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A uniform color space based on the Munsell system

By A. Kimball Romney and John P. Boyd

Institute of Mathematical Behavioral Sciences, University of California, Irvine, CA

92697-5100

ABSTRACT

We find that the similarity structure of the cube rooted reflectance spectra of

Munsell color chips obtained from a multidimensional scaling based only on

physical measurements is similar to the perceptual structure of normal human

observers. The perceptual structure is inferred by projecting the reflectance spectra

into the space of cone sensitivity curves derived from color matching functions. In

perceptual space the reflectance spectra are represented as linear transformations

of human cone sensitivity curves and the structure is similar to the Munsell color

space defined by three orthogonal axes (value or lightness, hue, and chroma or

saturation). A rigid rotation of the perceptual structure produces a match with the

Munsell color system. Any reflectance spectra whatsoever may be located in this

Euclidean representation of the uniform Munsell color system.

OCIS codes: 330.0330, 330.1690, 330.1710.

1. INTRODUCTION AND THE MUNSELL COLOR SYSTEM

We report an unanticipated discovery that unifies two previous results and strengthens

their validity. The first, by Romney [1] (Model 1), demonstrated that the similarity

structure of the cube rooted reflectance spectra of Munsell atlas color chips was well

represented in an Euclidean space resembling the Munsell color system. The second, by

Romney and Chiao [2] (Model 2), represented the same Munsell spectra as projected into

an orthonormalized cone sensitivity space that was designed to correspond to perceptual

color space. In this paper we show that the two structures are identical when transformed

by an appropriate rigid rotation matrix. We then show that it is possible to produce a set

of realizable reflectance spectra that fit perfectly in the Munsell color system. Since all

of these representations are isomorphic and represented in Euclidean space, we believe

that the Munsell color system could be the basis for a uniform color space. We present a

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brief summary of the Munsell color system as background information prior to our more

technical and mathematical discussion.

The Munsell color system was developed in the early years of the last century by

Albert H. Munsell. In many current accounts Munsell is described as an artist who

devised a color system. In fact he did extensive scientific experiments and published

scientific papers [3, 4]. He published a book [5] describing a color system together with an

atlas. The color system uses three independent dimensions to represent all possible

colors (as measured by reflectance spectra) in a distorted oblate shaped space, as

illustrated in Fig. 1.

Fig. 1. A view of the Munsell color solid and a sample page from the Atlas.

We cannot improve on Munsell's [3] own description, which "depends upon the

recognition of three dimensions: value, hue and chroma. These three dimensions are

arranged as follows. A central vertical axis represents changes in value from white at the

top to black at the bottom.The value of every point on this axis determines the level of

every possible color of equal value. Radial planes leading from this axis correspond each

to a particular hue [e. g., the 5 Yellow Atlas page shown in Fig. 1]. Opposite radial

planes correspond invariably to complementary colors: any three planes separated by

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120 form a complementary trio, etc. Thus the angular position of any hue is determined,

and the hues are balanced. Chroma, or intensity of hue, is measured by the perpendicular

distance from any point on the vertical axis, and the progression of chroma is an

arithmetical one.Thus is constructed a solid. In this solid every horizontal plane

corresponds to one and only one value. Every vertical plane extending radially from the

central vertical axis contains but one hue. Finally, the surface of every vertical cylinder

having the vertical axis as its principal axis contains colors of equal chroma."

Recent accounts of Munsell's color atlas usually fail to report on the extensive

empirical research that he carried out using instruments that he invented (U.S. Patents

640,792, 1900; 686,872, 1901, 717,596, 1903, and 824,374, 1906), including a

photometer and spinning top (Maxwell disk). Maxwell [6] and Helmholtz [7] did

extensive experiments with spinning disks and knew that complementary colored

surfaces would fuse to produce the appearance of an achromatic gray when spun at high

speeds. This was seen as analogous to the observation that complementary

monochromatic lights when mixed produce the appearance of white. Rood [8] presented

extensive experimental data resulting from his research with spinning disks in his 1879

book. Munsell was greatly influenced by his study of Rood and followed his example by

calibrating his color atlas with extensive experimentation. His complementary colors

were all tested to produce an achromatic gray when combined with his spinning top. His

equal chroma cylinders were produced by insuring that equal amounts (areas) of

complementary colors produced achromatic grays. His equal spacing of hues was tested

with triads of hues at 120 as well as complementary pairs. In addition he would take

one of two complementary hues and balance it against a pair of hues separated by 72 to

achieve an achromatic gray. In this way he could iterate to equally spaced hues as well as

to calibrated value and chroma scales.

