Similarity of Psychological and Physical Colour Space Shown by Symmetry Analysis Lewis D. Griffin Medical Imaging Science IRG, King’s College, London, UK

Received 20 March 2000; accepted 26 July 2000

Abstract: Data on the psychological structure of colour space was gathered from a dictionary of colour terms and by collecting responses to questions of the form, “which is the more similar pair: A and B or C and D?” and “which is lighter: A or B?” Where A, B, C, and D were drawn from the eleven Basic Colour Terms. Analysis showed that the data possessed an approximate symmetry corresponding to the exchange of Red and Orange, Blue and Green, Purple and Yellow, Black and White, Pink and Brown, and “darker” and “lighter.” Analysis of a physical model of colour vision showed that a similar symmetry exists in the structure of the space of possible colour sensations. © 2001 John Wiley & Sons, Inc. Col Res Appl, 26, 151–157, 2001

Key words: color distance; color diagrams; color spaces; mathematical psychology; lightness INTRODUCTION

Perceptual quality spaces, such as colour, brightness and pitch, have a psychological structure and a physical structure.1 Dimensionality and global shape are aspects of both types of structure.2,3 In some cases, agreement between the two structures is straightforward. For example, the psychological structure of the space of brightnesses is a one-dimensional continuum bounded at one end by a special brightness (that of black).4 The physical structure is determined by a sequence of processes beginning with illumination and ending with the physiological implementation of the photopic sensitivity function V␭. The output of this process, with which the psychological structure agrees in dimensionality and global shape, is modeled as a positive real number. A more difficult case is the space of illuminant colours.

Correspondence to: Lewis D. Griffin, Medical Imaging Science IRG, King’s College, London, UK (e-mail: [email protected]) © 2001 John Wiley & Sons, Inc.

Volume 26, Number 2, April 2001

An important landmark in the study of this space was Newton’s colour disc,5 which made explicit that illuminant colour is psychologically a two-dimensional continuous manifold with a simply connected boundary. This finding has since been confirmed with more repeatable methods6 and accords well with contemporary physical models of colour vision. The continuous manifold structure corresponds to the fact that the cone responses to a light stimulus vary continuously with the spectral content of the stimulus. The boundary corresponds to the colours of monochromatic illuminants in the visible range plus the colours of purple illuminants (which are the various mixtures of monochromatic light from either end of the visible spectrum). A yet more difficult case is the subject of this article, the space of body colours. Even if the range of body colours is restricted to the colours of Lambertian surfaces, thus eliminating, for example, metallic and gonio-chromatic surfaces, the psychological structure of this space is difficult to discern. Rather than being accessible to the introspection of everyman, current descriptions of the psychological structure have been arrived at only through a sustained process of refinement of previous descriptions.7 The most famous endpoint of this process of refinement is the Munsell colour solid.8 The general form of this solid was chosen by Munsell, while the exact shape was determined by multiple subjects’ judgments of the relative similarity of paint samples. Later studies9 have confirmed that the Munsell atlas captures psychological structure well. The physical structure of colour space was determined by Schro¨dinger10 and found to be similar to the Munsell solid. I aim here to show further evidence for agreement between the psychological and physical structures of body colour. This agreement is demonstrated by showing the similarity between a pair of approximate symmetries — one of the psychological structure and the other of the physical. Data on the psychological structure was gathered by questionnaire and from a colour-naming system. A contempo151

rary model of the processes of colour vision determined the physical structure.

