Putting aside metaphorical meanings of the term, color space is understood as a vector space, where lights having the same color (i.e., subjectively indistinguishable) are represented as a point. The CIE 1931 color space, empirically based on trichromatic color measurements, is a classical example. Its derivatives, such as CIELAB and sRGB, have been successfully used in many applications (e.g., in color management). However, having been designed for presenting the color of self-luminous objects, these spaces are less suitable for presenting color of reflecting objects. Specifically, they can be used to represent color of objects only for a fixed illumination. Here I put forward a color space to represent the color of objects independently of illumination. It is based on an ideal color atlas comprising the reflectance spectra taking two values: *k* or 1 − *k* (0 ≤ *k* ≤ 1), with two transitions (at wavelengths *λ* _{1} and *λ* _{2}) across the spectrum. This color atlas is complete; that is, every reflecting object is metameric to some element of the atlas. When illumination alters, the classes of metameric reflectance spectra are reshuffled but in each class there is exactly one element of the atlas. Hence, the atlas can uniquely represent the metameric classes irrespective of illumination. Each element of the atlas (thus, object color) is specified by three numbers: (i) *λ* = ( *λ* _{1} + *λ* _{2})/2, which correlates well with hue of object color (as dominant wavelength correlates with hue of light color); (ii) *δ* = ∣ *λ* _{1} − *λ* _{2}∣, which correlates with whiteness/blackness; and (iii) *α* = ∣1 − 2 *k*∣, which correlates with chroma of object color (as colorimetric purity correlates with saturation of light color). Using a geographical coordinate system, each element of the atlas (thus, each object color) is geometrically represented as a radius vector so that its length equals *α,* the latitude and longitude being proportional to *δ* and *λ,* respectively.

*color*

*signal*in response to a light with the spectral power distribution

*I*(

*λ*) as a triplet (

*φ*

_{1}(

*I*),

*φ*

_{2}(

*I*),

*φ*

_{3}(

*I*)), where

*t*(

*λ*) is the transmittance spectrum of the ocular media (e.g., the lens and macular pigment), and

*p*

_{ i}(

*λ*) is the spectral absorption of the

*i*th photopigment, (

*i*= 1, 2, 3), that is assumed to be positive throughout the visible spectrum interval [

*λ*

_{min},

*λ*

_{max}]. I will use throughout the paper the following values:

*λ*

_{min}= 380 nm,

*λ*

_{max}= 780 nm. Two lights

*I*

_{1}(

*λ*) and

*I*

_{2}(

*λ*) are assumed to be metameric if and only if the color signals for these lights are equal, that is,

*φ*

_{ i}(

*I*

_{1}) =

*φ*

_{ i}(

*I*

_{2}) for each

*i*.

*i*th

*cone fundamental*. The notations

*S*,

*M*, and

*L*will be used for the response of the cone fundamentals with the peak sensitivity in the short-, middle-, and long-wave range of the visible spectrum, respectively.

*S*,

*M*,

*L*), the color signals from all the lights form a convex cone in this space (referred to as the

*color*

*signal cone*). Each color signal represents a class of metameric lights. As metameric lights are assumed to have the same color, this cone can be considered as a geometrical representation of color in the 3D space that will be referred to as the

*SML color*

*space*.

*x*(

*λ*) illuminated by a light with the spectral power distribution

*I*(

*λ*) produces a color signal the

*i*th component of which is

*metameric reflecting objects*) will appear as having the same color. Therefore, under constant illumination each color signal will represent a class of metameric spectral reflectance functions. The color signals produced by all the reflecting objects under the same illuminant form a closed convex volume (referred to as the

*object-color*

*solid*) in the color signal cone (Koenderink & Doorn, 2003; Luther, 1927; Maximov, 1984; Nyberg, 1928; Schrodinger, 1920/1970; Wyszecki & Stiles, 1982).

*SML*color space the same way as the color signal cone does for light color. However, the analogy is rather superficial because such a representation depends on the illuminant. When the illuminant alters, metameric spectral reflectance functions might cease to be metameric. This phenomenon is known as

*metamer mismatching*(Wyszecki & Stiles, 1982). Conversely, spectral reflectance functions that produce different color signals under one illuminant may become metameric under the other. So, after an illumination change, some spectral reflectance functions will remain in the same metameric class while others will fall into different classes. Therefore, the reflecting object metamerism as defined above is different for different illuminants.

^{1}To be more exact, there is no spectral reflectance function metameric to the so-called

*optimal*spectral reflectance functions, that is, those which map to the boundary of the object-color solid. For example, there are infinitely many spectral reflectance functions mapping to the center of the object-color solid, one of them being a function

*x*(

*λ*) ≡ 0.5, which takes 0.5 at every wavelength within the visible spectrum (written

*x*

_{0.5}(

*λ*)). However, only one spectral reflectance function maps to the north (respectively, south) pole of the object-color solid, namely, the “perfect reflector”

*x*(

*λ*) ≡ 1 (respectively, the “ideal black”

*x*(

*λ*) ≡ 0), which takes 1 (respectively, 0) at every wavelength within the visible spectrum.

*S*,

*M*,

*L*) be a point on the boundary of the object-color solid obtained for the illuminant

*I*(

*λ*). There is just one optimal spectral reflectance function that produces this color signal. Denote it

*x*

_{opt}(

*λ*). As the set of optimal spectral reflectance functions remains the same for a very broad class of illuminants (see Optimal spectral reflectance functions and object-color solid section),

*x*

_{opt}(

*λ*) will map to some point in the boundary of the object-color solid obtained for the other illuminant,

*I*′(

*λ*), say, (

*S*′,

*M*′,

*L*′). A one-to-one map (

*S*,

*M*,

*L*) → (

*S*′,

*M*′,

*L*′) establishes natural correspondence between the object-color solid boundaries because it puts in correspondence color signals which represent the same classes of metamerism (though, they are singletons under both the illuminants). Note that this map is non-linear.

^{2}

*color atlas*. Each element of a color atlas represents a metameric class to which it belongs. A color atlas that represents all the classes of metamerism will be called

*complete*. As every optimal spectral reflectance function makes a metameric class on its own, a complete color atlas includes all the optimal spectral reflectance functions.