Munsell was aware of many complicating factors affecting color vision, including

context effects such as color contrast, color induction, changes in illumination, etc. For

example, in his first scientific publication [4] he was seeking to obtain the appearance of

equally spaced chroma levels and found that he could only do so by specifying the

observational context. He prepared two observational templates for a Maxwell disc as

illustrated in Fig. 2. Disk A is cut to produce a decreasing geometric progression of

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angles: 180, 90, 45, and so on, while the corresponding radii decrease arithmetically.

When a color (red in the Fig.) is placed on a white background and spun, a well

graduated progression of concentric rings is produced, diminishing in chroma and

increasing in value from the center to the circumference. A change that prevents any

variation in value may be obtained by changing the background to a neutral gray of the

same value as the color; in which case the spacing of the chroma rings becomes very

uneven. Disk B is now cut at five equal angles and with (as for disk A) an arithmetical

decrease of radii. Now when a color is placed on a gray background and spun an even

perceptual spacing of the chroma rings is observed. The finding is incorporated into the

Munsell color system and requires colors to be judged on a neutral background (an

achromatic gray of the same value). This reminds us of a very important general

scientific principle, frequently overlooked in color experiments, which is to always

isolate the variable (chroma in this case) being measured from possible interaction or

confounding effect with other variables (value in this instance).

Fig. 2. Munsell's two Maxwell disk templates.

When he produced his atlas, Munsell advised using as few pigments as possible

and grinding them finely and consistently. Later, when the Optical Society of America

[9] studied the Munsell color system, they produced samples exemplifying the Munsell

renotation, all sample color chips being painted with only six colors plus black and white

[10]. For reasons that are historically obscure the studies carried out by the Optical

Society of America [11-15] did not use Munsell's psychophysical methods and substituted

psychological judgments averaged over many subjects for their renotation system. We

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find only one replication of Munsell's careful experiments. It was written in 1940 by

Tyler (an undergraduate at MIT at the time) and Hardy [16] (who directed the preparation

of the classic Handbook of Colorimetry [17]). They comment that their work "calls

belated attention to the remarkable scientific insight of Professor A. H. Munsell. At a

time when there was little to suggest such a procedure, he formulated rules for the

construction of a psychophysical color system that could be used today without apology.

We believe also that the publication of this paper may call attention to the fact that the

psychophysical definitions of the terms hue, value, and chroma given in the Atlas of the

original Munsell Color System differ from purely psychological definitions used since

the death of Professor Munsell."

In the Munsell color structure, circles of equal chroma, but several different

values, form cylinders. Any valid model of perceptual color space should reflect this

fact. Research on scaling the similarity structure of the reflectance spectra of the Munsell

atlas chips has usually reported a conical rather than a cylindrical structure for the chroma

circles of different value, examples include Romney and Indow [18], Lenz, et al. [19],

Koenderink, et al. [20], and Burns et al. [21]. We view these earlier models as describing

the physical stimuli correctly but as misleading when interpreted as perceptual space.

The cylindrical structure only emerges when a cube root transformation is applied to the

original empirical measures of reflectance spectra. We follow the Romney and Chiao

procedure of cube rooting the reflectance spectra, since this is a critical element of their

model and has the effect of transforming the physical space of a cone to the perceptual

space of a cylinder. The use of the cube root transformation goes back to Plateau [22] and

Stevens [23]. It is also used in the CIE L*a*b* colorimetric system [24].

2. SUMMARY OF RESULTS FROM PREVIOUS RESEARCH

The previous analyses (Model 1[1] and Model 2[2]) were based on 1296 reflectance

spectra from the 1976 Munsell color atlas[25]. The color chips were measured as percent

reflectance (scaled 0 to 1) at each nm from 400 nm to 700 nm. These data and their

sources are described by Kohonen et al.[26] and in this paper are represented by matrix

. The cube root symbol is used to emphasize that all spectra have

been cube rooted element-wise (the only nonlinear operation in our analysis).

