually applying the nonlinearity to the L, M, and S values more closely emulates the human visual system. This option was chosen, since it is consistent with the premise of the paper.

Equations (1)–(3-1)–(3) contain the scaling values 116, 16, 500, and 200. These need to be modified to account for the fact that X, Y, and Z are functions of L, M, and S. The values for the constants k_n need to be determined so that the new space is as close to the same size as CIELAB as possible. These equations incorporate nonlinearity correction of LMS as well as normalization according to LMS by dividing by

L n

M n

, and

S n

$L * + 16 = k 1 f L L n + k 2 f M M n + k 3 f S S n$ (12)

$a * = k 4 f L L n + k 5 f M M n + k 6 f S S n$ (13)

$b * = k 7 f L L n + k 8 f M M n + k 9 f S S n$ (14)

Linear regression was used to find the scaling factors in Equations (12)–(14-12)–(14) that provided the best fit to a large collection of small ΔE_ab color differences based on the printing metamers database. Equation (15) is the equation for ConeLab2, in matrix form, based on the regression

$L c + 16 a c b c = 55.34 56.19 5.78 271.29 − 311.89 39.07 97.25 83.01 − 178.79 f L L n f M M n f S S n$ (15)

6.3 Comparison to previous cone-based versions of CIELAB

Use of the B&B color space for this test would answer the question of the magnitude of CIELAB's false color inconstancy due to normalization of XYZ. But ConeLab2 goes one step further, and applies the nonlinearity in a potentially more valid place, to the LMS values. The question answered with the use of ConeLab2 is broader: whether the use of XYZ in CIELAB for both normalization and nonlinearity correction is an appreciable source false color inconstancy.

A similar color space, named ConeLab, was previously introduced by this author.¹⁶ ConeLab and ConeLab2 are both based on LMS. ConeLab2 differs from ConeLab in the scaling parameters. ConeLab also included a variant on the nonlinearity function f in CIELAB which was both simpler than f and led to a perceptually linear analog to L*. In this paper, ConeLab2 will revert back to the use of the CIELAB nonlinearity function in order to (presumably) maintain a closer correspondence with CIELAB.

6.4 Demonstration of ConeLab2

The spectra for each grid point in the metamers database were averaged, and each averaged spectrum converted to L_ca_cb_c using Equations (9)–(15-9)–(15). Figure 10 shows the a_cb_c values of these averaged values for L* = 50. This grid represents the conversion of CIELAB a*b* values into ConeLab2.

The correspondence to a*b* is not perfect. The left side of the plot shows some compression, and vertical lines become increasingly bowed as you move to the left. These distortions are inevitable, since the nonlinear function f is involved.

One aspect of Figure 10 might be a bit troublesome: the origins of the two color spaces do not match. A perfect gray in CIELAB translates to a point slightly to the left and above the origin in the a_cb_c plot. The linear regression has found a best fit without any special regard for the position of the origin. As a result, the sums of the constants do not come out to zero: $k 4 + k 5 + k 6 ≠ 0$ and $k 7 + k 8 + k 9 ≠ 0$ . If this were an important feature, then this could have been enforced by substituting $k 6 = − k 4 − k 5$ into Equation (13), and $k 9 = − k 7 − k 8$ into Equation (14), and then proceeding with the regression using fewer variables.

Were ConeLab2 designed to be a color space to be used for purposes beyond this paper, this would be a concern. But forcing perfect agreement of the neutral points would very slightly decrease the overall match between ConeLab2 and CIELAB.

This new color space, ConeLab2 or L_ca_cb_c, is thus similar to L*a*b* in shape and magnitude, but is based on the assumption that the human visual system is better emulated when the computations are based on LMS rather than XYZ.

6.5 Comparison of ΔE in ConeLab2 and CIELAB

Before moving to the next section, it will be useful to quantify how well color differences in ConeLab2 approximate color differences in CIELAB. This evaluation was done by creating 58 000 pairs of spectra that were 1.0 ΔE_ab apart in CIELAB using D50/2. Each grid point from the metameric database was paired with up to 10 different spectra. These 10 spectra were a random weighting of three randomly selected neighbors and the grid point.

Note: There is no research to imply that any color difference formula applied to ConeLab2 is perceptual, nor would this be fruitful. Therefore, the decision was made to use the simplest color difference, ΔE_ab, to compare the CIELAB and ConeLab2 values. While this is a valid metric for the distance between two sets of color coordinates, it should not be interpreted as having any relation to our perception of the difference between colors. For clarity, the standard notation (ΔE_ab) is used for color differences in CIELAB, and the notation ΔE_c is used for differences computed between ConeLab2 value.