*invariant*with respect to this illuminant family. To summarize, given a family of illuminants, a

*complete illuminant-invariant color*

*atlas*is a set of spectral reflectance functions such that (i) each class of metamerism contains exactly one element of this set under any illuminant in the family, and (ii) any two spectral reflectance functions from the set do not become metameric under any illuminant in the family.

*x*(

*λ*) ≡ 1 is optimal. Another example of spectral reflectance functions that can be optimal is a

*step function*(i.e., a piecewise constant function taking 1 or 0) with transition at the wavelength

*λ*

_{1}:

*x*(

*λ*) ≡ 1 is a particular case of

*x*(

*λ*;

*λ*

_{1}) when

*λ*

_{1}=

*λ*

_{min}. Given

*λ*

_{2}≥

*λ*

_{1}, the step function with transition wavelengths

*λ*

_{1}and

*λ*

_{2}

*rectangular*spectral reflectance function. Note that

*x*

_{1}(

*λ*;

*λ*

_{1}) is a particular case of

*x*

_{2}(

*λ*;

*λ*

_{1},

*λ*

_{2}) when

*λ*

_{2}=

*λ*

_{max}. Generally, the step function with transitions at the wavelengths

*λ*

_{min}<

*λ*

_{1}<

*λ*

_{2}< … <

*λ*

_{m}<

*λ*

_{max}

*x*

_{opt}(

*λ*), the spectral reflectance function 1 −

*x*

_{opt}(

*λ*) has proven to be also optimal. Particularly, if

*x*

_{ m}(

*λ*;

*λ*

_{1}, …,

*λ*

_{ m}) is an optimal spectral reflectance function, the spectral reflectance function 1 −

*x*

_{ m}(

*λ*;

*λ*

_{1}, …,

*λ*

_{ m}) is also optimal. A pair of optimal spectral reflectance functions

*x*

_{opt}(

*λ*) and 1 −

*x*

_{opt}(

*λ*) will be called

*complementary*. The optimal spectral reflectance functions

*x*(

*λ*) ≡ 1 and

*x*

_{ m}(

*λ*;

*λ*

_{1}, …,

*λ*

_{ m}) for any integer

*m*will be referred to as of type I, the optimal spectral reflectance functions

*x*(

*λ*) ≡ 0 and 1 −

*x*

_{ m}(

*λ*;

*λ*

_{1}, …,

*λ*

_{ m}) as of type II.

*two-transition assumption*). However, this is not, strictly speaking, true. In fact, the number of transitions depends on the shape of the cone fundamentals (Maximov, 1984; West & Brill, 1983). More specifically, a theorem has been proved (Logvinenko & Levin, 2009) from which it follows that, for continuous cone fundamentals

*s*

_{1}(

*λ*),

*s*

_{2}(

*λ*), and

*s*

_{3}(

*λ*), and an illuminant with integrable spectral power distribution

*I*(

*λ*), if

*λ*

_{1}, …,

*λ*

_{m}are the only roots of the following equation

*k*

_{1},

*k*

_{2}, and

*k*

_{3}are arbitrary real numbers (at least one of which is not equal to zero), then a step function (Equation 6) with transitions at the wavelengths

*λ*

_{1}, …,

*λ*

_{m}will be an optimal spectral reflectance function.

*I*(

*λ*) > 0 for

*λ*

_{min}≤

*λ*≤

*λ*

_{max}, Equation 7 is equivalent to the following

*λ*is given by (

*s*

_{1}(

*λ*),

*s*

_{2}(

*λ*),

*s*

_{3}(

*λ*)). When

*λ*runs through the interval [

*λ*

_{min},

*λ*

_{max}], a point (

*s*

_{1}(

*λ*),

*s*

_{2}(

*λ*),

*s*

_{3}(

*λ*)) makes a curve in the

*SML*color space which is usually referred to as the

*spectral curve*( Figure 1). The cone through the spectral curve is referred to as the

*spectral cone*( Figure 1). It represents color of the monochromatic lights. The color signal cone proves to be the convex hull of the spectral cone. The roots of Equation 8 are the points where the spectral curve intersects the plane defined by the equation

*k*

_{1}

*S*+

*k*

_{2}

*M*+

*k*

_{3}

*L*= 0.

*spectrum locus*). Let us complete this contour with the interval joining the ends of the spectrum locus (the

*purple interval*). If the resultant contour (referred to as the

*completed spectral contour*) is convex, then, as noted by West and Brill (1983), the two-transition assumption holds true.

*λ*

_{min},

*λ*

_{max}] (i.e.,

*t*(

*λ*) > 0 in Equation 2), Equation 8 amounts to

*S*-cone's, are very low in the long wavelength end of the spectrum, they differ from zero. (This property is of great theoretical importance. For instance, letting

*p*

_{1}(

*λ*) = 0 for

*λ*

_{min}>

*λ*′ ≥

*λ*≥

*λ*

_{max}in Equation 5, will result in metamerism in the object-color solid boundary.) The cone fundamentals based on these photopigment spectra are presented in Figure 3.

*regular object-color*

*solid*( Figure 8). It is a closed volume nested into the real object-color solid.

*λ*

_{1}and

*λ*

_{2}. Specifically, each such line represents the color signals induced by the spectral reflectance functions

*x*

_{2}(

*λ*;

*λ*

_{1},

*λ*

_{2}) ( Equation 5) when either

*λ*

_{1}or

*λ*

_{2}is fixed. Two coordinate lines, corresponding to the two particular cases when either

*λ*

_{1}=

*λ*

_{min}or

*λ*

_{2}=

*λ*

_{max}, will be referred to as the

*meridians*of the regular object-color solid. When

*λ*

_{2}=

*λ*

_{max}, the meridian (referred to as the

*main meridian*) is an image of the spectral reflectance functions

*x*

_{1}(

*λ*;

*λ*

_{1}) ( Equation 4). When

*λ*

_{1}=

*λ*

_{min}, the meridian (referred to as the

*opposite meridian*) is an image of the optimal spectral reflectance functions 1 −

*x*

_{1}(

*λ*;

*λ*

_{2}). The main and opposite meridians are marked with red and blue, respectively, in Figure 2, the main meridian being almost hidden. The space between the coordinate lines corresponds to (

*λ*

_{max}−

*λ*

_{min}) / 30. As can be seen, the coordinate lines are not evenly distributed across the boundary surface. They can be made equally spaced after reparameterizing the wavelength interval (see Wavelength reparameterization section).