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Model 1 showed that the Munsell reflectance spectra were well represented in a

three dimensional Euclidean space. Singular value decomposition (SVD) was used to

obtain estimates of the orthonormal coordinates (in matrix B) as follows:

(1)

where, as usual for the SVD, the columns of B and F are orthonormal and L is the

diagonal matrix of singular values in decreasing order. The three dimensional

representation of the reflectance spectra (the matrix ) accounts for .9992% of the

variance (of matrix ), indicating a very close fit to a Euclidean structure. The

structure was shown to be close to that of the Munsell color system. The second and

third dimensions of the Euclidean structure as represented in matrix are illustrated

in Fig. 3A along with the corresponding dimensions of matrix F (the basis functions of

the reflectance spectra).

Model 2 showed that the Munsell reflectance spectra when transformed by a

projection matrix derived from cone sensitivity curves (then analyzed with SVD) resulted

in a three dimensional Euclidean structure closely resembling CIE L*a*b* [24]

coordinates. The second and third dimensions of the Euclidean structure represented in

matrix (computation details appear below) are illustrated in Fig. 3B along with

the corresponding dimensions of matrix W (the basis functions of the cone sensitivity

curves) which represent the monochromatic spectral locus. It is obvious that the

similarity structures of the location of the color chips are remarkably similar. It was

curiosity about this similarity that led us to the discovery that they are in fact

mathematically identical after rigid rotation. Before demonstrating this identity we

present a brief summary (full details are in the original article) of the Model 2.

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Fig. 3. The second and third dimension scaling results showing locations of the reflectance spectra of Munsell color chips using two different methods. Munsell chips of hue 5 red shown in red. A. Location of chips using Model 1 in which reflectance spectra are scaled with SVD together with the basis functions of the reflectance spectra. B. Location of chips using Model 2 in which reflectance spectra are scaled after projection into cone sensitivity space together with the basis functions (spectral locus) of the cone sensitivity curves.

Model 2 is designed to extract the maximum amount of color information possible

(using mathematical procedures) from a set of spectra to estimate the similarity structure

of the color space and the associated monochromatic spectral locus (basis functions). We

call this perceptual color space because the empirical spectra are converted into linear

combinations of the cone sensitivity curves. Any set of empirical spectra has an invariant

similarity structure embedded in the cone sensitivity space.

. The model may be summarized in four equations in standard matrix notation [27].

(2)

(3)

(4)

(5)

Matrix R in Eq. 2 contains the data of the cone receptor curves as defined by

Stockman and Sharpe [28] and derived from the color matching functions of Stiles and

Burch [29]. The model calculates an orthonormal projection matrix ( ) from the

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cone sensitivity curves using the SVD in Eq. 2 followed by the matrix multiplication of

Eq. 3. Though derived differently, this projection matrix (matrix P in Eq. 3) is

mathematically identical to Cohen's matrix R [30,31] (not to be confused for matrix R in

Eq. 2) except for the fact that Cohen does not use the cube root transformation.

Multiplying the empirical spectra by the projection matrix using Eq. 4 converts each

spectrum into a linear transformation of the cone sensitivity curves. The projected

reflectance spectra in matrix S represent what the receptors "see" in the literal sense that

they are weighted combinations of the receptors. Since the 1269 spectral curves are

represented as actual combinations of the cone receptors it is assumed that the similarity

among those curves may be interpreted as the color appearance similarity structure for

the specified observer. Note the identity . The SVD

of Eq. 5 provides a mathematical procedure for obtaining a three-dimensional Euclidean

representation of the similarity structure (matrix M) plotted in Fig. 3B. Fig. 3B shows

the structure of the reflectance spectra after being projected into the space of the cone

sensitivity curves. The spectra are now represented in the model as linear combinations

of the cone sensitivity curves (matrix U or W).

The projection matrix plays such a critical role in the model that it will be useful

to explain its function in more detail. Fig. 4 will serve as a helpful visual aid in clarifying

the importance and implications of projecting reflectance spectra into cone sensitivity

space. The first six panels (Fig. 4AF) show the Stiles and Burch CMFs [29], the

Stockman and Sharpe [28] cone sensitivities, and an approximation of Wald's [32]

measurements of the cone sensitivities. The three data sets are plotted in two formats,

first in natural form and second in orthogonal form as calculated with SVD. That is, if

we let the data in Fig. 4A, 4C, or 4E represent matrix R in Eq. 2, then the curves in Fig.

4B, 4D, and 4F are matrix U in Eq. 2. All six of these panels contain equivalent

information and are linear transformations of each other. Thus any of the three would

produce identical projection matrices.

The projection matrix is a canonical formulation that perfectly expresses the

unique communality among the infinity of linear transformations sharing identical

information with different shapes when viewed from different perspectives. It has a

variety of interesting properties. It is symmetric and idempotent under multiplication,

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i.e., . Its rows (or columns since it is symmetric) contain transformed perceptual

representations of physical stimuli of monochromatic spectra (of equal units of energy)

and constitute the spectral locus produced by the model. In Fig. 4G we have plotted 31

of these perceptual spectra (every 10th row) of the projection matrix P. It is a remarkable

fact that any (reasonably spaced) selection whatsoever of three of the curves plotted in

Fig. 4G contains all the information necessary to reproduce the projection matrix; and

when multiplied by their transpose (as in Eq. 3) they form the same projection matrix P.

This is analogous, in color matching experiments, to being able to match any

monochromatic test light with any reasonably spaced selection (no two lights should be

so close as to make the problem computational ill-conditioned) of three fixed primary

lights.

Fig. 4H shows the column sums of the projection matrix. This two hump shaped

figure is shown in MacAdam [33, 34] as the sum of the CIE 1931 color matching functions.

In his 1920 article Schrdinger [35] derived this curve but rejected it by saying it was a

"hideous camel's back with two pronounced maxima" [35, p. 164]. Fig. 4J shows the

canonical form of the basis functions of the projection matrix. Note that the shape of the

column sums plot is identical to that of the first canonical basis function and differs only

by a scale factor. In our model this first basis function is the achromatic axis

corresponding to value in the Munsell system. We have synthesized the double hump

shape spectra in an OL490 Agile Light Source [www.boochandhousego.com] instrument

and it appears completely achromatic. Having an achromatic-appearing first basis

function seems to us superior to one with a saturated yellow-green appearance with a

single hump ( ) as used in the CIE [24] chromaticity calculations. Any flat physical

spectra (a constant at all wavelengths), when processed by model 2, has the two hump

shape.

Fig. 4I shows the diagonal values of the projection matrix. These values are

proportional to the strength of the color (as opposed to the achromatic) signal at each

wavelength. MacAdam [33, 34] derived the same shaped curve as the moment per watt of

spectrum colors. For pairs of complementary monochromatic lights the curves estimate

the relative amounts of each required for an achromatic balance. In Figs. 3B and 5A it

corresponds to the distance of the spectral locus boundary from the origin.

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Finally, we note that Model 2 uses the projection matrix to project any spectra

into a linear combination of orthonormalized color matching functions (or cone

sensitivity curves) that specify a location in terms of three coordinate positions on three

orthogonal axes in Euclidean space.

We now turn to the task of demonstrating that the structures shown in Fig. 3A and

Fig. 3B are identical after an appropriate rigid rotation.

Fig. 4. Various views of the information in the projection matrix. A. Color matching functions. B. Color matching function in orthonormal form. C. Cone sensitivity curves normed to maximum equal to one.. D. Cone sensitivity curves in orthonormal form. E. Wald's cone sensitivity curves. F. Approximation of Wald's cone sensitivity curves in orthonormal form. G. Plot of every 10th row (or column) of projection matrix. H. Column sums of projection matrix. I. Diagonal values of projection matrix. J. The canonical form of the basis functions of the projection matrix (solid line is the achromatic axis).

3. DEMONSTRATING IDENTITY AMONG COLOR STRUCTURES

The task is to show that matrix B in Eq. 1 is identical to matrix M in Eq. 5 after

appropriate rotation of, say, B. The solution is to find a rotation matrix such that

BX is nearest (with respect to the Frobenius norm, defined for any matrix A as

) to M and check for identity, meaning that the Frobenius norm of the

difference is zero. Eq. 6 shows how to calculate matrix X and the numerical results.

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(6)

After rotation by the matrix X, the Frobenius norm , which demonstrates

that matrix B is identical to matrix M after rotation by X. This follows the mathematical

procedure outlined by Ben-Isreal and Greville [36] (Ex. 33, p. 217). The results of the

calculations are shown in Fig. 5A where the coordinates in the second and third

dimensions contained in matrix BX are plotted with those contained in matrix M. In

addition we have plotted the corresponding natural basis functions (open horseshoe

shape) in matrix F (Eq. 1) and the orthonormal basis functions (representing

monochromatic spectral locus) of the receptors (closed triangular shape) in matrix W

(Eq. 5).