The spectra in each of the 58 000 pairs were converted to ConeLab2 using D50, and the ΔE_ab formula was used to compute the Conelab2 color difference for the two points. Figure 11 shows a histogram of those 58 000 ConeLab2 color difference values (ΔE_c).

The average of the 58 000 ΔE_c values is 1.00245, with a standard deviation of 0.072 ΔE_c. A total of 91.1% of the data points are between 0.90 ΔE_c and 1.10 ΔE_c, and 97.5% of the data points are between 0.80 ΔE_c and 1.20 ΔE_c.

This demonstrates that when all color values are computed using D50 illumination, there is a close correspondence between ΔE_ab and ΔE_c. So, for the purposes of comparing color differences under D50/2 illuminant, ConeLab2 is a very good approximation to CIELAB.

7 COLOR CONSTANCY IN ConeLab2

The immediately previous subsection looked at pairs of color coordinates computed under the same illuminant in CIELAB and ConeLab2. In this section, we consider cross-illuminant color differences, where one set of color coordinates are computed under one illuminant and the other set under another illuminant. In other words, this section looks at implied color inconstancy in the two color spaces.

The computations from Figure 7 were duplicated with ConeLab2 instead of CIELAB. Each spectrum from the database was converted to ConeLab2 under D50 and D65. The plot at the left side of Figure 12 is a copy of Figure 7, which shows the effect in CIELAB of D50/2 to D65/2 conversion. The plot at the right shows those same conversions in ConeLab2. Figures 13-16 show close ups of portions of the graphs in Figure 12.

Figure 13 shows a modest decrease in the computed color shifts with ConeLab2. In other figures, such as Figure 16, the decrease in color shifts is dramatic. It is clear from these plots that ConeLab2 predicts considerably smaller color change between the two illuminants than CIELAB does.

The ConeLab2 plots in Figures 12-16 demonstrate the consequence of the decision to use a set of LMS functions that are linear combinations of the tristimulus functions. A pair of spectra will have the same CIELAB values (i.e., be metameric) if and only if the XYZ values are also the same. And if the LMS values of metamers under XYZ are computed from a set of LMS functions which are linear combinations of XYZ, then these values will be metameric whenever the original CIELAB values are. In other words, metamers in CIELAB are also metamers in ConeLab2. This is a convenient advantage of using the Bradford chromatic adaptation method.

The Euclidean distance formula was next used to quantify the amount of implied color inconstancy in CIELAB and in ConeLab2. Spectra were selected from the metameric database where the D50/2 CIELAB chromatic values were in the range $− 40 ≤ a * , b * ≤ 40$ with any L* value, resulting in a total of 22 678 spectra from the database. The CIELAB values under D50/2 and D65/2 were computed, and the ΔE_ab color differences between the positions in color space were computed for each pair. The corresponding Euclidean distance (ΔE_c) was determined for ConeLab2 computations.

For CIELAB, the mean color difference between D50 and D65 color values was 2.50 ΔE_ab and the 95th percentile was 4.19 ΔE_ab. For ConeLab2, the mean color difference between D50 and D65 color values was 1.08 ΔE_c and the 95th percentile was 1.86 ΔE_c. For both measures, the ConeLab2 implied color inconstancy was about 43% of that of CIELAB. The magnitude of this difference cannot be ascribed to an inherent mismatch in the magnitude of ΔE_c versus ΔE_ab as described in the previous section.

Thus, it has been established that a large portion of the implied color inconstancy in CIELAB is not real color inconstancy, but merely an artifact of the use of XYZ values for the CIELAB calculations.

8 CONCLUSION

8.1 Summary

It was pointed out that the equations in CIELAB do a poor job of emulating the lower mechanisms for color constancy in the human visual system. One of the emulation failures, the normalization by XYZ values, has been termed a “wrong von Kries transform.” The other poor emulation is in the application of the nonlinearity correction to XYZ rather than LMS.

Two methods were used to assess the implied ensemble color inconstancy of CIELAB.

The Bradford CAT was used to investigate the change in CIELAB a*b* values in the L* = 50 plane when the illuminant changes from D50/2 to D65/2. The result showed a warping of an a*b* grid. This transform is an estimation of the implied ensemble color inconstancy, but is suspect since it does not take spectra into account.