*I*(

*λ*) > 0, for each spectral reflectance function

*x*(

*λ*) there is (i) a unique optimal spectral reflectance function

*x*

_{opt}(

*λ*) and (ii) a unique number 0 ≤

*α*≤ 1 such that spectral reflectance function

*x*(

*λ*). (Recall that

*x*

_{0.5}(

*λ*) ≡ 0.5.)

*x*(

*λ*), the optimal spectral reflectance

*x*

_{opt}(

*λ*) that determines the spectral waveform of ( Equation 10) can be found from the condition that the color signal (

*φ*

_{1}(

*x*

_{opt}),

*φ*

_{2}(

*x*

_{opt}),

*φ*

_{3}(

*x*

_{opt})) is the boundary point of the object-color solid lying on the same radius as (

*φ*

_{1}(

*x*),

*φ*

_{2}(

*x*),

*φ*

_{3}(

*x*)).

*x*

_{opt}(

*λ*), this atlas will be referred to as the

*optimal color*

*atlas*. A member ( Equation 10) of the optimal color atlas metameric to

*x*(

*λ*) will be called the

*optimal metamer*of

*x*(

*λ*). It takes two values: 0.5(1 ±

*α*), the difference between which being

*α*. Therefore,

*α*indicates how much the spectral reflectance function (1 −

*α*)

*x*

_{0.5}(

*λ*) +

*αx*

_{opt}(

*λ*) =

*x*

_{0.5}(

*λ*) +

*α*(

*x*

_{opt}(

*λ*) −

*x*

_{0.5}(

*λ*)) deviates from the level of 0.5. I will call the quantity

*α*the

*chromatic amplitude*of the optimal metamer ( Equation 10). It can be evaluated as

*α*goes from 0 to 1, the color signal of Equation 10 moves from the center to the boundary of the object-color solid remaining on the same radius.

*x*

_{opt}(

*λ*) (see Optimal spectral reflectance functions and object-color solid section). Generally, a spectral reflectance function will be referred to as of type I (respectively, II) if its optimal metamer is of type I (respectively, II).

*x*(

*λ*) ≠

*x*

_{0.5}(

*λ*), there is a unique optimal metamer that differs from

*x*(

*λ*) only in type. Two such optimal metamers will be called

*complementary*. They map to the points symmetrical with respect to the object-color solid center.

^{3}For the sake of generality, the optimal metamer

*x*

_{0.5}(

*λ*) will be assumed to be complementary to itself. The colors of two complementary optimal metamers will be called

*complementary*. For each object color, there is only one complementary color.

_{opt,}the optimal spectral reflectance function complementary to an optimal spectral reflectance function

*x*

_{opt}(i.e.,

_{opt}= 1 −

*x*

_{opt}), we get

*x*

_{0.5}(

*λ*) +

*α*(

_{opt}(

*λ*) −

*x*

_{0.5}(

*λ*)) =

*x*

_{0.5}(

*λ*) −

*α*(

*x*

_{opt}(

*λ*) −

*x*

_{0.5}(

*λ*)). This equation allows us to extend formally the chromatic amplitude range treating an optimal metamer with

*negative*chromatic amplitude as that of positive chromatic amplitude of the same magnitude and complementary spectral waveform. For example, the optimal spectral reflectance function

*x*(

*λ*) ≡ 0 can be interpreted as the optimal spectral reflectance function

*x*(

*λ*) ≡ 1 with negative chromatic amplitude −1. Hence, using negative chromatic amplitude, we can restrict our consideration to only optimal spectral reflectance functions of type I. In this case, the spectral reflectance functions of negative chromatic amplitude will be exactly the set of the spectral reflectance functions of type II. I will use the term

*purity*for the absolute value of chromatic amplitude.

*chromatic base*of the optimal color atlas. It will be shown in the Perceptual correlates of the

*αδλ*color descriptors section that the chromatic base specifies the object-color hues. In the case of the two-transition assumption, the chromatic base comprises only rectangular spectral reflectance functions. It will be called the

*rectangle chromatic base*.

*x*(

*λ*) producing the color signal (

*φ*

_{1}(

*x*),

*φ*

_{2}(

*x*),

*φ*

_{3}(

*x*)) (see Equation 3), consider simultaneous equations

*x*

_{2}(

*λ*;

*λ*

_{1},

*λ*

_{2}) is a rectangular spectral reflectance function with the transition wavelengths

*λ*

_{1}and

*λ*

_{2}( Equation 5) (

*i*= 1,2,3). These equations can always be resolved with respect to

*α, λ*

_{1}, and

*λ*

_{2}. If the color signal (

*φ*

_{1}(

*x*),

*φ*

_{2}(

*x*),

*φ*

_{3}(

*x*)) lies within the regular object-color solid, the solution will be such that ∣

*α*∣ ≤ 1. For those color signals that are between the boundary surfaces of the real and regular object-color solids, we will get ∣

*α*∣ > 1. In this case (1 +

*α*) / 2 exceeds 1, and (1 −

*α*) / 2 is negative. While such profiles cannot be treated as spectral reflectance functions, I will consider them as

*improper spectral reflectance functions*. Taking them into consideration allows one to represent the optimal color atlas by rectangle spectral reflectance functions with not more than two transitions.

*x*(

*λ*), there is a (perhaps, improper) spectral reflectance function

*x*

_{0.5}(

*λ*) +

*α*(

*x*

_{2}(

*λ*;

*λ*

_{1},

*λ*

_{2}) −

*x*

_{0.5}(

*λ*)) that meets Equation 12. It will be called the

*rectangular metamer*of the object-color stimulus

*x*(

*λ*). The rectangular metamer will be referred to as improper if ∣

*α*∣ > 1. Although the set of all the rectangle metamers for all the object-color stimuli cannot be a proper colors atlas (as it contains improper spectral reflectance functions), I will refer to it as the

*rectangle color*

*atlas*. To avoid confusion, I will call

*improper*those color atlases that contain improper spectral reflectance functions. As a rectangular metamer can be specified by only three numbers (

*λ*

_{1},

*λ*

_{2}, and

*α*), using the rectangle color atlas will allow a three-coordinate representation of the optimal color atlas.