Fig. 5A reveals that even though the similarity structures of the reflectance

spectra are identical, the basis functions are very different. For human observers these

basis functions constitute the monochromatic spectral locus and ultimately derive from

color matching functions based on experiments with monochromatic spectral lights that

may vary widely from individual to individual. There is a duality between reflectance

spectra of surfaces (the rows of matrix A) and monochromatic spectral lights (the

columns of matrix A) artificially produced with prisms or various optical instruments.

The perceptual similarity among the Munsell chips (based on reflectance spectra) is

represented by the vertical and horizontal symbols (crosses) in Fig. 5A and is invariant

for all trichromatic observers regardless of the shape of the spectral locus. In contrast the

perceptual similarity among the monochromatic spectral lights is represented by the open

circles and is valid only for the Stockman and Sharpe observer. We emphasize the fact

that humans with a different color matching function will have a different projection

matrix and a different shaped spectral locus.

A further implication of the duality between reflectance spectra and

monochromatic spectral lights involves the difference in the rules for combining them to

produce new perceptual colors. Reflectance spectra obey the rules of convex

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combination while monochromatic lights obey the rule of vector addition. The physical

structure of reflectance spectra (cube rooted) is identical to the perceptual structure in

Fig. 5A and the rule of convex combination of reflectance spectra constrains the

similarity structure of the rows (Munsell chips) to be invariant for all trichromatic

observers (including the null case of Model 1) and is the rule of mixtures on the Maxwell

spinning color disks. The rule of vector addition applies to the values obtained for the

monochromatic spectral locus which are different for observers with different color

matching functions (which specify how much of each of three primary lights are needed

to produce a monochromatic test light of one watt).

Fig. 5. The empirical location of Munsell reflectance spectra and the uniform perceptual space of Munsell color. Munsell chips of hue 5 red shown in red. A. Model 1 points (after rotation by matrix X) indicated by vertical line and Model 2 points indicated by horizontal line (a cross shows the location of a single spectrum in both models). Natural basis functions are shown as open circles and the monochromatic spectral locus as squares. B. The uniform perceptual space of Munsell color and the monochromatic spectral locus (after rotation by matrix ).

We now turn to deriving realizable reflectance spectra that may be represented in

a Euclidean space that is isomorphic with the Munsell color system. We begin by

generating Munsell Cartesian coordinates for the sample of 1269 reflectance spectra in

matrix A from the Munsell color variables of value (lightness), hue (as an angle in the

color circle), and chroma (saturation) as follows: Each of the 40 pages of the Atlas was

assigned an appropriate angle on the 360 color circle beginning with 5 Red at 0 and

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incrementing each page 9 up to 2.5 Red at 351 (by convention hue degrees ascend

counter-clockwise). A final adjustment is required to bring the Munsell color system into

an adequate uniform model. From psychophysical evidence Indow [37,38] has shown that

one value step in the Munsell system is roughly equivalent to two chroma steps. To

adjust for Indow's finding we multiply Munsell value by two to obtain a uniform color

space in which the perceptual distance in any direction is the Euclidean distance. The

three dimensional coordinates for the uniform Munsell color space (designated as matrix

) are computed as follows:

= value 2 (theoretical coordinates range from 0 to 20)

= (theoretical coordinates range from -20 to +20)

= (theoretical coordinates range from -20 to +20)

In order to show that human perceptual space is identical to Munsell conceptual

space we need to produce a set of realizable reflectance spectra that fit perfectly in the

Munsell conceptual space. The method that we use to obtain such idealized data is to

transform the reflectance spectra into the space of the Munsell conceptual system by

multiplying the orthornormalized conceptual coordinates by the singular values and basis

functions of the original reflectance spectra (using values obtained with Eq. 1). We begin

by computing an orthonormal representation of the Munsell conceptual space using SVD

to get:

. (7)

We than compute 1269 idealized reflectance spectra as follows (using matrices L and F

from Eq. 1.):

(8)

All the idealized realizable reflectance spectra in matrix (distinguished from

A by asterisk) have positive values showing that they could, in principle, be replicated in

practice. The spectra in matrix are similar to the spectra in matrix A, for example, the

sum of squares of divided by the sum of squares of A is 0.9992. It can be

demonstrated (using the procedures outlined above) that if we substitute matrix in

place of matrix A in Eq. 4 that the idealized spectra, when analyzed by Model 2 produce

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an identity between matrix E in Eq. 8 and the new matrix (distinguished from M by

asterisk). Furthermore a simple linear transformation will transform matrix into the

Munsell coordinate system derived above as matrix C of Eq. 7, and illustrated in Fig. 5B.