A large database of spectra that are metameric under D50/2 was also used to investigate the implied ensemble color inconstancy in CIELAB. Transforming the spectra to D65/2 clearly showed metamerism, but the general directions of color changes were not random. The set of metameric D65/2 values were averaged to reduce the effect of metameric variation, revealing a second estimate of the implied ensemble color inconstancy of CIELAB. The results of this method are also suspect, since there is an underlying assumption that the collection of metameric septuplets is a unbiased sampling of all possible metamers.

The two methods agreed fairly well. Since the shortcomings of the two methods are orthogonal, the agreement between the two bolsters the assertion of a consistent change in CIELAB values under a change in illumination. Either set of results can be taken as a reasonable estimate of the implied ensemble color inconstancy.

A third experiment was conducted to estimate the extent that the emulation failures in CIELAB contribute to false ensemble color inconstancy. A new color space, ConeLab2, was developed to apply a similar test to spectra of the metameric database. The calculations of ConeLab2 are computed on LMS values instead of XYZ, so ConeLab2 presumably eliminates the improper normalization and nonlinearity correction in CIELAB. The calculations in ConeLab2 emulate CIELAB with coefficients chosen to provide tight correspondence to ΔE_ab color differences between the two color spaces.

Since color differences in ConeLab2 are a very close approximation to color differences in CIELAB, and color differences are one of the major practical limitations to CIELAB, ConeLab2 is explicitly NOT recommended to be used as an alternative to CIELAB.

Estimates of implied color inconstancy of the metameric database were computed using CILEAB and ConeLab2. ConeLab2 showed only 43% as much color inconstancy as CIELAB, demonstrating that the improper use of XYZ in CIELAB is responsible for more than half of the implied color inconstancy in CIELAB.

The conclusion is that cross-illuminant color difference calculations, where the two CIELAB values to be compared have been computed under different illuminants, will provide misleading results. Cross-illuminant color differences tell more about the failings of CIELAB than about the color inconstancy of the sample.

Cross-illuminant color difference calculations performed in a color space where normalization occurs in LMS do not have this artifact. When compared with CIELAB, such color spaces will provide a more accurate estimate of color inconstancy simply because of the removal of this artifact. Whereas this paper does not advocate ConeLab2, it does support the use of any color space where the calculations are done on LMS values instead of XYZ.

This paper did not look at the extent that an LMS-based color space accurately predicts our perception of color inconstancy. This paper has not investigated this idea on tests of human subjects.

Discussions of metrics for color inconstancy are also outside the scope of this paper, other than the general statement that color constancy metrics which make use of LMS, are potentially more accurate than those which make use of XYZ.

8.2 Implications

When choosing between potential formulations of a given color, it would be useful to have a metric which assays the degree that a color will change appearance under a change in illumination. The conclusion of this paper is that CIELAB should not be used by a color manufacturer to determine how much a sample will change color under different illuminants. Color spaces which normalize LMS values rather than XYZ values are likely to be more reliable in estimating the color change caused by a change in illuminants.

Another potential industry benefit from this work is in colorant formulation, the manufacturer is given a target color and is asked to develop a combination of pigments which achieves this color. It is common for another manufacturer to be asked to do the same, and that the two products would reside next to each other and be expected to match. For example, one company may make the rubber bumper of a car and another may make the metal fender with a matching coating. The set of pigments is likely to differ.

To avoid matching problems down the road, it would be useful if both manufacturers had a way to determine the extent that a given color is likely to change under a different illuminant. This concept differs from the metameric index in that it only considers the color change of one spectrum, rather than the relative color change of a pair of spectra.

If the bumper and the fender manufacturers both minimize the color change, the resulting match will likely be improved. Cross-illuminant color differences in an LMS-based color space are potentially useful for this. The concept of metameric proclivity has been investigated by this author in a previous paper.²¹

Various papers have looked at how CIELAB values change when the illuminant is changed.^{2, 3} The results of this paper suggest that this previous work may not show a true color change, but largely reflect an artifact in CIELAB. Researchers should avoid the use of CIELAB when attempting to quantify color inconstancy.

Appendix A: MECHANISMS FOR COLOR CONSTANCY

The most salient of the mechanisms for color constancy is the dilation and constriction of the pupil in response to the overall brightness. The recently-discovered mediator of this response is the photo-responsive retinal ganglion and not in the rods or cones.^{23, 24} This is an involuntary response which occurs in a few tenths of a second.⁸

The pupil's quick modulation of the overall intensity of light is clearly not the only mechanism. Color vision functions over a range of perhaps six orders of magnitude, whereas the area of the pupil can change by only a factor of 10. Furthermore, the dilation of the pupil does not explain the adaptation that occurs when the spectral balance of light changes.