*x*

_{1}(

*λ*),

*x*

_{2}(

*λ*), and

*x*

_{3}(

*λ*), any element of the optimal color atlas

*x*(

*λ*) is metameric to some linear combination of these functions:

*k*

_{1}

*x*

_{1}(

*λ*) +

*k*

_{2}

*x*

_{2}(

*λ*) +

*k*

_{3}

*x*

_{3}(

*λ*). The weights

*k*

_{1},

*k*

_{2}, and

*k*

_{3}can be considered as linear coordinates of the element of the optimal color atlas. When

*x*(

*λ*) runs over the whole optimal color atlas, the resultant set of the functions, {

*k*

_{1}

*x*

_{1}(

*λ*) +

*k*

_{2}

*x*

_{2}(

*λ*) +

*k*

_{3}

*x*

_{3}(

*λ*)}, makes an improper color atlas that will be referred to as a

*linear color*

*atlas*based on the basis functions

*x*

_{1}(

*λ*),

*x*

_{2}(

*λ*),

*x*

_{3}(

*λ*).

*λ*

_{1},

*λ*

_{2}, and

*α*are, obviously, independent of illumination. Still, a number of optimal spectral reflectance functions with three and four transitions have been found. These spectral reflectance functions are, generally, metameric to different rectangular metamers under different illuminants. The rectangle metamers have been calculated for the optimal spectral reflectance functions with more than two transitions for a few illuminants. Their purity has been found to fall into the narrow band between 1 and 1.01. Hence, if one measures the distance between the boundaries of the real and regular object-color solids along radii, then the maximum distance does not exceed 1%. It follows that the regular object-color solid is a good approximation to the real object-color solid. Furthermore, the variability of the coordinates

*λ*

_{1},

*λ*

_{2}, and

*α*for the optimal spectral reflectance functions with more than two transitions has been found to be negligibly small. Therefore, while being of theoretical importance, the dependence of the coordinates

*λ*

_{1},

*λ*

_{2}, and

*α*on illuminant is unlikely to have a significant impact in most practical applications. Thus, the rectangle color atlas can be use to provide the three-coordinate representation of the optimal color atlas which is quite easy to compute.

*λ*

_{1}and

*λ*

_{2}, and its type. If it is of type I, it takes up 1 within the interval [

*λ*

_{1},

*λ*

_{2}], its complementary spectral reflectance function taking up 1 exactly outside it. The interval [

*λ*

_{1},

*λ*

_{2}] can be specified by its width

*δ*= ∣

*λ*

_{1}−

*λ*

_{2}∣ and the position of its center

*λ*

_{1}+

*λ*

_{2})/2. I refer to

*δ*as the

*spectral bandwidth*and

*central wavelength*for a rectangular spectral reflectance function of type I. Thus, the rectangular metamer of a spectral reflectance function of type I is completely specified by the three numbers: purity, spectral bandwidth, and central wavelength ( Figure 9).

*λ*

_{min}and

*λ*

_{max}, turning the visible spectrum interval into the visible spectrum circle ( Figure 10). Two points,

*λ*

_{min}≤

*λ*

_{1}<

*λ*

_{2}≤

*λ*

_{max}, determine two complementary arcs on this circle. The rectangular spectral reflectance function of type I takes 1 on the arc where

*λ*

_{1}≤

*λ*≤

*λ*

_{2}. The spectral bandwidth and central wavelength are the length and the center of this arc, respectively. Its complementary spectral reflectance function (of type II) takes 1 on the complementary arc. It is the length and center of this complementary arc that are taken, by definition, as the spectral bandwidth and central wavelength for spectral waveforms of type II.

*λ*

_{max}−

*λ*

_{min}. This is in line with our intuition of complementarity.

*x*(

*λ*) ≡ 0 has zero spectral bandwidth; however, central wavelength is not defined for it. Likewise, for

*x*(

*λ*) ≡ 1, the spectral bandwidth

*δ*=

*λ*

_{max}−

*λ*

_{min}, the central wavelength being not defined.

*δ*< (

*λ*

_{max}−

*λ*

_{min}) and

*λ*

_{min}<

*λ*

_{max}, a pair (

*δ,*

*λ*

_{1}and

*λ*

_{2}of which are given by

*δ*= 0) defines the optimal spectral reflectance function

*x*(

*λ*) ≡ 0. Likewise, the spectral bandwidth

*δ*=

*λ*

_{max}−

*λ*

_{min}defines the optimal spectral reflectance function

*x*(

*λ*) ≡ 1.

*δ,*

*αδλ*

*color*

*descriptors*of the spectral reflectance function. The

*αδλ*color descriptors will be used to represent spectral reflectance functions in a 3D space ( Object-color space section).

*αδλ*color descriptors

*αδλ*color descriptors for the spectral reflectance functions of 1600 Munsell papers (retrieved from http://spectral.joensuu.fi/databases) were evaluated for the CIE illuminant

*D*65 using the cone fundamentals based on the photopigment spectra ( Figure 3). The papers having the same spectral waveform and differing only in purity were found to have similar color appearance. Specifically, they differ in a perceptual dimension, which will be referred to as

*apparent purity*. Intuitively, apparent purity indicates the strength of the chromatic quality (specified by the spectral waveform of the rectangular metamer). It should be mentioned, however, that apparent purity is different from Munsell Chroma. Indeed, having zero Munsell Chroma, all achromatic Munsell papers, nevertheless, differ in purity, and thus in apparent purity. For example, the purity of the gray paper with the spectral reflectance

*x*

_{0.5}(

*λ*) is zero whereas that of the white paper (

*x*(

*λ*) ≡ 1) is maximal (i.e., equal to 1). The apparent difference between various spectral reflectances that have the same purity will be referred to as that in

*chromaticity*. In other words, the chromatic base specifies chromaticity of object-color stimuli.

*blackness and whiteness*) that seems to correlate with the spectral bandwidth. Indeed, whiteness was found to increase and blackness to decrease with the spectral bandwidth. Consider, for example, a series of Munsell papers of the same central wavelength. When the spectral bandwidth is close to zero, the papers look black. When the spectral bandwidth gradually increases, the papers get tinged with some chromatic hue, loosing their blackness. For some value of spectral bandwidth, the hue becomes purely chromatic (containing no blackness at all). Further increase of spectral bandwidth results in a tinge of whiteness. Whiteness becomes maximal for the spectral bandwidth value close to (

*λ*

_{max}−

*λ*

_{min}).