MAPPING THE PSYCHOLOGICAL STRUCTURE

Data on the psychological structure of quality spaces can been acquired by the irreproducible method of introspection11 or by the reproducible method of averaging the judgments of multiple subjects.9 It has been shown that subjects give similar responses (97% correlation) to questions of relative colour similarity posed using colour names or actual colour samples.12 The method of colour names was used here. Unlike previous studies, a set of object colour names rather than hue names was used. The colour names used were the eleven Basic Colour Terms13 (BCTs): Black, Gray, White, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Brown. Three aspects of colour space were mapped: the similarity and lightness structures and the topology. Preliminary findings from this mapping process were presented previously.14 The similarity structure was mapped by collecting responses from 195 subjects to questions of the form “which is the more similar pair: A & B or C & D,” where A, B, C, and D were BCTs. The data were collected using randomly generated questionnaires, typically containing 200 questions. The lightness structure was also mapped by questionnaire: responses were collected from 47 subjects to questions of the form “which is lighter: A or B?”* All subjects had self-assessed normal colour vision and spoke English as their main language. Their mean age was 27 years (s.d. ⫽ 10 years), and 59% were female. In analyzing the responses, ordering within a pair of BCTs and within a question was ignored. Thus, there were 1,485 distinct similarity questions and 55 distinct lightness questions. A total of 47,557 similarity judgments were collected, resulting in an average of 32 responses to each distinct question. For lightness judgments, 2,560 responses were collected, resulting in an average of 47 per distinct question. The proportion of responses according with the majority, averaged across distinct questions, was 79% for similarity judgments and 89% for lightness. To map the topological structure of colour space, an adjacency model was derived from a colour naming system15 for the Munsell atlas8 of colour samples. In this model, two BCTs X and Y are adjacent if and only if some Munsell sample is named “X-Y,” “Y-ish X,” etc., or there are adjacent Munsell samples with the labels X and Y. This rule gives rise to 32 pairs of adjacent BCTs.† For subsequent analysis, the topology data was treated in the same manner as the similarity and lightness data, but with only one response per question.

* Although it was not checked, I assume in subsequent analysis that (on average) a subject who answers A to “which is lighter: A or B?” would answer B to “which is darker: A or B?” † The rule about adjacent Munsell samples adds Gray-Black, GrayWhite, and Pink-Red to the list of adjacent BCT pairs.

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SYMMETRIES OF THE PSYCHOLOGICAL STRUCTURE

A structure possesses symmetry, if it retains its form under a group of transformations.16 It possesses an approximate symmetry, if it approximately retains its form. The transformations that are considered here are permutations of the BCTs and permutations of {“darker,” “lighter”}. Thus, the possible symmetries of psychological colour space (as measured here) are identified with groups of permutations of the BCTs, with or without the exchange of “darker” and “lighter.” To appreciate what a symmetry of the database of responses would amount to, imagine data being collected in the following manner. Suppose that questions are posed in the forms “which is more similar: A & B or C & D?” and “which is ␣: A or B?” And further suppose that accompanying each questionnaire is a key that gives (i) a translation from uppercase letters into BCTs, and (ii) says whether ␣ should be read as “darker” or “lighter.” The existence of a symmetry could be investigated by having two cohorts of subjects complete questionnaires, with each cohort using a different key. If the judgments of the two cohorts were found to be statistically distinguishable, the proposed symmetry, encoded in the transformation between the two keys, would be rejected. Fortunately, given the number of potential symmetries, this laborious method of data collection is unnecessary, because the assessment can be made by comparing the response database to a transformed version of itself. Given a permutation of the BCTs, and a permutation of {“darker,” “lighter”}, the transform of a particular question is obtained by (i) transforming each BCT in the question, and (ii) transforming any occurrence of “darker” or “lighter.” The observed response pattern to the full list of questions can be transformed by replacing the responses to each question by the responses to the transformed question. Each transformation can be assessed by comparing the observed response pattern to a transformed version of it. Each possible symmetry group can be assessed by comparing the response patterns corresponding to each pair of elements of the group. Statistical testing rejects all symmetries in their exact form (␹2 test, p ⬍ 10⫺400), but there are symmetries that are approximate. The quality of an approximate symmetry is measured by the degree to which form is retained under transformation. As fuzzy data is being analyzed, it is appropriate to normalize the degree to which the data matches a transformed version of itself, by the degree to which the data matches an untransformed version of itself. For a given question, this normalized degree of match under transformation can be assessed from the probabilities that the question will receive the same answer if asked of (i) two subjects and (ii) a subject and a hypothetical subject with a transformed response pattern. The ratio between the latter and the former probabilities is termed agreement and can be calculated for similarity, lightness, and topology data. For all three aspects of structure, the agreement is averaged across all questions COLOR research and application

TABLE I. Good approximate symmetries of the basic colour terms (BCTs).