Where does the rest of the chromatic adaptation occur? It has been pointed out^{4, 6} that there have historically been two schools of thought on how chromatic adaption happens: (1) the retinal mediation explanation inspired by von Kries,²⁵ and (2) the cognitive mediation explanation inspired by Helmholtz.

von Kries postulated in 1905 that the sensitivity of each of the three cones drops in proportion to increasing overall illumination.⁵ This has been referred to as the dark glasses effect.⁶ Wandell⁷^(p.314) goes so far as to state: “There is an emerging consensus in many branches of color science that the von Kries coefficient law explains much about how color appearance depends on the illumination.”

Thus, it is reasonable to suspect that at least some portion of our mechanisms for color constancy occur within the individual cones or directly on the signals from the cones before they are combined with signals from dissimilar cones.

Mediation within the cones

The process of a cone sensing light begins with a chromophore molecule capturing a photon, causing he molecule to isomerize. This sets off a sequence of chemical actions that ultimately result in the closure of a channel in the cell membrane. This limits the intake of positively charged Na⁺ ions. This combined electrical potential is then registered by the neurons.

Chromophore depletion is one mechanism by which the gain for a cone is reduced as the light intensity increases. This is the result of two things, first, chromophore molecules are a discrete and finite resource. Second, when a chromophore molecule is activated by a photon, there is a time lag of tens of seconds before the chromophore can once again capture a photon. Thus, under high light conditions, there is a dearth of chromophores and a resulting lower probability of an individual photon capture.⁶

In the steady-state condition—when the deactivation of chromophores by a constant influx of photons is at equilibrium with the delayed reactivation of photons—the probability of capture of a photon has also stabilized. The varying probability works as an automatic gain control of the form of Equation (16).

$p = 1 I + I 0$ (16)

where

I

is the intensity of incoming light and

I 0

is a constant which represents the intensity level where one-half of the chromophore molecules are deactivated.

As a result, the output of this stage of the system, that is, the rate of photon capture, is the Michaelis and Menten equation:

$S = I I + I 0$ (17)

This is part of the automatic gain control in the cones, but it is not the only factor, and likely is not even the major contributor. First, the value of

I 0

is too high to account for a gain control at a lower intensity of light. Second, direct measurement of the voltage at the cones as a function of intensity of incoming light fits well to an empirical relationship that provides more significant compression at lower light levels:

$V V max = I n I n + I 0 n$ (18)

where

V

is the output voltage and

V max

is the maximum output voltage. The coefficient was determined to be

n = 0.7

in some experiments.

⁶

The stages in the sequence of chemical reactions subsequent to the isomerization of the chromophore are also subject to reduction in gain due to depletion in the necessary chemicals. Also, the number of ion channels is limited. A similar “gain change due to depletion” may occur in multiple stages, each being approximated by an equation of the form of Equation (17).^{11, 12} The combined result, the composition of multiple compression functions, can be approximated by the generalized Michaelis and Mention equation of Equation (18).

Postretinal mediation

An object may appear to change color due to subconscious assignment of the white point, as has been aptly demonstrated by the recent social media furor over “the color of the dress”.²⁶ The color of objects can also be affected by the colors of other objects in the visual field, an affect known as simultaneous contrast.

Many authors have stated that the predominant cause of chromatic adaptation occurs after the signals leave the retina. In 1866, Helmholtz argued color constancy is achieved through a cognitive process. “Seeing the same objects under these different illuminations, we learn to get a correct idea of the object colors in spite of difference of illumination. We learn to judge how such an object would look in white light, and since our interest lies entirely in the object color, we become unconscious of the sensations on which the judgments rests.” (quoted from Jameson and Hurvich⁵) Reddish orange stop signs appear to be red because we recognize them and know that stop signs are red. While this may occur, the fact that paint swatches show color constancy shows that Helmholtz's proposed mechanism is not the only mechanism.

Walraven et al.¹⁰^(p.59) acknowledge that chromophore depletion is one mechanism for chromatic adaptation, but they argue that most of the gain control occurs downstream from the photoreceptors. They base this on the claim that bleaching (deactivation) only occurs at relatively high level of illumination. Walraven et al. do not consider the possibility that there are additional compression mechanisms inside the cones.

Ebner²⁷ comes to the same conclusion as Walraven, but with different rationale: “Given that color constancy also occurs using exposure times less than 0.01s (Land and McCann 1971), it seems unlikely that an adaptation mechanism that uses eye movements is actually used by the visual system. D'Zmura and Lennie suggest that the first color stage is a simple scaling of the color channels using space average color.”