*λ*

_{min}and

*λ*

_{max}, have been inappropriately defined. Note that the length of the visible spectrum interval (

*λ*

_{max}−

*λ*

_{min}) is involved in the definition of both spectral bandwidth and central wavelength for spectral reflectance functions of type II ( Equations 13 and 14). Any change of

*λ*

_{min}and/or

*λ*

_{max}will affect the

*αδλ*color descriptor values. Therefore, if, say, the long wavelength end,

*λ*

_{max}, is unduly in excess, this will result in a central wavelength shift for spectral reflectance functions of type II similar to that observed in Figure 12.

*λ*

_{1}< 405 nm and reaches its maximum approximately at 650 nm. To be more exact, the length of the fragments of the opposite meridian for 380 ≤

*λ*

_{1}≤ 405 is less than 0.01 and more than 0.99 for 380 ≤

*λ*

_{1}≤ 650. Therefore, given a 1% accuracy, the values 405 and 650 nm can be taken as effective ends of the visible spectrum interval.

*σ*of the arc of the opposite meridian between the points (

*s*

_{1}(

*λ*

_{min}),

*s*

_{2}(

*λ*

_{min}),

*s*

_{3}(

*λ*

_{min})) and (

*s*

_{1}(

*λ*),

*s*

_{2}(

*λ*),

*s*

_{3}(

*λ*)) (

*λ*

_{min}≤

*λ*≤

*λ*

_{max}) is given by

*ω*. Specifically, given a rectangular spectral reflectance function

*x*

_{2}(

*λ*;

*λ*

_{1},

*λ*

_{2}) ( Equation 5), define

*ω*

_{ i}=

*λ*

_{ i}) (

*i*= 1,2). Similarly, define

*α,*

*α,*one can use spectral bandwidth and central wavelength,

*δλ*

*diagram*. It shows which element of the rectangle chromatic base should be taken to produce (in the sense of Equation 10) the rectangular metamer of the spectral reflectance in question. Figure 16 presents in the

*δλ*diagram the same Munsell papers as Figure 12. Note that, unlike Figure 12, there is almost no misalignment between the papers of different type.

*αδλ*color descriptors of the Munsell and NCS papers

*δλ*diagram. As one can see, the Munsell papers are rather unevenly distributed. There are large “lacunas” void of the Munsell papers. The Munsell collection is believed to be uniform. Specifically, adjacent papers in the Munsell tree are supposed to be separated from each other by small, presumably, equal perceptual distances. It follows that even after wavelength reparameterizing the

*δλ*diagram remains perceptually non-uniform.

*αδλ*dimensions. Figure 18 shows that purity gravitates to values more than 0.5. Most of the Munsell papers have the spectral bandwidth value less than 0.25 ( Figure 19). Central wavelength is also distributed rather unevenly ( Figure 20). There are four well-pronounced peaks—around the

*δλ*diagram is a particular feature of the Munsell collection, the

*αδλ*color descriptors have been also evaluated for the 1950 NCS papers (Hard & Sivik, 1981) for the same illuminant and cone fundamentals. Figure 21 presents the Munsell and NCS papers in the

*δλ*diagram. Both the Munsell and NCS collections are distributed remarkably similarly.

*δλ*diagram allows one to see how these two different color systems relate to each other. Moreover, this allows for transforming the Munsell notations into that of NCS and vice versa, giving us a way to go between the system. The practical importance of such a transformation is generally acknowledged (e.g., Nayatani, 2004).

*δλ*diagram has two major shortcomings. Firstly, it does not represent purity. For purity to be represented, one needs one more spatial dimension. Secondly, mapping the rectangle chromatic base onto a rectangle area in the Cartesian plane is incorrect from the topological point of view. It is discontinuous at both the perfect reflector (

*x*(

*λ*) ≡ 1) and the ideal black (

*x*(

*λ*) ≡ 0). Indeed, for any central wavelength, a rectangular spectral reflectance function converges to

*x*(

*λ*) ≡ 0 when the spectral bandwidth goes to zero. However, a point (

*δ*

_{0},

*λ*

_{0}) in the

*δλ*diagram will converge to (0,

*λ*

_{0}) when

*δ*

_{0}→ 0. Thus, the points with different

*λ*

_{0}will converge to different points in the bottom side of the rectangle area in Figure 21. One can avoid such a break of continuity mapping the rectangle chromatic base onto a sphere as described below.

*α,*spectral bandwidth

*ρ*of the point from the origin equals the purity, i.e.,

*β*and longitude

*θ*being

*αδλ*color descriptors by Equations 20 and 21 will be referred to as the

*αδλ*

*object-color*

*space*.

*x*(

*λ*) ≡ 1 (i.e., the perfect reflector) and

*x*(

*λ*) ≡ 0 (i.e., the ideal black), respectively. Spectral bandwidth and central wavelength serve as the internal coordinates on the spherical surface

*ρ*= 1; thus, every rectangular spectral reflectance function can be uniquely located on it. Therefore, the rectangle chromatic base maps onto the unit sphere

*ρ*= 1, which will be called the

*chromaticity diagram, β*and

*θ*in Equation 21 being referred to as the

*chromaticity coordinates*.

*αδλ*object-color space as a 3D unit ball. The rectangle color atlas induced by cone fundamentals not satisfying the two-transition assumption is represented in the

*αδλ*object-color space as a 3D volume deviating from the 3D unit ball. As this volume can be considered as a smooth transformation of the object-color solid, it will be referred to as the

*αδλ*object-color

*solid*.

*αδλ*object-color solid is invariant with respect to any non-singular linear transformation of the cone fundamentals. Furthermore, although it is not invariant relative to an illuminant change, it remains almost the same for any illuminant that takes non-zero value across the visible spectrum (see Optimal metamers and optimal color atlas section).

*αδλ*object-color solid for the cone fundamentals based on the photopigment spectra ( Figure 3) is practically indistinguishable from the unit ball ( Figures 22 and 23). Since the Munsell papers have purity of the magnitude less than 1 ( Figure 18), they all lie inside of the unit ball. For each Munsell paper, a point on the

*αδλ*object-color solid boundary (i.e., the chromaticity diagram) was determined, which lies on the same radius. In other words, all the Munsell papers were projected on the unit sphere dropping down the information of their purity ( Figures 22 and 23). As in Figure 17, symbol shape encodes Munsell Value and symbol color Munsell Hue. In spite of the metric difference between the

*δλ*diagram and the spherical chromaticity diagram,

^{4}Munsell papers are also quite unevenly distributed over the spherical surface in Figures 22 and 23. As can be seen in

*illuminant-induced color*

*stimulus shift*) results in an alteration of object color. For instance, as mentioned in the Optimal metamers and optimal color atlas section, object-color stimulus having complementary colors may get non-complementary colors because of the illuminant-induced color stimulus shift.