The “Symmetry” column of the table gives the permutation transformation that generates the symmetry, unmentioned BCTs being unaffected. Ticks show the symmetries for which “darker” and “lighter” are exchanged. “Agreement” columns give rates as percentages, followed by the rank of these rates within the set of possible symmetries. Agreement rates are normalized as described in the text. “Worst Flaw” columns show the question (upper line), the responses to which are most dissimilar to the responses to the transformed question (lower line). The two numbers following each question are the responses (agreeing/disagreeing). For similarity flaws, “⬍” should be read as “are more similar than.” For the lightness flaws, “⬎” should be read as “is lighter than,” and “⬍” as “is darker than.” The “Example Flaw” column shows a violation of the symmetry (selected as the most serious by the author) according to the topological model derived from the colour-naming system. Pairs of colours with a tick are adjacent in the model; those with a cross are not. The symmetries of the shaded row agree well with the approximate rotational symmetry of the body colour solid.

that are affected by the transformation being assessed. Unaffected questions are ignored, because their inclusion artificially raises the apparent quality of symmetries that affect few BCTs. Agreement rates for all possible symmetries were calculated. The symmetries were separately ranked by the agreement rates for symmetry, lightness, and topology (S-, L-, and T-ranks). From these ranks, a composite SLT-ranking was computed according to the rule 共S 1 ⬍ S 2 ⵩ L 1 ⬍ L 2兲 ⵪ 共S 1 ⬍ S 2 ⵩ T 1 ⬍ T 2兲 ⵪ 共L 1 ⬍ L 2 ⵩ T 1 ⬍ T 2兲 N SLT 1 ⬍ SLT 2.

A widely used model18 of the physical processes of body colour perception is as follows. An illuminant with spectral energy described by the nonnegative function I is reflected by a surface with spectral reflection described by the function C, which takes values in the interval [0,1]. The reflected light is filtered by ocular structures with spectral transmission described by the function F, and detected by the linear operation of three classes of retinal cone with spectral absorption efficiencies described by the nonnegative functions S, M, and L, resulting in the triple of numbers:

(1)

Table I shows a selection of the best symmetries. The overall best (top row) is the exchange of Red and Orange, Purple and Yellow, Green and Blue, Black and White, Pink and Brown, and “darker” for “lighter.” The degree to which the symmetry is approximate rather than exact can be assessed from its agreement rates of 91.4%, 96.5%, and 80.0% for similarity, lightness, and topology, respectively. By making a spatial diagram of the BCTs, following in the tradition of such diagrams,7,17 the transformation can be understood as a 180° rotation about an axis through Orange/ Red-Gray-Green/Blue (Fig. 1). Volume 26, Number 2, April 2001

PHYSICAL STRUCTURE

cជ SML ⫽

冕冉 冊

S M .F.C.I. L

(2)

Conventionally, this cone response space is referenced with a coordinate system different from the S, M, and L axes. Instead, the linearly related coordinate system, defined by the action on the unfiltered reflected light of the colourmatching functions X, Y, and Z, is used, i.e.: cជ XYZ ⫽

冕冉 冊

X Y .C.I. Z

(3)

This convention is followed here. 153

FIG. 1. The psychological structure of the Basic Colour Terms (BCTs). The spheres represent BCTs and the thick rods their adjacencies. The configuration follows previous colour diagrams in layout, but also has the symmetry identified in the top row of Table I. The associated transformation is manifest as a 180° rotation about the Orange/Red-GreyGreen/Blue axis shown by the thin rod.

For a given illuminant energy function and varying object reflection function, not all cone responses are possible. The possible responses form a bounded convex subvolume of cone response space referred to as the body colour solid [Fig. 2(a)]. The surface of the solid corresponds to the cone responses due to bodies with reflectance functions that are {0,1}-valued and with at most two wavelengths where the reflectance changes between 0 and 1.10,19,20 These reflectance functions are called optimal, because they produce highly chromatic colours. Almost all the surface is due to optimal reflectance functions with two transitions, and at such points the surface is smooth, due to the smoothness of the cone response functions. In contrast, at the two surface points corresponding to optimal reflectors with no transitions, the surface is not smooth. On point is the black point (0ជ ) corresponding to a reflection function that is zero everywhere, and the other the white point (w ជ ) corresponding to a reflection function that is unity everywhere. At both of these points, the surface is locally like the vertex of an elliptical cone. The points corresponding to optimal reflectors with only one transition are also points where the surface is not smooth. These points lie on a pair of curves, both running between the black and white points. One curve corresponds to short-wavelength-reflecting single-transition optimal reflectors and the other to long-wavelength-reflecting singletransition optimal reflectors. There is a discontinuity in the