The evidence provided by Ebner shows that there is indeed some chromatic adaptation that occurs at a time interval faster than a strictly chemical process would allow. On the other hand, Ebner neglects prior research that shows that there is additional chromatic adaptation that occurs well beyond the 0.1-s time frame. For example, Fairchild and Reniff¹⁴ state: “Single or piecewise exponential decay functions cannot be fitted to the data. However, sum-of-two-exponentials functions provided accurate descriptions of the data. The results suggest two stages of adaptation: one extremely rapid (a few seconds) and the other somewhat slower (approximately 1 min).” Rinner¹⁵ performed a similar experiment and found three separate time constants.

Combination

Other authors argue that there are both retinal and cognitive components involved. Shevell²⁸^(p.181) states: “Theories can be distinguished by the emphasis placed on peripheral sensory mechanisms versus central processes (including cognitive processes). Most theories include both retinal and cortical mechanisms.”

According to Valberg²⁹^(p.266): “Color induction in adaptation were for a long time explained at the retinal level by a relative reduction of the sensitivity of the cones, but more recent work points also to lateral interactions at several stages of the visual pathway.”

From this review it seems likely that an automatic gain in the cones is at least partially responsible for chromatic adaptation. Thus, some form of cone-level normalization in a color space is beneficial in order for a that color system to accurately portray how our perception of colors change under different illuminants.

Appendix B: LIMITS TO COLOR CONSTANCY

Chromatic adaptation works well enough that any change in apparent color of an object is likely to go unnoticed by an uninitiated observer. One thing that hides any potential inconstancy is our relatively poor memory for colors.^30-32 To investigate limits to color inconstancy, researchers often resort to asking subjects to do color matches under unusual conditions. For example, an apparatus is often used which shows a different scene to the two eyes so that the right and left eyes are adapted differently.^{20, 33-35}

But perfect color constancy is not always possible

One inevitable cause of color inconstancy is caused by degenerate light sources which have little or no energy over an appreciable part of the spectrum. If the world is viewed under the illumination of a red LED (with a peak wavelength of 637 nm, and no appreciable energy below 600 nm), it is impossible to decipher what goes on in the lower two-thirds of the spectrum. An object that would reflect a brilliant blue under white light will look the same as one which is black under red LED illumination. In this case, try as they might, it is not possible for the cones to adapt. No amount of gain can make up for the fact that there is no light in that part of the spectrum.

Illuminants do not need to be fully degenerate to have an effect on color vision. Moonlight is an example of a broadband light source, which is to say, it has the potential to be sensed in all three types of cones. The intensity of the light is, however, insufficient for the visual system to distinguish between actual incoming light and the noise floor of the detector. This noise floor is caused by spontaneous deactivation of chromophore molecules caused by heat or mechanical action.

Color inconstancy can also be noticeable in less extreme cases, for example, when a pastel color is viewed under daylight versus warm light. The spectral information at the blue end of the spectrum is suppressed so hue shifts are possible.

The presence of metamerism demonstrates a second cause of color inconstancy. Two objects are metameric if they have dissimilar spectra and match under one lighting, but do not match under another. It stands to reason that if they no longer match, at least one of them has changed color.

Metamerism is due to the fact that the eye is not registering the spectrum of incoming light, but rather reducing the spectrum to three channels of information. Different illuminants will emphasize different parts of the spectrum, and hence and hence change the response of the cones. The color shift due to spectral reduction is generally too small to notice but can be appreciable under certain unusual conditions.

As shown in Figures 13-16, there is a more fundamental color inconstancy, even under nondegenerate illumination and when the effects of metamerism are discounted. This paper defined this as ensemble color inconstancy. A mathematical definition is proposed in Appendix D.

There is likely another mechanism that limits color constancy. If a sheet of white paper is viewed under D50 or D65, it appears white. When viewed under incandescent or other warm white light source, that same sheet will appear slightly yellowed, even after sufficient time for accommodation.

Appendix C: PROVENANCE OF THE PRINTING METAMER DATABASE

A database of metamers in the print industry has been created by this author.²¹ The database includes spectra from seven different types of print, as listed:

Pantone formula guide
Pantone GOE guide
Flexographic printing with CMYK
Flexographic printing with expanded gamut, each with one of orange, green, or violet ink
Indigo 7900 printing with CMYK
Indigo 7900 printing with expanded gamut, each with one of orange, green, or violet ink
Epson expanded gamut.