*D*65 with the CIE illuminant

*A*has been examined for Munsell and NCS papers. Specifically, using Equation 12 the

*αδλ*color descriptors of 1600 Munsell papers were calculated for the illuminants

*D*65 and

*A*using the cone fundamentals depicted in Figure 3. Figure 24 shows how purity evaluated for the CIE illuminant

*A*covaries with that evaluated for the CIE illuminant

*D*65. Likewise, Figures 25 and 26 present the spectral bandwidths and central wavelengths as evaluated for the CIE illuminants

*D*65 and

*A,*respectively.

*α,*

*x*(

*λ*) under one illumination, and

*α*′,

*x*(

*λ*) can be quantified by the difference

*α*−

*α*′, the illuminant-induced spectral bandwidth shift by

*D*65 to

*A*have been derived from the data presented in Figures 24– 26. Specifically, the color descriptors

*α,*

*D*65 and then

*α*′,

*A*. The distribution of the purity shift magnitude, ∣

*α*−

*α*′∣, is given in Figure 27. Purity of the Munsell collection turns out to be rather robust to the illuminant change. Specifically, it has been found that 1023 Munsell papers (64%) change their purity by 0.01 or less, 1404 papers (88%) changing it by not more than 0.025. Similarly, Figures 28 and 29 present the distributions of the spectral bandwidth shift magnitude, ∣

^{5}through the points. More specifically, consider two object-color stimuli lying on the same sphere in the

*αδλ*object-color space (i.e., of the same purity) with coordinates (

*c,*

*c,*

*β, β*′,

*θ,*and

*θ*′ be the geographical latitudes and longitudes of the points (

*β*=

*π*

*π*/2,

*β*′ =

*π*

*π*/2,

*θ*= 2

*π*

*θ*′ = 2

*π*

*c*is given by

*d*is proportional to purity,

*c*. This is in line with our intuition that the chromaticity difference between the object-color stimuli decreases with their purity. Equation 22 will be used to quantify the chromaticity difference between the object-color stimuli with color descriptors (

*c,*

*c,*

*c*and

*c*′, provided that the purity difference ∣

*c*−

*c*′∣ is not large. In this case, the average purity, (

*c*−

*c*′) / 2, can be put as

*c*in Equation 22. More specifically, to measure the chromaticity difference between the object-color stimuli with color descriptors (

*c,*

*c*′,

*c*−

*c*′ is small the following formula can be used:

*π*in Equation 23, the maximum chromaticity difference equals 1.

*αδλ*object-color solid (i.e.,

*β*=

*β*′ = 0 and

*θ*−

*θ*′ =

*π*): one of full purity (

*c*= 1) and the other with the purity close to zero, that is,

*c*′ ≅ 0. In this case, Equation 23 yields

*D*≅

*c*/2. Therefore, despite that these object-color stimuli differ practically only in purity, the Equation 23 gives a measure substantially different from zero. To avoid such a “pathology,” Equation 23 can be modified as follows

*c*−

*c*′ is small, both Equations 23 and 24 give approximately the same value. However, when

*c*′ → 0,

*D*in Equation 24 approaches zero for any (

*D*65 to

*A*) has been evaluated for 1600 Munsell papers ( Figure 30) and 1950 NCS papers ( Figure 31). The mean chromaticity shift for 1600 Munsell papers is 0.0118; the maximum chromaticity shift being 0.1283. For 57.6% of the Munsell papers, the chromaticity shift does not exceed 0.01, for 83% 0.02, and for 92.3% 0.03. A similar result has been observed for NCS papers ( Figure 31). Specifically, the mean and maximum chromaticity shifts are found to be 0.0135 and 0.1349, respectively. Nearly half the NCS papers (48.5%) undergo chromaticity shift of not more than 0.01, 79.1% changing their chromaticity by not more than 0.02, and 92% not more than 0.03.

*D*65 to

*A*) observed for all the 1600 Munsell papers. Hence, on average, a change in color for Munsell papers induced by the shift from the illuminants

*D*65 to

*A*is, by and large, at the limit of our ability to discern chromatic differences.

*αδλ*object-color space vary with illumination as well (because of illuminant-induced color stimulus shift). Yet, there is an important difference between these two cases. In the case of the CIE 1931 color stimulus representation, the coordinate system itself changes with illumination. For example, a sheet of paper under (i) the direct sun light and (ii) the light from blue sky will reflect different lights. These lights will be represented by two different points in the CIE 1931 color space. This difference is, generally, produced by both a possible change of the sheet color (due to the illuminant-induced color stimulus shift) and by the change in the coordinate system. In the

*αδλ*object color, these two contributions are well separated because the coordinate system in this space does not depend on illuminant. That is, the new space is invariant of illumination change. Moreover, any change of the coordinates of a reflecting object evaluated for two illuminants indicates the change of the object color induced by the illuminant alteration. Hence, the new space can be used to predict the effect of the illumination on object color that is hard to achieve with the CIE 1931 color space and its derivatives.

^{6}For instance, 200 years ago Runge, with his color sphere, anticipated the geometrical form of the object-color manifold

^{7}(Kuehni, 2003, pp. 59–62). However, one cannot locate a particular spectral reflectance in Runge's sphere. Nor can one compute Munsell Hue, Chroma, and Value for an arbitrary spectral reflectance function.

^{8}On the other hand, the dimensions of the physical color order systems do not lend themselves readily to perceptual interpretation. For example, representing a spectral reflectance as a weighted combination of three basis spectral reflectance functions, the linear models specify each arbitrary spectral reflectance by the three numbers (for a review, see Brainard, 1995; Hurlbert, 1998; Maloney, 2003). However, a phenomenological interpretation of these numbers is hardly possible.