FIG. 2. Symmetries of the body colour solid. (a) The body colour solid for the daylight-approximating illuminant D65. The lower vertex corresponds to a black body, the upper vertex to a white. The black curve girdling the solid is the full colour locus. Other features are explained under (b). (b) The purple and yellow curve segments show the intersection of the plane of approximate affine reflection with the solid. The orange-red and blue-green line shows the axis of approximate affine rotation. As in (a), the black curve is the full colour locus; the red curve is its reflected or rotated image.

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COLOR research and application

tangent plane of the surface across these two curves. The condition for this discontinuity to occur at a point corresponding to an optimal reflector with a single transition at the wavelength ␭ is that

冉

X共 ␭ 兲 Y共 ␭ 兲 Z共 ␭ 兲 X共 ␭ ⫺兲 Y共 ␭ ⫺兲 Z共 ␭ ⫺兲 X共 ␭ ⫹兲 Y共 ␭ ⫹兲 Z共 ␭ ⫹兲

冊

(4)

has full rank (where ␭⫺ and ␭⫹ are the wavelength limits of visual sensitivity). This condition holds for all wavelengths within the visible range, and, hence, the curves are ridgelike throughout their lengths. Between the black and white points, the achromatic axis runs through the interior of the solid. Girdling the solid is a locus of points where the surface tangent plane is parallel to the achromatic axis. Because the corresponding colours are maximally saturated, this is called the full colour locus.21 SYMMETRIES OF THE PHYSICAL STRUCTURE

Both the location of the white point (w ជ ) and the precise shape of the body colour solid are dependent on the illuminant. However, whatever the illuminant, the solid always has an exact symmetry21: E共cជ 兲 ⫽ w ជ ⫺ cជ

(5)

which can be interpreted as inversion in the mid-gray point bisecting the achromatic axis. This symmetry exists, because the inversion transformation exchanges the cone responses to a spectral reflectance function ␭ 3 C(␭) with the cone responses to the complementary spectral reflectance function ␭ 3 1 ⫺ C(␭). This transformation maps optimal reflectance functions to optimal reflectance functions. For illuminants with flattish energy spectra, the body colour solid has the same approximate symmetries as the parallelepiped has exact22 (i.e., the RGB cube is not far from the truth). The transformations corresponding to these symmetries are approximately the permutations of the scaled signals from the three cone classes. There are three transformations that swap a pair of cone classes and two transformations that permute all three. The scaling is necessary to equate the response to white when permuting channels. For example, the exchange of the S and M cone classes is given by

冉冊 S M L

3

冢 冣

S white M M white M white . S S white L

(6)

Of these five approximate symmetries, the one (K) that is approximately the exchange of the M and L cone classes is particularly exact. In XYZ space, K is an affine reflection with a plane of reflection containing the achromatic axis. The reflection K depends on the illuminant, though it is similar for all flattish spectra. I have calculated K for the daylight-approximating illuminant standard D65. D65 was Volume 26, Number 2, April 2001

chosen, rather than, say, the equal energy illuminant, because it best represents natural viewing conditions. To find K, I identified the affine reflection (in a plane containing the achromatic axis) that best preserves the shape of the body colour solid. A natural measure of the degree of shape preservation is the overlap (intersection volume divided by union volume) of the body colour solid and its image. The significance of the overlap measure is that it is an affine invariant and so does not depend on the use of XYZ coordinates, which are after all arbitrary. The overlap is, however, difficult to compute. Instead, I use the RMS distance between the full colour locus and its image, the justification being that sketch arguments show that minimizing the RMS distance (to a first approximation) maximizes the overlap measure. Once the best transformation according to this surrogate measure has been identified, the corresponding overlap measure can then be estimated to evaluate the quality of the symmetry found. For the case of D65, the best (in the surrogate measure sense) K is given by the matrix transformation: K共cជ XYZ兲 ⫽ H 䡠 cជ XYZ,