The linear transformation of matrix into matrix C is shown in Eq. 9.

(9)

Thus the idealized reflectance spectra are transformed back into the Munsell coordinate

system in terms of the original scales of matrix C. This is possible since matrix C is

orthogonal. This procedure could be generalized to convert any collection of cube rooted

reflectance spectra into the Munsell coordinate system.

Note that we would have arrived at the same similarity structure for the Munsell

color chips regardless of the choice of color matching functions or cone sensitivity curves

used for matrix R of Eq. 2. In fact the geometric constrains are so strong that any

orthonormal matrix whatsoever may be substituted for matrix U (in Eq. 2 and 3) or

matrix W (in Eq. 5) and the similarity structure of the reflectance spectra would be the

same within a rigid rotation of the structure. What would vary under these different

choices is the shape of the basis functions with empirical consequences for predicting the

fit between color perception and wavelength. Even though the similarity structure (of the

reflectance spectra) is invariant for different observers, the shape of the monochromatic

spectral locus (which determines the perceptual nature of monochromatic lights) may

vary radically among different observers. This means that the fact that the Stockman and

Sharpe cone sensitivity curves used in Model 2 lead to a good fit is a necessary but not

sufficient requirement to prove its validity. Additional empirical requirements are

required such as predicting complementary pairs (and in what proportion) of

monochromatic lights that combine to form an achromatic white.

4. COMMENT ON MODEL DEVELOPMENT

The discovery that human perceptual space is identical to Munsell conceptual space may

appear surprising. Still, one reason we might expect the two structures to be identical is

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that, even though the Munsell conceptual system and the color matching functions are

based on totally different data and methods of analysis, they are two independent

attempts to model the same phenomena, namely the structure of human color space. It

would be surprising and disturbing if they did not arrive at similar solutions.

The measurement and modeling of color involves two distinct levels of

phenomena. At the physical level there are beams of light consisting of electromagnetic

waves (spanning the visible range from 400 nm to 700 nm) that instruments measure as

light spectra (for black body radiation or monochromatic spectral lights) or as reflectance

spectra (for object color). At the psychological level color can be represented in a

perceptual color structure consisting of three independent dimensions, value (lightness),

hue, and chroma (saturation). Two major types of experiments provide data to model the

relation between these two levels. Color matching experiments provide a bridge between

monochromatic spectral lights as physical stimuli and the perceptual color structure of

mixtures of three primary lights. Experiments with mixing color samples on a Maxwell

spinning top to produce achromatic pairs or triads of complementary colors provide a

bridge between reflectance spectra as physical stimuli and perceptual color structure.

Our goal has been to build a bridge that goes from the domain of physical stimuli

to that of perception. The bridge consists of a model written in mathematical terms that

specifies how to transform the physical structure into the perceptual structure. It is

understood that physical stimuli are measured by instruments and perceptions are

measured in psychophysical experiments such as color matching and spinning Maxwell

disks. The model must be consistent with known facts of conventional color science.

We began with the notion of representing all components of the model of color

perception in three dimensional Euclidean space with three orthogonal axes.

The major components are: First, the physical stimuli (matrix A) consist of the

reflectance spectra as measured with instruments. Second, an observer with known

receptor characteristics as measured, for example, in color matching experiments. Third,

the Munsell color system describes the structure of colors as perceived by human

observers. We observe that each component is well represented in Euclidean space. A

natural next question is to determine if the axes in the various components could be

matched with one another. In the final model we are willing to ignore the small

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discrepancy between the empirical Munsell color chip locations and the theoretical

Munsell color coordinates.

The model is consistent with the findings of trichromacy. Trichromacy is based

on the fact that whatever arrangement of color matching experiment is set up to produce a

complete visual match between stimuli of unequal spectral energy distributions, the

observer never needs to operate more than three independent controls. So far as is known

the observation is true for all stimulus intensities and for all conditions of the eye,

including dark adaptation and adaptation to excessively bright lights [39]. The color

matching functions (and the cone sensitivity curves derived from them) summarize our

knowledge of how much of each of three primary monochromatic lights is required to

match one unit of energy of a monochromatic test light. The classic rules of trichromacy

were first formalized by Grasssmann[24, 40] and more recently put in axiomatic form by

Krantz [41, 42]. As mentioned above, the psychophysical relationship between wavelength

and perceptual colors vary among observers (the shapes of the spectral locus differ) and

therefore Grassmann's laws and Krantz's axioms are different for different observers [43].