*a*and

*b,*which are not necessarily symmetric with respect to value 0.5 (that is,

*a*+

*b*can differ from 1). Secondly, Ostwald's color atlas does not include the whole set of rectangular spectral reflectance functions. More specifically, Ostwald employed a set of rectangular spectral reflectance functions of special form which he called semichromes (Vollfarben) (Koenderink & Doorn, 2003). The set of semichromes included only those rectangular spectral reflectance functions for which either their transition wavelengths were complementary to each other; or one transition wavelength coincided with the visible spectrum interval end (i.e.,

*λ*

_{min}or

*λ*

_{max}), and the other lay between the wavelengths complementary to

*λ*

_{min}and

*λ*

_{max}. The semichromes prove to map into a closed curve on the object-color solid boundary (Koenderink & Doorn, 2003). As a result, Ostwald's color atlas turned out to be not complete; that is, it did not encompass all the object colors. Moreover, as the complementarity of wavelengths depends on illuminant, Ostwald's definition of semichromes depends on illuminant. Therefore, Ostwald's color atlas is not invariant of illumination.

*x*

_{0.5}(

*λ*) ≡ 0.5 always belongs to the class of metamerism, which is the center of the manifold of all the classes of metamerism for any illuminant. Note that this way of encoding is similar to that which the visual system uses to encode color. Achromatic and chromatic hues encode the optimal object-color stimulus, and purity encodes the distance from the center. Therefore, the way of encoding object color underlying the proposed object-color space is a rough sketch of the end product of the human color vision.

*αδλ*color descriptors of the optimal rectangular spectral reflectance functions remain practically constant with respect to illumination. Particularly, the chromaticity coordinates and purity of the perfect reflector (

*x*(

*λ*) ≡ 1) and the ideal black (

*x*(

*λ*) ≡ 0) are the same for all the illuminants (with positive spectral power distribution). This is not the case when reparameterization of the optimal color atlas is based on a linear representation unless the basis spectral reflectance functions themselves include the perfect reflector or ideal black. Generally, the linear coordinates of the optimal rectangular spectral reflectance functions will change with illumination. Furthermore, the basis spectral reflectance functions move from one class of metamerism to another when illumination alters. Therefore, the linear representation of a metameric class is performed in terms of the basis that is subject to uncontrollable change when illumination alters. In other words, the linear models do not provide a frame reference robust to illumination change. From this point of view, using the linear models for representing spectral reflectance provides no advantage to the common practice of using the CIE 1931 color space to represent object color.

^{9}The same way as the Munsell paper in question.

*Object-color*

*stimulus*: An object-color stimulus is spectral reflectance.

*Illuminant*: An illuminant is spectral power distribution.

*Remark*: Object-color stimulus times illuminant amounts to color stimulus as defined by Wyszecki and Stiles (1982, p. 723).

*Sensor*: A sensor

*φ*is a linear device the response,

*φ*(

*w*), of which to a color stimulus

*w*(

*λ*) is given by

*s*(

*λ*) is the spectral sensitivity of the sensor and [

*λ*

_{min},

*λ*

_{max}] is the visible spectrum interval.

*Color*

*signal*: The color signal produced by sensors

*φ*

_{1},

*φ*

_{2}, and

*φ*

_{3}in response to a color stimulus

*w*(

*λ*) is a triplet (

*φ*

_{1}(

*w*),

*φ*

_{2}(

*w*),

*φ*

_{3}(

*w*)).

*Color*

*signal space*: Given sensors

*φ*

_{1},

*φ*

_{2}, and

*φ*

_{3}, a color signal space is a 3D affine space in which any color stimulus

*w*(

*λ*) is represented as a point with the coordinates (

*φ*

_{1}(

*w*),

*φ*

_{2}(

*w*),

*φ*

_{3}(

*w*)).

*Metameric object-color*

*stimuli*: Two physically different object-color stimuli producing the same color signals under some illuminant are called metameric under this illuminant.

*Object-color*: An object-color is a class of metameric object-color stimuli.

*Object-color*

*solid*: The object-color solid is a set of points in the color signal space produced by all possible object-color stimuli for a fixed illuminant.

*Optimal object-color*

*stimuli*: An optimal object-color stimulus is a spectral reflectance the color signal of which lies on the boundary surface of the object-color solid.

*Remark*: If

*x*(

*λ*) is optimal object-color stimulus, then 1 −

*x*(

*λ*) is optimal object-color stimulus as well.

*Complementary optimal object-color stimuli*: Given an optimal object-color stimulus

*x*(

*λ*), the optimal object-color stimulus 1 −

*x*(

*λ*) is called

*complementary*to

*x*(

*λ*).

*Step function*: A step (spectral reflectance) function is a piecewise constant function taking only two values: 0 or 1.

*Remark*: Each optimal object-color stimulus is a step spectral reflectance function.

*Transition wavelengths*: Transition wavelengths of a step spectral reflectance function are those where transition between 1 and 0 occurs.

*Rectangular function*: A rectangular (spectral reflectance) function is a step function taking 1 on the interval [

*λ*

_{1},

*λ*

_{2}] (

*λ*

_{min}≤

*λ*

_{1}≤

*λ*

_{2}≤

*λ*

_{max}) and 0 outside it.

*Complete color*

*atlas*: A complete color atlas is a set of (not metameric) object-color stimuli such that each object-color stimulus is metameric to one of its element.

*Illuminant invariant color*

*atlas*: A complete color atlas is said to be invariant with respect to the family of illuminants if and only if (i) each class of metamerism contains exactly one element of the atlas under any illuminant in the family and (ii) any two elements of the atlas do not become metameric under any illuminant in the family.

*Optimal color*

*atlas*: The optimal color atlas is a set of spectral reflectance functions expressed as

*x*

_{0.5}(

*λ*) is the spectral reflectance function taking 0.5 at every wavelength

*λ*within the visible spectrum interval [

*λ*

_{min},

*λ*

_{max}] and

*x*

_{opt}(

*λ*) runs over the whole set of optimal object-color stimuli.

*Remark*: The optimal color atlas is complete and invariant with respect to illuminants with positive spectral power distribution.

*Rectangular metamer*: The rectangular metamer of an object-color stimulus

*x*(

*λ*) is a piecewise constant spectral reflectance function, metameric to

*x*(

*λ*), which is given by

*x*

_{0.5}(

*λ*) is defined as above ( Equation A2);

*x*

_{2}(

*λ*;

*λ*

_{1},

*λ*

_{2}) is the rectangular spectral reflectance function with the transition wavelengths

*λ*

_{1}≤

*λ*

_{2};

*α*is a real number.