(7)

where

冉

冊

⫺0.416 1.165 0.165 0.640 0.473 ⫺0.075 H⫽ . 0.493 ⫺0.406 0.943 That this describes an affine reflection is shown by H2 ⫽ I3 and 兩H兩 ⫽ ⫺1. That the plane of reflection contains the achromatic axis is shown by ជ D65. H䡠w ជ D65 ⫽ w

(8)

The exact inversion symmetry E can be composed with K to produce an approximate affine rotational symmetry R ⫽ K ⴰ E ⫽ E ⴰ K. Since E is exact, R is as exact or approximate as K. The axis of rotation of R is given by 1 2

冉 冊

0.869 w ជ D65 ⫹ ␣ ⫺0.392 , ⫺0.302

(9)

which goes through the mid-gray point with the direction of the eigenvector of H with associated eigenvalue ⫺1. Figure 2(b) shows the axis of rotation of R and the intersection of the body colour solid with the plane of reflection of K. It also shows the full colour locus and its image under either of the approximate symmetry transformations (they both give the same result). The quality of the approximate symmetries in terms of the overlap measure defined above can be visually gauged from the full colour locus and its image. From the figure, I estimate the overlap to be in the range 90 –95%. Because the symmetries were found using the RMS measure rather than the overlap, it is possible that there may be similar transformations with slightly higher overlap scores. 155

SIMILARITY OF THE PSYCHOLOGICAL AND PHYSICAL SYMMETRIES

It has been shown that both the psychological structure and the physical have a symmetry that can be interpreted as a rotation — the top row of Table I and R, respectively. It remains to be shown that they are in some sense the same rotation. To show qualitative similarity between the two rotations, I consider the patches from the Macbeth Color Checker identified as best corresponding to the hue BCTs.23 The cone responses to these patches when viewed under D65 are projected by the transformation

冉冊 X Y Z

3

冉冊

1 X X⫹Y⫹Z Y

(10)

into the standard CIE xy-diagram in Fig. 3. Also plotted are the projections of the plane of reflection of K and the axis of rotation of R, both of which project to lines. For ideal correspondence with the psychological structure, the reflection plane would run through purple, the achromatic point, and yellow, and the rotation axis would run between green and blue, through the achromatic point, and between red and orange. Given that ideal correspondence would require the vanishingly unlikely circumstance of numerically fitted lines passing exactly through the points corresponding to the Macbeth Color Checker, more plausible criteria of correspondence need to be established. These can be decided by considering the possible results of perturbing the ideal correspondence configuration. Of these perturbed diagrams, only the projectively invariant aspects should be considered. This is because the xy-projection is an arbitrary choice from the possible projections, just as XYZ coordinates are arbitrary. So, finally, the criteria for realistic nonarbitrary correspondence are that the reflection axis should pass (i) between purple and blue or purple and red, (ii) through the achromatic point, and (iii) between yellow and green or yellow and orange. And that the rotation axis should pass (iv) between red and orange, (v) through the achromatic point, and (vi) between blue and green. Of these criteria, (ii) and (v) are true by design. Of the remainder, Fig. 3 shows that (i), (iii), and (vi) hold in fact, while (iv) fails. In conclusion, I would describe the correspondence between the psychological and physical symmetries as rough. Because the agreement between the two symmetries is only rough, other symmetries derived from the analysis of the psychological structure are worth considering. The symmetry with SLT-rank 22 (fifth row of Table I), with agreement rates of 90.6%, 91.9%, and 70.8% for similarity, lightness, and topology, respectively, can also be readily interpreted as a rotation.24 The associated transformation is the exchange of Black and White, Yellow and Blue, Orange and Purple, Pink and Brown, and “darker” for “lighter.” This implies a Blue-Gray-Yellow line of reflection and a Red-Gray-Green axis of rotation, consistent with a Hering opponent-colours model of colour space.25 There is little to choose between the two candidate psy156