The spectra for the two formula guides were measured from one of the respective booklets. Each of these were solid coverage of some blend of the appropriate base inks. The base inks are different between the two systems, so the spectra are dissimilar.

Spectra from the two flexographic subsets were based on a tradition halftone print run at Clemson University, courtesy of Liam O'Hara.³⁶ The data consisted of characterization data for (1) CMYK, and (2) for expanded gamut, which includes combinations of CMYK with orange ink, with green ink, and with violet ink. All spectra in the expanded gamut were selected to have at least one of the three inks EG inks.

Spectra from the Indigo press was collected in a similar manner and provided courtesy of Abhay Sharma of Ryerson University.³⁷ The Epson print included only expanded gamut spectra.

The database includes spectra that are a perfect match to CIELAB grid points, where the grid points are in steps of five in L*, in a*, and in b*. The number of spectra for each grid point depends on the number of print modalities where that grid point is within the gamut. Some grid points have only a pair of metameric spectra, whereas many grid points have metameric triplets, quadruplets, and so forth, up to metameric septuplets. All told, there are 38 222 spectra in the database.

For each CIELAB grid point, each of the subsets of spectra were interrogated to find the closest match in terms of D50/2 CIELAB calculations. The CIELAB values were not, in general, a perfect match to the grid points. Principal component spectra were determined from the relevant printing condition using singular value decomposition. The first three principal components were selected as stand-ins for the process specific primaries. The assumption is that the addition of any small linear combination of these process specific primaries could theoretically be attained by judicious choice of the pigments available to the process. A similar process was tested by Li and Berns,³⁸ where they compared a variety of methods for creating metamers, including principal component analysis. PCA performed well.

For most of the grid points and print conditions, there are multiple combinations that could have been used to reach that CIELAB value. No attempt was made to emulate any of the disparate method use in real-life software to do this color separation. As such, the spectra in the database may not represent spectra that may occur in normal practice, but rather should be considered spectra that could occur on a given press and ink combination. The choice of “closest match to the grid point” should provide some randomization to the selections and hence a somewhat more diverse set of spectra.

The database is available upon request from this author: john@johnthemathguy.com.

Appendix D: MATHEMATICS OF IMPLIED ENSEMBLE COLOR INCONSTANCY

Precise definition

For the purposes of this appendix, define a physically realizable spectrum to be any spectrum where the reflectance is between 0% and 100%, inclusive, at all wavelengths. Define a physically realizable color in a given color space and under a given illuminant as any set of color coordinates that can be derived from at least one physically realizable spectrum.

All physically realizable colors have an associated ensemble of metameric spectra. Metameric spectra are further defined to be constrained to be physically realizable. In the case of pure black (0% reflectance at all wavelengths) and pure white (100% reflectance at all wavelengths), the metameric ensembles contain a single spectrum. All other physically realizable colors have an infinite number of spectra in their ensembles.

The ensemble color inconstancy is defined for a given color space, set of color coordinates in that color space under the primary illuminant, and a secondary illuminant. The first step is to determine the average spectrum from the metameric ensemble associated with the set of color coordinates under the primary illuminant. The color coordinates for this average spectrum are then determined under the secondary illuminant. The vector from the set of coordinates under the primary illuminant to the set of coordinates under the secondary illuminant is defined as the ensemble color inconstancy.

Calculation of ensemble color inconstancy

The ensemble color inconstancy can theoretically be calculated, although the actual calculation involves multiple steps which are inherently difficult to compute.

The first step is to find one starting spectrum that yields the required set of color coordinates in the color space under the primary illuminant. Any such spectrum is acceptable. If one is not already available, one can be computed by defining a set of three dissimilar basis spectra, and then finding a weighting of the three which yields the desired XYZ values. Being a linear problem, this is not terribly difficult.

The second step is to determine a complete set of spectral vectors which

are orthogonal to the product of the spectrum of the primary illuminant and each of the tristimulus functions relevant to the color space (typically the XYZ or LMS functions).
are mutually orthogonal to each other, and
which span the space of all spectral vectors which are orthogonal to aforementioned product.

The number of orthogonal vectors will be three less than the number of spectral channels, that is, typically either 28 or 33. This set of vectors can be accomplished with a variation on the Gram–Schmidt process known in linear algebra.

All metameric spectra can be written as the sum of the starting spectrum and a linear combination of the orthogonal vectors. The converse is not true, however. Some of said sums will violate the constraint of physical realizability.

The ensemble color inconstancy is defined to be the centroid of this irregular, approximately 30-dimensional object constrained by physically realizable reflectance values at all wavelengths. This is a difficult direct computation.

Direct integration could be used, but is generally inefficient for dimensions greater than 3 or 4. A Monte Carlo method would be recommended.

Relation to centroid of the ensemble in color coordinates

The ensemble color inconstancy is defined as an average in reflectance space. For any color space which is nonlinear, this will not, in general be equal to an average computed from the color coordinates under the second illuminant.

On the other hand, the range of points in the color space will not generally be large. Over the range of points included, the color space may be fairly linear. In this way, it is expected that the average in color coordinates will be a reasonable estimate of the ensemble color inconstancy.

Relation to estimates in this paper

This paper demonstrated two estimates of the ensemble color inconstancy. The first was based on applying a CAT to XYZ values. To the extent that the individual CAT can be interpreted as being an average of metamers, it serves as an estimate to the ensemble color inconstancy.

The second estimate to the ensemble color inconstancy is closer to the actual definition. It is an estimate, since the spectra of only seven metamers served as the average. One could argue, on the other hand, that this estimate is far closer to reality than the average of an infinite collection of spectra. The sets of seven spectra used in the paper were not only physically realizable, but physically realizable with commonly used pigments. Using the collection of all possible metamers means that spectra that cannot be obtained with available pigments will be included. These include spectra with sharp transitions, zig-zags, and those which deviate from the normal in a way to best accentuate the difference in illuminants.

Utility of ensemble color inconstancy

The concept of ensemble color inconstancy was developed for the purpose of this paper to put a name to the concept that the color shift for all metamers tend to head in the same direction. Given that and the fact that the computation of this is involved, this may be relegated to the basket of useful but impractical theoretical concepts.

On the other hand, there is one potential application of the ensemble color inconstancy, or more precisely, for the spectrum associated with the ensemble color inconstancy. When a color manufacturer develops a recipe for a color, there are general numerous possible combinations of colorants which could achieve a specific color. In the event that there is not an existing physical sample to match under various illuminants, it may be beneficial to choose a formulation that comes close to the spectrum associated with the ensemble color inconstancy. This will have the best chance of matching a future formulation under the secondary illuminant.

Biography

John Seymour is an applied mathematician and color scientist. He is a professor at Clemson University, teaching color science and process control in the Graphic Communication school. He has worked as a consultant since 2012 under the name “John the Math Guy”. John currently holds 30 U.S. patents, has authored 30 technical papers, has presented at 33 conferences, and has keynoted at 6 conferences. He is an expert on the Committee for Graphic Arts Technologies Standards and ISO TC 130. He is currently on the board of the Technical Association of the Graphic Arts. He writes a blog which is described as “applied math and color science with a liberal sprinkling of goofy humor.”

DATA AVAILABILITY STATEMENT

The metameric database used in this study is available upon request to the author. Permission is granted to make use of the database provided proper attribution is given to to the author.

REFERENCES

1Brill M. Is CIELAB one space or many? Color Technol. 2021; 137: 83- 85.
2Safi M. A study of relationship between color inconstancy level and CIELAB color coordinates. Prog Color Color Coat. 2014; 7: 165- 175.
3Dorukalp D, Davis W. Evaluation of hue shift formulae in CIELAB and CAM02. Paper presented at: Proceedings of 29th CIE Session. 2019.
4Foster D. Color constancy. Vision Res. 2011; 51: 674- 700.
5Jameson D, Hurvich LM. Essay concerning color constancy. Annu Rev Psychol. 1989; 40: 1- 22.
6Kaiser P, Boynton R. Human Color Vision. 2nd ed. Optical Society of America; 1996.
7Wandell B. Foundations of Vision. Sinauer Associates; 1995.
8Feinberg R, Podolak E. Latency of Pupillary Eeflex to Light Stimulation and Its Relationship to Aging. Federal Aviation Agency, September 1965.
9Normann RA, Perlman I, Hallet PE. Cone photoreceptor physiology and cone contributions to colour vision. In: P Gouras, ed. The Perception of Colour, Vision and Visual Dysfunction. Vol 6. CRC Press; 1991.
10Walraven J, Enroth-Cugell C, Hood DC, MacLeod DIA, Schnapf JL. The control of visual sensitivity: receptoral and postereceptoral processes. In: L Spillman, JS Werner, eds. Visual Perception: The Neurophysiological Foundations. Academic Press; 1990.
11Brill MH, Bergeron DW, Stoner WW. Retinal model with adaptive contrast sensitivity and resolution. Appl Opt. 1987; 26(23): 4993- 4998.
12Hood DC, Birch DG. Models of human rod receptors and the ERG. In: MS Landy, J Anthony Movshon, eds. Computational Models of Visual Processing. MIT Press; 1994.
13D’Zmura M, Lennie P. Mechanisms of color constancy. J Opt Soc Am A. 1986; 3(10): 1662- 1672.
14Fairchild MD, Reniff L. Time course of chromatic adaptation for color-appearance judgments. J Opt Soc Am A. 1995; 12(5): 824- 833.
15Rinner O, Gegenfurtner KR. Time course of chromatic adaptation for color appearance and discrimination (Technical Report No. 69). Max-Planck-Institut für Biologische Kybernetik; 1999.
16Seymour J. Why does the a* axis point toward magenta instead of red? Color Res Appl. 2020; 45: 1040- 1054.
17Fairman H, Brill M, Hemmendinger H. How the CIE 1931 color-matching functions were derived from Wright-Guild data? Color Res Appl. 1997; 22: 11- 23.
18Terstiege H. Chromatic adaptation; a state-of-the-art report. J Color Appearance. 1972; 1(4): 119- 123.
19Fairchild M. Color Appearance Models. 2nd ed. John Wiley & Sons; 2005.
20Süsstrunk S, Finlayson GD. Evaluating chromatic adaptation transform performance. Paper presented at: Color and Imaging Conference. 2005.
21Seymour J. Metameric proclivity. Paper presented at: Technical Association of the Graphic Arts. 2020.
22Berns RS. Billmeyer and Saltzman’s Principles of Color Technology. 4th ed. Wiley; 2019.
23Zaidi FH, Hull JT, Peirson SN, et al. Short-wavelength light sensitivity of circadian, pupillary, and visual awareness in humans lacking an outer retina. Curr Biol. 2007; 17: 2122- 2128.
24Markwell EL, Feigl B, Zele AJ. Intrinsically photosensitive melanopsin retinal ganglion cell contributions to the pupillary light reflex and circadian rhythm. Clin Exp Optom. 2010; 93: 137- 149.
25von Kries J. Die Gesichtsempfindungen. In: W Nagel, ed. Handbuch der Physiologie der Menschen. Vol 3. Vieweg und Sohn; 1905.
26Seymour J. What color is the dress? John the Math Guy Blog. 2015. Accessed January 7, 2022. https://johnthemathguy.blogspot.com/2015/02/what-color-is-dress.html
27Ebner M. Color Constancy. Wiley; 2007.
28Shevell SK. The Science of Color. Elsevier; 2003.
29Valberg A. Light Vision Color. Wiley; 2005.
30Newhall SM, Burnham RW, Clark JR. Comparison of successive with simultaneous color matching. J Opt Soc Am. 1957; 47: 43- 56.
31Bartleson CJ. Memory colors of familiar objects. J Opt Soc Am. 1960; 50(1): 73- 77.
32Bae GY, Olkkonen M, Allred SR, Flombaum JI. Why some colors appear more memorable than others: a model combining categories and particulars in color working memory. J Exp Psychol Gen. 2015; 144(4): 744- 763.
33Luo MR, Clarke AA, Rhodes PA, Schappo A, Tait CJ. Quantifying colour appearance. Part 1. Lutchi colour appearance data. Color Res Appl. 1991; 16(3): 166- 180.
34Smet KAG, Zhai Q, Luo MR, Hanselaer P. Study of chromatic adaptation using memory color matches, Part I: neutral Illuminants. Opt Express. 2017; 25(7): 7732- 7748.
35Hwang J, Hong Y, Lee DH, et al. Experimental method for measuring color appearance shifts in high-dynamic-range luminance conditions. J Opt Soc Am A. 2019; 36: 1940- 1948.
36O'Hara L, Congdon B, Lindsey K. Optimizing chroma and lightness of OGV inks for expanded color gamut. Technical Association of the Graphic Arts. 2019.
37Sharma A, Seymour J. Evaluation of expanded gamut software solutions for spot color reproduction. Color Res Appl. 2019; 31: 315- 324.
38Li Z, Berns RS. Comparison of methods of parameric correction for evaluating metamerism. Color Res Appl. 2007; 32(4): 293- 303.

from Hacker News https://ift.tt/XQPVhAW

Wednesday, February 23, 2022

Color inconstancy in CIELAB: A red herring?