*Proper and improper rectangular metamers*: The rectangular metamer ( Equation A3) is called proper (respectively, improper) if ∣

*α*∣ in Equation A3 is such that ∣

*α*∣ ≤ 1 (respectively, ∣

*α*∣ > 1).

*Rectangle color*

*atlas*: The rectangle color atlas is the set of the rectangular metamers for all the object-color stimuli.

*Central wavelength*: The central wavelength of a rectangular spectral reflectance function with the transition wavelengths

*λ*

_{1}and

*λ*

_{2}is given by (

*λ*

_{1}+

*λ*

_{2}) / 2. The central wavelength of an element of the rectangle color atlas (1 −

*α*)

*x*

_{0.5}(

*λ*) +

*αx*

_{2}(

*λ*;

*λ*

_{1},

*λ*

_{2}) is the central wavelength of the rectangular spectral reflectance function

*x*

_{2}(

*λ*;

*λ*

_{1},

*λ*

_{2}). The central wavelength of an object-color stimulus

*x*(

*λ*) is the central wavelength of its rectangular metamer.

*Spectral band*: The spectral bandwidth of a rectangular spectral reflectance function with the transition wavelengths

*λ*

_{1}and

*λ*

_{2}is given by ∣

*λ*

_{1}−

*λ*

_{2}∣. The spectral bandwidth of an element of the rectangle color atlas (1 −

*α*)

*x*

_{0.5}(

*λ*) +

*αx*

_{2}(

*λ*;

*λ*

_{1},

*λ*

_{2}) is the spectral bandwidth of the rectangular spectral reflectance function

*x*

_{2}(

*λ*;

*λ*

_{1},

*λ*

_{2}). The spectral bandwidth of an object-color stimulus

*x*(

*λ*) is the spectral bandwidth of its rectangular metamer.

*Chromatic amplitude*: The parameter

*α*of an element of the rectangle color atlas (1 −

*α*)

*x*

_{0.5}(

*λ*) +

*α*

*x*

_{2}(

*λ*;

*λ*

_{1},

*λ*

_{2}) is its chromatic amplitude. The chromatic amplitude of an object-color stimulus

*x*(

*λ*) is the chromatic amplitude of its rectangular metamer.

*Purity*: The purity of an object-color stimulus is the absolute value of its chromatic amplitude.

*Complementary object-color*

*stimuli*: Two object-color stimuli are complementary if their rectangle metamers differ only by sign of the chromatic amplitude

*α*.

*Chromaticity coordinates*: The chromaticity coordinates of an object-color stimulus

*x*(

*λ*) are the quantities

*β*and

*θ,*which are related to the spectral bandwidth and the central wavelength of

*x*(

*λ*) as

*β*=

*π*

*π*/2 and

*θ*= 2

*π*

*Object-color*

*space*: Object-color space is a 3D space with the geographical coordinate system such that each object-color stimulus

*x*(

*λ*) is represented as a point at a distance from the origin equal to its purity, the latitude and longitude being equal to the chromaticity coordinates of

*x*(

*λ*).

*Chromaticity difference*: Chromaticity difference between object-color stimuli with the chromaticity coordinates

*β, β*′,

*θ,*and

*θ*′ and purities

*c*and

*c*′ is given by Equation 24.

*x*

_{1}(

*λ*) and

*x*

_{2}(

*λ*) be two different spectral reflectance functions given by Equation 10, that is,

*x*

_{1}(

*λ*) = (1 −

*α*′)

*x*

_{0.5}(

*λ*) +

*α*′

*x*

_{opt}′(

*λ*) and

*x*

_{2}(

*λ*) = (1 −

*α*″)

*x*

_{0.5}(

*λ*) +

*α*″

*x*

_{opt}″(

*λ*), where

*x*

_{opt}′(

*λ*) and

*x*

_{opt}″(

*λ*) are optimal spectral reflectance functions (with respect to some set of cone fundamentals) such that 0 <

*α*′,

*α*″ < 1, and either

*x*

_{opt}′(

*λ*) ≠

*x*

_{opt}″(

*λ*) or

*α*′ ≠

*α*″, or both the inequalities hold true. As illuminants with positive spectral power distribution do not change the optimal spectral reflectance function set, for any such illuminant,

*x*

_{opt}′(

*λ*) and

*x*

_{opt}″(

*λ*) will remain optimal. Given an arbitrary illuminant with positive spectral power distribution, let us denote

_{opt}′ and

_{opt}″ the points in the object-color solid boundary into which

*x*

_{opt}′(

*λ*) and

*x*

_{opt}″(

*λ*) map. Let

_{0.5},

_{1}, and

_{2}be the points in the object-color solid into which

*x*

_{0.5}(

*λ*),

*x*

_{1}(

*λ*), and

*x*

_{2}(

*λ*) map, respectively. Assume that

_{1}=

_{2}, that is, (1 −

*α*′)

_{0.5}+

*α*′

_{opt}′ = (1 −

*α*″)

_{0.5}+

*α*″

_{opt}″. Then, it implies that

*α*′ =

*α*″, then it follows from Equation A4 that

_{opt}′ =

_{opt}″. As there is no metamerism in the object-color solid boundary,

_{opt}′ =

_{opt}″ implies

*x*

_{opt}′(

*λ*) =

*x*

_{opt}″(

*λ*) that is impossible because

*x*

_{1}(

*λ*) and

*x*

_{2}(

*λ*) are different. Therefore,

*α*′ ≠

*α*″.

*α*′ >

*α*″ and rewrite Equation A4 as

_{opt}′ is an interior point of the object-color solid that is impossible because

*x*

_{opt}′(

*λ*) is optimal.

_{1}and

_{2}are different points in the object-color solid. So we conclude that different elements

*x*

_{1}(

*λ*) and

*x*

_{2}(

*λ*) of the optimal color atlas cannot become metameric under illuminants with positive spectral power distribution.

^{7}Needless to say that at the time, no experimental data corroborated Runge's guess. Yet, more recently, it was supported by multidimensional analysis of Munsell papers (Izmailov, 1995).

*Sources of color sciences*(pp. 134–182). Cambridge, MA: The MIT Press. (Reprinted from

*Annalen der Physik,*vol. 63, pp. 481–520, 1920).