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Estimating physical reflectance spectra from human colormatching experiments

Contributed by A. Kimball Romney
Abstract
This article presents methods for estimating Munsell reflectance spectra, measured physically with a spectrophotometer, from psychophysically derived colormatching functions. The method is general and may also be used to estimate the reflectance spectra from human cone photoreceptor sensitivities. The colormatching functions and the cone sensitivities were found to contain almost identical information and may be considered to give equivalent estimates. The physical description of the Munsell color structure under D65 illumination was compared to the structure estimated from the human perceptual judgments. The results are virtually indistinguishable.
This article investigates the relationship between two distinct levels of color representation, namely, the reflectance spectra of Munsell color samples as measured by a spectrophotometer and human perceptions of color obtained in psychophysical experiments. Two measures derived from human perceptual judgments are considered: colormatching functions and spectral sensitivities of human cone photoreceptors. The two major goals of this article are: first, to present a method for estimating physical reflectance spectra with measures from psychophysical data, and, second, to compare the physical description of the Munsell color structure under D65 illumination to the structure estimated from measures derived from human judgments.
A previous paper (1) demonstrated that most of the variation among spectral reflectance curves of 1,269 matte Munsell color chips was well represented in a 3D Euclidean space. In a subsequent paper (2) we modeled both the color samples and the spectral wavelengths (430–660 nm) of 370 Munsell color samples under D65 illumination in a common 3D Euclidean space, oriented to yield the most interpretable structure with respect to the Munsell color structure.
The Munsell system consists of painted standard color chips arranged according to cylindrical coordinates (H, V/C) where the vertical axis in the center represents lightness V (value) from black to white, the polar angle from the center represents H (hue), and the distance from the center represents saturation C (chroma). Along each axis, adjacent chips are arranged to differ in intervals of equal perceptual size δ, although the size of δ of an attribute is not defined to be the same as that of another attribute. The scale V is 0–10 V (black to white) and how far the scale of C extends from 0 depends on H and V. The hue circle is divided into 10 H sectors, five principal hues (red, yellow, green, blue, purple) and five intermediate hues such as YR, GY, etc. Each H sector is further divided into four finer categories, namely, 2.5H, 5H, 7.5H, and 10H (0 for the next H). The ideal color of each H sector is taken to be 5H. Based on very extensive visual examination, the Optical Society of America determined the colorimetric data (x, y, Y) of each chip, and the chips are painted to meet these specifications (3).
The previous paper (2) was based on a subset of spectral reflectance measurements from the 1,269 chips in the Munsell color book from 380 to 800 nm at 1nm resolution (geometry 0/d) made with a spectrophotometer. The data were obtained from the web site www.cs.joensuu.fi/∼spectral/databases/download/munsell_spec_matt.htm. The spectra selected for analysis, 430–660 nm, approximate the range of human vision. The analysis was limited to a sample of the 360 most representative Munsell hues, namely, (5R, V/C), (5YR, V/C), … , (5PR V/C), where V covers 2, 2.5, 3, 4, 5, 6, 7, 8, 8.5, 9V and C covers the whole range of chroma. The data were denoted as S = (s_{jμ}), where j = 1, 2, … , N represents the Munsell chips and μ = 1, 2, … , M represents wavelength λ from 430 nm to 660 nm (M = 231). Ten flat reflectance vectors were added in which s_{jμ} is constant over λ, and s_{V} corresponds to Munsell V, where ideal white (10V) s_{10} = 1.0 and the other nine values are 0.78660 (9V), 0.59100 (8V), 0.43060 (7V), 0.30050 (6V), 0.19770 (5V), 0.12000 (4V), 0.06555 (3V), 0.03126 (2V), and 0.01210 (1V). Thus, the total sample size, including the 10 achromatic samples, was N = 370. Finally, each column μ of S_{370×231} = (s_{jμ}) was multiplied by e_{μ}, the spectral radiant power distribution of D65 light (with an arbitrary unit) and the data set analyzed were denoted as SE_{370×231} = (s_{jμ}e_{μ}) = (se_{jμ}).
The present research depends on the major results of the previous paper (2), which presents the color chips in a 3D Euclidean space with coordinates, (P̄_{370×3}), oriented to give the most interpretable structure with respect to the Munsell color structure. The criteria for the orientation were to match the 10 achromatic chips to the first dimension and then to minimize the thickness of the Munsell value plates when viewed from the side (see Fig. 3B). In this orientation axis 1 roughly corresponds to the mean power of the spectral reflectance (approximate Munsell value), axis 2 goes from Munsell red to bluish green, and axis 3 goes from Munsell greenish yellow to purple.
In the present article, we compare the earlier obtained Euclidean structure to results estimated from the 10° colormatching functions of Stiles and Burch (4) and the cone photoreceptor sensitivity curves reported by Stockman and Sharpe (5). We used cone fundamentals in 1nm steps in terms of energy (linear) from the web site http://cvrl.ioo.ucl.ac.uk/database/data/cones/linss2_10e_1.txt. The Stiles and Burch curves are based on colormatching data from 49 observers with monochromatic primary stimuli R, G, B (wavelength number m = 15,500, 19,000, and 22,500 cm^{−1}). The data we used are in 5nm intervals of wavelength from table 3 of ref. 5.
The data in this article are a subset of the earlier study (2) and designated as SE_{370×47} = (se_{jμ}), that is, the spectra of reflected light from selected Munsell chips under D65 illumination in 5nm intervals. To illustrate what these spectra look like, Fig. 1A shows the reflectance spectra of the five primary Munsell hues for V = 7 and C = 8.
Fig. 1B shows the three colormatching functions of Stiles and Burch (4) measured at 10° that we will designate as Θ_{3×47} = (θ_{pμ}) with p = 1, 2, 3 [θ_{1} = r̄(λ), θ_{2} = ḡ(λ), θ_{3} = b̄(λ)] for monochromatic primary stimuli, R, G, B, respectively. The solid lines in Fig. 1C show the Stockman and Sharpe cone sensitivity curves, which we will designate as C_{3×47} = (c_{qμ}) with q = L, M, S for long, medium, and short, respectively. The Stockman and Sharpe curves are normalized to be 1.0 at their peaks. The curves, θ_{pμ} and c_{qμ}, both derived from psychophysical experiments, are interpreted as follows. A monochromatic light μ is matched by the additive mixture of θ_{1μ} of R, θ_{2μ} of G, and θ_{3μ} of B, and the light μ excites cone L with the degree k_{L}c_{Lμ}, cone M with the degree k_{M}c_{Mμ}, and cone S with the degree k_{S}c_{Sμ}. Values of coefficients k_{q} are not specified yet.
This article demonstrates a general procedure for estimating the spectra for any color sample by the use of multiple regression coefficients from any set of three spectra such as the Stiles and Burch colormatching functions or the Stockman and Sharpe cone sensitivities. We can illustrate the method by estimating the cone curves from the Stiles and Burch colormatching functions. The dotted curves in Fig. 1C are the estimated values of c_{q} obtained by multiple regression on θ_{p}, using the formula, The canonical correlation between Θ and C is one to better than four significant digits and the Pearson correlations between the solid lines and the dotted lines in Fig. 1C are all above r = 0.99. We interpret this to mean that the cone sensitivity curves have very nearly the same information in them as do the colormatching functions. In fact, any linear combination of variables, based on colormatching functions or cone sensitivities, such as opponent processes based on combinations of cone sensitivities, would give similar results. All the results in the remainder of the article are based entirely on Θ since it would be redundant to use C. We have made calculations based on C and find no noticeable differences with those reported below based on Θ.
Two related methods are used to estimate the reflectance spectra of each Munsell chip from the colormatching functions. The first method is to estimate each spectra separately, as illustrated in the calculation estimating cone sensitivity curves above. In this case we obtain individual spectra of Munsell color samples by estimating values of se_{jμ} by multiple regression on θ_{pμ}, where We included the constant a_{j} since estimates were more accurate than without it.
The second method is to estimate all the spectra simultaneously by using information derived from basis functions (spectral eigen vectors) obtained from the original raw data, SE_{370×47} = (se_{jμ}). To do this, a little notation will be useful. The best rankk approximation of SE is obtained from a singular value decomposition that gives SE_{(k)} = UΔV^{T} where Δ = () is a k × k diagonal matrix and U^{T}U and V^{T}V are identity matrices. The values of λ_{α} and k columns of V are obtained as the first k largest eigenvalues and eigenvectors of SE^{T}SE, and U = SE_{(k)}VΔ^{−1}. In the previous two articles (1, 2) it was shown that a 3D solution is sufficient, so k = 3, α = 1, 2, 3. We now define SE_{(3)} = (se_{jμ(3D)}) =PW^{T}, P = (p_{jα}) =UΔ, W = (w_{μα}) =V. The configuration of points representing N Munsell colors in a 3D space is governed solely by P and (se_{jμ(3D)}) is obtained as the modulation of W by P.
To satisfactorily represent the Munsell color system it is necessary to perform a rigid Procrustus rotation of P to P̄ (P̄ obtained from ref. 2). Coordinates for P̄ are available as a web supplement to ref. 2 at www.pnas.org. After an appropriate rotation matrix R was obtained, we defined (instead of P̄), P̃_{N×3} = PR and W̃ = WR so that SE_{(3)} = PW^{T} = PRR^{T}W^{T} = P̃(WR)^{T} = P̃W̃^{T}. In Fig. 2 the colored solid lines represent the spectra of the five primary Munsell hues, and the colored dotted lines represent the 3D reconstruction contained in SE_{(3)}. It can be seen that the fit is quite close.
The values of W̃ may be estimated in the same way from the Stiles and Burch colormatching functions as were the cone sensitivity curves earlier. This is done by estimating values of w̃_{αμ} by multiple regression on θpμ, where Fig. 1D presents the comparison of the actual w̃_{αμ} (solid lines) with the w̃ (dotted lines) estimated from the Stiles and Burch colormatching functions. This result may be used to obtain an aggregate estimate of SE by substituting W̃^{Θ} for W̃ to obtain SE^{w̃Θ} = (se) =P̃(W̃^{Θ})^{T}. Fig. 2 shows a comparison of this aggregate measure se with the previously obtained individual estimate se from the Stiles and Burch colormatching functions for the five primary Munsell hues. The aggregate estimates are the solid black lines and the individual estimates are dotted black lines. The two estimates are very close.
An examination of Fig. 2 shows that for three of the primary Munsell hues (red, green, and purple) the actual spectra (and their 3D reconstruction) are quite well estimated with estimates derived from the Stiles and Burch colormatching function (and the cone sensitivity curves). Two facts are noticeable in Fig. 2. One is that agreement between se_{jμ(3)} and se_{jμ} is good in all the cases, which implies that the 3D representation is sufficient. The other is that both se and se are in agreement with se_{jμ} in one opponent direction (redgreen, Fig. 2 A and C) but not quite in the other opponent direction (yellowblue, Fig. 2 B and D). The same deviations from se_{jμ} were found with estimates obtained from the cone sensitivity curves C. These deviations seem to stem from θ_{3μ} in Θ as well as c_{Sμ} in C. The effects of these differences may be examined in more detail by looking at the effects they have on the relative placement of the Munsell color samples viewed in a spatial representation.
By a singular value decomposition of SE^{w̃Θ}, we obtain new P^{Θ} and W^{Θ}, SE^{w̃Θ} =P^{Θ}W^{Θ}, with Δ^{Θ}. This P^{Θ} can be converted to P̃^{Θ} = P^{Θ}R by the same rotation matrix R used before. Then, P^{Θ} and W^{Θ} as well as P̃^{Θ} and W̃^{Θ} can be plotted in the same space. Because these stem from P and W that are defined in different scales [P^{T}P = Δ^{2}, and W^{T}W = I (identity matrix), and diagonal elements of Δ^{2}, λ_{α}, is much larger than 1], we used W̃Δ and W̃^{Θ}Δ^{Θ} for these plots.
Fig. 3 shows a comparison of the placement of the Munsell color chips by the two methods simultaneously with the corresponding spectral basis factor plots. The black vectors for the color sample plots originate at the symbol representing P̃ based on the physical measures, while the unmarked end of the vector represents the P̃^{Θ} based on the Stiles and Burch regression estimates. The corresponding spectral values are represented by red vectors. Notice that the greatest discrepancies in Fig. 3 occur on the redyellow to blue axis, which is consistent with the discrepancies in the estimates of the color spectra shown in Fig. 2. The discrepancies occur with both the location of the color samples and in the location of the spectral estimates, although the differences W̃ and W̃^{Θ} are much more prominent than P̃ and P̃^{Θ}. The latter is almost invisible. Overall, the correspondence between the location of the Munsell color samples based on the two estimates are quite close. The axis of the discrepancies is not quite vertical so it does not quite match with the orientation obtained previously (Fig. 3B).
A method for estimating Munsell reflectance spectra from cone sensitivity curves or colormatching function has been presented. It was found that cone sensitivity curves and colormatching functions provide virtually identical estimations and may be considered substitutes for each other. They provide quite accurate estimations of the Munsell reflectance spectra, particularly with respect to the location of the color samples. The estimates of the location of the spectra reveal somewhat greater discrepancies. In both cases the errors occur in the redyellow to blue axis of the Munsell color structure with virtually perfect correspondence in the opposing purple to yellowgreen axis. It is noteworthy that the colormatching function based entirely on psychophysical measurements should correspond so closely with the spectrophotometer measures that are derived solely from physical measures.
It should be noted that the estimation of any single spectra from three colormatching or three cone sensitivity spectral functions may be generalized over a wide variety of other functions or subsets of three spectral functions. Examples would include dispersed sets of Munsell color sample spectra, selection of three phosphors in color monitors, etc.
Acknowledgments
The research was funded in part by National Science Foundation Grant SBR9631213 to A.K.R. and William H. Batchelder.
 Accepted September 16, 2002.
 Copyright © 2002, The National Academy of Sciences