FIG. 3. Symmetries shown in the CIE diagram. Shows the projection of the body colour solid into the standard CIE diagram. Whatever the illuminant, the body colour solid projects one-to-one onto the gray parabolic shape, the boundary of which is the spectral locus corresponding to the responses to monochromatic lights. Other features of the diagram are specific to the D65 illuminant. The achromatic axis of the body colour solid maps to the black-graywhite bull’s eye. The projections of the full colour locus and its transform are shown by the white and black curves, respectively. The plane of affine reflection projects to the purple and yellow line, and the axis of affine rotation to the orange-red and blue-green line. These two lines are close to though not exactly perpendicular. The coloured discs correspond to the cone responses to chips from the Macbeth Colour Checker when viewed under D65; the dotted line links them into the hue circle. Qualitative agreement between the symmetries of the psychological and physical structures is shown by the position of the reflection plane relative to purple-gray-yellow and by the rotation axis relative to blue/green-gray-red/orange.

chological symmetries either in terms of their match to the physical or in their quality of approximation as shown by the agreement rates. So both are approximate symmetries of the psychological structure, and the approximate rotational symmetry of the physical is similar to both of them. Hence, agreement between the psychological and the physical structure of colour space is shown.

ACKNOWLEDGMENTS

I gratefully acknowledge the contribution of the reviewing referees to the final state of this article. I also thank the subjects, who filled out colour questionnaires, and my colleagues, who were sufficiently encouraging about this research to spur me to finish it. COLOR research and application

1. Clark A. Sensory qualities. Oxford: Clarendon; 1993. 2. Riemann A. On the hypotheses which lie at the foundations of geometry. 1854. In: A source book of mathematics. New York: Dover; 1929. 3. Richards W, Koenderink JJ. Trajectory mapping: a new nonmetric scaling technique. Perception 1995;24:1315–1331. 4. Russell B. The Principles of mathematics. London: Routledge; 1903. 5. Newton I. Opticks (Book 3). London: Smith Walford; 1704. 6. Shepard RN. The analysis of proximities: multidimensional scaling with an unknown distance function, I. Psychometrika 1962;27:125– 140. 7. Gerritsen F. Evolution of the color diagram. Color Res Appl 1979;4: 33–38. 8. Munsell AH. A color notation. New Winsor, NY: Munsell; 1905, revised 1971. 9. Indow T. Multidimensional studies of the Munsell color solid. Psych Rev 1988;95:456 – 470. 10. Schro¨dinger E. Theorie der pigmente von grosster leuchtkraft. Ann Physik 1920;62:603– 622. 11. Wittgenstein L. Remarks on colour. Oxford: Blackwell; 1977. 12. Shepard RN, Cooper LA. Representation of colors in the blind, colorblind, and normally-sighted. Psych Sci 1992;3:97–104.

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13. Berlin B, Kay P. Basic color terms: Their universality and evolution. Berkeley: Univ CA; 1969. 14. Griffin LD. Empirical assessment of colour symmetries. Behav Brain Sci 1999;22:952–953. 15. The ISCC-NBS method of designating colors and a dictionary of color names. National Bureau of Standards Circular 553; 1955. 16. Weyl H. Symmetry. Princeton: Princeton Univ; 1952. 17. Pope A. The language of drawing and painting. New York: Russell; 1929. 18. Wyszecki G, Stiles WS. Color science. New York: Wiley; 1982. 19. Ostwald W. Neue forschungen zur farbenlehre. Phys Z 1916;17:322– 332. 20. MacAdam DL. The theory of the maximum visual efficiency of materials. J Opt Soc Am 1935;25:249 –252. 21. Koenderink JJ. Color atlas theory. J Opt Soc Am A 1987;4:1314 – 1321. 22. Koenderink JJ. Personal communication; 1999. 23. Sturges J, Allan Whitfield TW. Locating basic colours in the Munsell space. Color Res Appl 1995;20:364 –376. 24. Palmer SE. Color, consciousness and the isomorphism constraint. Behav Brain Sci 1999;22:923–943. 25. Hering E. Outlines of a theory of the light sense. Hurvich LM, Jameson D, translators. Cambridge, MA: Harvard Univ; 1964 (trans. of 1920 ed.).

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