5. DISCUSSION

The mathematical model we have presented is simple in concept and shared by all

trichromatic observers. Both of these features arise from geometric constraints of three

dimensional Euclidean space. If perceptual color space can be described in three

dimensional Euclidean space, then by definition it has three orthogonal axes. If one of

these axes is described as achromatic and as going from black to white, while the other

two axes are described as a chromaticity planes contained color circles of varying

chroma, and if color circles of equal chroma and different value form cylinders, then

there is only one geometric similarity structure possible. The structure has infinitely

many orientations but all are related by a simple rigid rotation matrix. In this sense there

is a single color system (describing spectra in the 400 nm to 700 nm range) common to

all trichromatic observers (biological or robotic).

The idea of orthogonal color space has a history that goes back over 100 years.

Kuehni [44] recently discovered and described in detail a 1901 paper by Ludwig Pilgrim

that represents color in an intended orthogonal space derived from cone sensitivity

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curves. Pilgrim's analysis was based on the color matching functions of Knig and

Dieterici of 1892. The curves for the three axes used by Pilgrim were computed from

formulas obtained from the work of Helmholtz and were meant to represent three

orthogonal axes with an achromatic dimension and two chromatic axes forming a

chromaticity plane. These curves are shown in Fig. 8 (p. 9) of Kuehni's article and are

virtually indistinguishable from the curves shown if our Fig. 4J including the double

hump camel shape of the achromatic dimension. The degree to which Pilgim's axes are

orthogonal is remarkable for work done before the existence of computers.

The next attempts to obtain orthogonal axes were made in the pre-computer age

and had to be based on some simple assumption. The simplest assumption was to assume

that the middle color matching function would be the achromatic axis. This choice was

codified into de facto law in the 1931 CIE chromaticity diagram [24] and later detailed in

Handbook of colorimetry [17] where the function was made to correspond exactly to

the "visibility" function (now called ). MacAdam thought that normal and orthogonal

functions would be useful and simplify many color calculations [34, 45, 46]. In the

MacAdam work the color-mixture functions were made proportional to the luminosity

function (as in the CIE system). Since the luminosity function is not achromatic this

means these attempts all failed to obtain a uniform perceptual color system. The CIE

chromaticity diagram is not orthogonal to the value or achromatic axis, it is orthogonal to

some kind of yellow-green. Since this confounds luminosity and chromaticity it makes a

uniform color system impossible [24].

When computers were perfected it became possible to actually compute

orthogonal axes mathematically from color matching functions, cone sensitivity curves,

or reflectance spectra. Cohen47 was the first to compute the basis function of a large set

of reflectance spectra representing the Munsell atlas color chips showing the low

dimensionality of color space. Later he developed his Matrix R theory [30, 31, 48, 49] and

demonstrated how reflectance spectra (non-cube rooted) of Munsell color chips fit into

color matching function space using a projection matrix. Our work owes a great deal to

Cohen's pioneering work. We added the critical step of the cube root transformation (of

the reflectance spectra) to Cohen's model.

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A potential weakness of the model is that the mathematical procedures are

unrealistically effective and fail to allow for error. Clearly actual biological systems are

not error free and a way to allow for the effects of various sorts of error needs to be

incorporated in the model. For example, if cone receptivity curves are abnormally

closely spaced and any errors are present, the matrix operations become ill-conditioned

and the errors are magnified.

The model has some novel advantages. The most important may be that it

provides the basis for a uniform perceptual color system that allows for color difference

calculations to be made on a simple Euclidean distance basis. In this space the MacAdam

[34] error ellipses are all circles of equal size. The fact that the perceptual distance is

related in a known way to the physical stimuli (reflectance spectra) is a signal advantage.

Another advantage is that the triangular heart shape of the spectral locus provides a direct

linkage to industrial application used in color reproduction [50] since it accounts for the

location of the prime colors [51-53]. Finally, the model might unify the task of color

specification using color matching functions [24] with the task of building an ideal color

atlas [54] into a single coherent endeavor.

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