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

*metamerism*, but we prefer the less ambiguous term

*metamer mismatching*(Logvinenko et al., 2013). This article explores the relationship between metamer mismatching and how accurately observers can distinguish pairs of very similar colors.

*how*neural mechanisms implement the computation of color discrimination. In contrast, the metamer mismatching hypothesis aims to help explain

*why*the visual system computes the differences between colors the way it does. These are parallel, but fundamentally different, questions. However, progress in answering one can be expected to be useful to answering the other.

*R*, which is known as the object color solid (OCS). In contrast to the gray, which is at the center of the OCS, for any color signal on the boundary of the OCS, there is only one possible reflectance creating it, so the volume of the MMB drops to zero for such color signals. This is illustrated by the plot in Figure 1.

^{n}*. The MMB of*

**C**_{D65}*for a change of illumination to CIE A, for example, represents the set of all colors that could result when S is lit by CIE A instead of D65. From the reverse perspective, under CIE A, any color*

**C**_{D65}*in that MMB is a candidate for matching*

**C**_{A}*under D65. Now suppose that we observe*

**C**_{D65}*; does it correspond to S? The answer is “almost certainly not” because*

**C**_{A}*could have arisen from any one of an infinite set of metameric reflectances, only one of which is reflectance S. Hence, the MMB represents the “uncertainty” in being able to identify a specific surface such as S by its color under some other illuminant (i.e., CIE A in this example).*

**C**_{A}*that it matches*

**C**_{D65}*. The uncertainty then becomes the union of the MMB of*

**C**_{D65}*and the MMB of \({\boldsymbol C}_{{\boldsymbol D}65}^{{\boldsymbol \prime}}\) along with the MMBs of all colors in between. In other words, it is the set of all colors that either*

**C**_{D65}*or \({\boldsymbol C}_{{\boldsymbol D}65}^{{\boldsymbol \prime}}\) (or those in between) could become under CIE A. Given a threshold for an acceptable level of uncertainty (keeping in mind that metamer mismatching means that some uncertainty is unavoidable), how does the tolerance for error in color discrimination vary as a function of color? To provide some further intuition, Figure 2 shows the trend from gray to both blue and red (Munsells 5B 5/6 and 5R 5/8). The figure is based on keeping the volume of the convex hull of the two MMBs fixed. That choice of volume is quite arbitrary other than needing to be somewhat larger than the volume of the MMB for flat-spectrum gray under a change in illuminant from D65 to CIE A. The convex hull of the two MMBs provides a good approximation to the union of the infinite set of MMBs for all points between the two colors. The qualitative upward trend in the Figure 2 (bottom) is unaffected by the precise number. Note that this example is only an illustration, not a complete model (e.g., it models distance, not volume, and it will fail for colors approaching the boundary of the object color solid where in the limiting case, the volume of the MMB tends to zero). The figure is intended to provide some intuition as to how the uncertainty reflected in metamer mismatching could affect the size of discrimination ellipsoids, but intuition only. A formal statistical analysis of the evidence of the relationship between metamer mismatching and ellipsoid volume is presented in the next section.*

**C**_{D65}*∆*\({\boldsymbol E}_{{\boldsymbol ab}}^*\) units (CIELAB color space). They asked the observers to compare the magnitude of the color difference of the sample pair to that of the anchor pair. Probit analysis was then used to compute 156 median tolerances around 19 color centers in different directions. Melgosa et al. (1997) then used the 156 median tolerances reported in the RIT DuPont data set (Berns et al., 1991) to compute the ellipsoid parameters in x, y, Y/100 color space. For our purposes, we convert the Melgosa ellipsoids’ coefficients to XYZ color space. The four data sets explained above are summarized in Table 1.

*, in XYZ, as described in the Appendix. For each color center, the volume, M, of the corresponding MMB for a change in illuminant from CIE D65 to CIE A is computed directly in XYZ space using the algorithm of Logvinenko et al. (2013).*

**E**_{vol}*is then normalized by*

**M***, the cube of the Euclidean distance,*

**C**^{3}*, from the origin to the given color center. This normalization eliminates the effect of the intensity/luminance on the volumes.*

**C***and*

**E**_{vol}

**C**^{3}/

*. Note that*

**M**

**C**^{3}/

*is dimensionless. The jackknife method is used to examine the accuracy of the correlation coefficient estimates. Jackknife uses a leave-one-out strategy to derive the bias in an estimator, resulting in a bias-corrected estimate of the original statistic.*

**M***E*and

_{vol}

**C**^{3}/

*for the Melgosa (i.e., 19 color centers based on the Berns et al. [1991] data) and Huang (17 color centers) data sets are shown in Figure 3. The figure caption includes the Pearson correlation coefficient (*

**M***r*), mean jackknife estimate of

*r*, bias, and standard error (

*SE*) in jackknife replicates. A

*y*-intercept is included in the linear regression model. The null hypothesis is rejected at the 5% significance level, with

*p*values in all cases being less than 10

^{−5}. The results are as follows:

*r*= 0.83 with a mean jackknife estimate of

*r*= 0.83, bias = 0.03, and

*SE*= 0.13 for the Melgosa data set; and

*r*= 0.9 with a mean jackknife estimate of

*r*= 0.9, bias = −0.05, and

*SE*= 0.11 for the Huang data set. The correlation between

*E*and

_{vol}

**C**^{3}/

*includes the nonlinearity of the inverse 1/*

**M***. The corresponding, simpler (negative) linear correlation results between*

**M***E*and

_{vol}*/*

**M**

**C**^{3}are significantly weaker: −0.52 and −0.7, respectively.

**S**) to adjust the ellipses onto a common scale. They showed that adjusting the individual ellipses with \({\bf{\bar R}}\) values results in a more consistent plot than using the group mean for each data set.

*r*= 0.88 and a

*p*value of 7e-16.

*in CAM16-UCS around each of the color centers included in the four data sets, convert them to XYZ coordinates, and fit the volumes of the resulting ellipsoids in XYZ to those of the color discrimination ellipsoids. Figure 5 shows the Pearson correlation coefficient (*

**E***r*) between the unit Δ

*spheres in CAM16-UCS color space (ellipsoids in XYZ space) and the discrimination ellipsoid volumes reported in the data sets, the mean jackknife estimate of*

**E***r*, and the bias and standard error (

*SE*) in jackknife replicates.

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**g**

_{ik}, the ellipsoid parameters in one coordinate system, such as the instrument's RGB, the following relation recommended by Brown and MacAdam (1949) can be used to convert them to another color space such as XYZ:

*G*

_{jl}values are the ellipsoid parameters in XYZ color space, and the partial derivatives are obtained using the transformation equations:

*R*,

*G*,

*B*in Equation 2 to Equation 4 should be replaced with

*x*,

*y*,

*Y*. The transformation between

*xyY*and XYZ is given by

*G*

_{11},

*G*

_{12},

*G*

_{13},

*G*

_{22},

*G*

_{23},

*G*

_{33}coefficients are equal to \(\frac{1}{{\sqrt {{{\boldsymbol{\sigma }}_{\bf{i}}}} }}\) when

**σ**

_{i}(

**i**= 1, 2, 3) are the roots of the following equation:

_{1}*σ

_{2}*σ

_{3}). The roots of Equation 21 (

**σ**

_{i}) are in fact the eigenvalues of the matrix Γ. The product of the eigenvalues of a matrix is equal to the determinant of that matrix. Therefore, rather than solving Equation 21, the determinant of matrix Γ can be used in the ellipsoid volume calculations: