A Theory of Unique and Colour Categories in the Human Colour Vision

Alexander D. Logvinenko* Department of Vision Sciences, Glasgow Caledonian University, Cowcaddens Road, Glasgow G4 0BA, United Kingdom

Received 12 July 2010; revised 13 September 2010; accepted 21 September 2010

Abstract: A novel approach to assessing colour appearance is tary (component) hues.3 For example, orange consists of yel- described. It is based on a new technique—partial -match- low and , purple of red and . An important issue is a ing—which allows for measuring colour in terms of compo- nomenclature of the component hues, particularly, their nent hues objectively, without resorting to verbal definitions. number. Hering believed that the component hues happened The new method is believed to have a potential to be as exact to be these six unique hues that Leonardo described. as colorimetric techniques. In contrast to classical colour Curiously enough, there is no objective definition of hue in matching, which implies visual equivalence of lights, partial general and unique hues in particular. For instance, in their hue-matching is based on judgements of whether two lights that definitive book, Wyszecki and Stiles defined hue as ‘‘the at- are different in colour have some hue in common. The major tribute of a colour perception denoted by blue, , , difference between classical colour matching and partial hue- red, purple, and so on,’’4 reducing it to corresponding verbal matching is that the latter is intransitive, whereas the former is categories, for which there is no operational definition. More- generally believed to be transitive (though see Logvinenko, over, their definition of unique hues is evidently circular. For Symposium on 75 years of the CIE Standard Colorimetric example, the definition of unique blue as a hue containing Observer, Vienna, Austria, 2006). Formally, partial hue-match- neither red nor green4 implies the definition of unique red ing can be described as a reflexive and symmetric binary rela- and green. But, unique red, in turn, is defined as a hue con- tion (i.e., tolerance). The theoretical framework of tolerance taining neither blue nor yellow. In fact, specifying unique spaces is used for developing a theory of partial hue- hues as nonreducible is rather a declaration of intention than matching. Ó 2011 Wiley Periodicals, Inc. Col Res Appl, 00, 000 – 000, a proper definition. Some naı¨ve observers believe that orange 2011; Published online in Wiley Online Library (wileyonlinelibrary.com). DOI or violet is unique colour (see Experiment below). One has to 10.1002/col.20661 admit that at the present there is no objective method to check whether this is the case, or this is simply resulted from Key words: colour vision; colour matching; component the lack of experience in introspective analysis. As a result, hues; unique hues; tolerance space when it comes to experiment, it is left to observers them- selves to decide which hue is ‘‘unique.’’ However, the con- INTRODUCTION tinuing debate* concerning the nature of unique hues (e.g., Refs. 6 and 7) and large interindividual variability in experi- It has been noticed since the ancient times that there are mental estimates of unique hues8 show that this problem is 1 unique colours. The distinctive feature of a unique colour is far too involved to be left to inexperienced observers. 2 its ‘‘nonreducibility’’ to any other colour. As far back as It should be also mentioned that the relationship between 500 years ago, Leonardo da Vinci voiced the fact that there unique and component hues is not always clearly understood. were four chromatic unique hues—yellow, blue, red and A component hue should not be necessarily unique. For green—and two achromatic—black and white. Thereafter, instance, in principle, only a perfect reflector could be per- Ewald Hering laid the foundations of his colour theory ceived as unique white. As real surfaces are not ideal reflec- assuming that every colour can be decomposed into elemen- tors, even the ‘‘whitest’’ of them are always tinged with some chromatic component hues. Therefore, white can, strictly Correspondence to: Alexander D. Logvinenko (e-mail: a.logvinenko@ speaking, never be experienced as unique. Yet, white is cer- gcu.ac.uk). tainly a component hue. On the other hand, our experiments Contract grant sponsor: EPSRC; contract grant number: EP/C010353/1. *For instance, it is still unclear whether brown can be reduced to the VC 2011 Wiley Periodicals, Inc. classical six unique hue, or it is a unique hue on its own (e.g., Ref. 5).

Volume 00, Number 0, Month 2011 1 show that grey is not a component hue (Logvinenko and Beat- Following the traditional terminology one can say that a tie, submitted for publication); however, it is unique by gen- component hue is unique if it is the unitary hue for some col- eral opinion. our stimulus. A hue structure is simple if every component Although, by and large, there is a consensus concerning hue is unique. In a complete hue structure every combination the six unique hues, the situation with nomenclature of com- of the component hues is assigned to some colour stimulus. ponent hues is less certain. As a matter of fact, the different There is every indication that a complete hue structure can- sets of component hues have been used to specify colour.9 not be a model of the human colour vision because such pairs For example, the natural colour system is based on six classi- of hues as red and green, and yellow and blue can hardly be cal unique hues,10 whereas, the Munsell colour system experienced at the same place and in the same time.3,13 11 involves purple as an additional component hue despite Definition 2 Given a hue structure hS; H;#i, let us that it is hardly unique. A few important questions immedi- define a binary relation s on S (partial hue-matching) ately arise. What is the minimal set of component hues suffi- cient to specify colour as perceived by normal trichromats? xsy , #ðÞ\x #ðÞy 6¼ [; 8x; y 2 S: (1) How are they related to the unique hues? How many unique and component hues exist in the human colour vision? The pair hiS; s will be called a partial hue-matching In this article, I present a new method (partial hue-match- structure associated with hS; H;#i. ing), which has been recently used to address experimen- tally some of these questions12 (Logvinenko and Beattie, In other words, colour stimuli x and y partially match submitted for publication). The method aims at establishing each other in hue if and only if they have the hues with the minimal set of component hues (i) not presupposing at least one common component hue. Partial hue-match- ing is a reflexive and symmetric binary relation, thus a their number, (ii) without resorting to verbal categories and 14 (iii) not assuming that they are unique. Some important the- tolerance. Note that in experiment one can observe only s W oretical issues arising with respect to this method are the tolerance , the hue function remaining inaccessible. explored in detail in the following section. Then, the results In this article, I investigate what hue structures can be s of a pilot experiment on partial hue-matching are described uniquely recovered from the tolerance . to show how the fundamental theoretical notions (such as In an experiment on partial hue-matching, observers are component hues and unique hues) can be operationalised. supposed to decided whether two colour stimuli partially Finally, the relevance of the presented theory to colour cat- match each other in hue, that is, if these stimuli have a egories and the limitations of the partial hue-matching tech- shade of some hue in common. Although this task is con- nique are discussed in the last section. sidered simple and easy by all observers, there is no guar- antee that the observers’ judgements are always based on sharing common hues. Moreover, the results obtained in THEORY an experiment described below as well as in the other studies12 (Logvinenko and Beattie, submitted for publica- In what follows by colour stimuli, I mean coloured surfaces tion) indicate the otherwise, that is, that some observers that are matt, flat, of the same shape, spatially homogene- rest their judgements on different grounds. Therefore, it ous, presented against a neutral background and illuminated becomes important to ascertain, first, if the empirical tol- by the same illuminant. Two such surfaces may either erance registered in an experiment is really based on par- match each other in colour or be different. In the latter case, tial hue-matching, and then, if it is established that this is they may share some common hue (as yellow and orange), the case, to retrieve the component hues of the underlying or they may have nothing in common at all (as red and hue structure. green). If two colour stimuli have at least one common hue, A departure point will be a tolerance s defined on the we shall say that they partially match in hue each other. colour stimulus S, which will be referred to as partial col- Definition 1 Let us consider a set of colour stimuli, S, our matching (see Definition 3). Every partial hue-match- and an abstract set H¼fgh1; ...; hm , the elements of ing is a partial colour matching, but the converse is cer- which will be referred to as component hues. Let us define tainly not true. The major objective is to find out the a function W: S ! 2H, which will assign a nonempty subset properties of partial colour matching, which guarantee of H to each colour stimulus x 2 S. We will call W a hue that it is induced by some hue structure. A theory of par- function, and W(x) ¼ (hi ,...,hi ) the hue of x (1 i1 \, ..., tial colour matching presented below is based on the 1 p 14–16 \ ip m). We say that a colour, stimulus x possesses a uni- theory of tolerance spaces. W W ... Given a tolerance s, define an equivalence relation, s, tary hue, if (x) ¼ (hi), a binary hue if (x) ¼ (hi1, hi2), , W ... S; ;# on S as a p-nary hue if (x) ¼ (hi1, ,hip). The triplet h H i will be referred to as the hue structure. If there is x 2 S with p- [ W21 ... 8x; y; z 2 S x s y , ðÞxsz , ysz : (2) nary hue, and for every q p (hi1, ,hiq) ¼ Ø, we say that p is the degree of chromaticity of hS; H;#i. A hue structure hS; H;#i will be called simple if for each compo- It proves that x s y , s(x) ¼ s(y), where sðÞ¼x 21 S s nent hue hi W (hi) = Ø. A hue structure hS; H;#i will be fgy 2 jx y . The equivalence s determines a partition called complete if the function W is surjective. of S into equivalence classes.

2 research and application When s is a partial hue-matching, s(x) will be referred function t: S ! 2K, which will assign to each x 2 S the to as the hue match for x. Therefore, for a partial hue- subset of K comprising all the s-classes containing x.As matching s, s takes place between two colour stimuli any x 2 S belongs to at least one s-class, t(x) = Ø for only if they have the same hue match. It follows from (1) every x in S. Proposition 2.1 implies that for any x and y that stimuli have the same hue match when they are in S xsy ,t (x) \ t (y) = Ø. We will refer to this fact assigned exactly same component hues. For this reason saying that the set of s -classes K induces the partial col- when s is a partial hue-matching, the equivalence s will our matching s. be referred to as hue equivalence. It determines a partition As shown elsewhere,16 given a tolerance s, there may of S into equivalence classes (referred to as hue equiva- exist proper subsets of K, which also induce s (e.g., see lence classes). These prove to be nonempty preimagesy of Example 1 below). W the hue function . Definition 6 Let hiS; s be a partial colour matching It is clear that if a partial colour matching is induced structure. A set K of s-classes will be called a basis if it by a hue structure, then the number of hue equivalence meets the following two conditions. First, for any x; y 2 S classes is finite. Therefore, one can exclude from consid- that partially match each other (i.e., x s y), there exists a s eration those tolerances , which result in an infinite num- s-class K 2Kcontaining x and y. Second, K is the mini- ber of s-equivalence classes. mal set with this property, that is, removing any K in K Definition 3 Let S be the colour stimulus set. We say will result in that for some x and y there will be no that a reflexive and symmetric binary relation s on S is s-class in K which both x and y belong to. S; s partial colour matching and a pair hiis a partial col- Next proposition is an immediate consequence of the S= our matching structure if a quotient set s is finite, definition of basis. where s is defined in (2). Proposition 2 Let hiS; s be a partial colour matching s S; s As is a tolerance and hiis a tolerance space, we structure, and K is its basis. Define a function t: S ! 2K can define tolerance classes and some other standard (for assigning to each x S the subset of comprising all 16 2 K tolerance spaces) notions. the s-classes containing x. Then, Definition 4 Given a partial colour matching structure 8x; y 2 S xsy , tðÞ\x tðÞy 6¼ [: (3) hiS; s , a set P S is called a s-preclass in hiS; s if each pair of colour stimuli x and y in P partially match each It turns out that there is only one basis in a partial hue- other, i.e., x s y. matching structure associated with a simple hue structure. As shown elsewhere,16 given a tolerance s, for any x and Definition 7 In a hue structure hS; H;#i, a nonempty y to be in relation s there must be a s-preclass containing x subset of stimuli each of which is assigned a hue containing and y. Therefore, to be in the same s-preclass is the neces- a particular component hue will be called the component sary and sufficient condition to partially match each other. hue class. More specifically, given a component hue h 2H, the stimulus subset H defined as Definition 5 Let hiS; s be a partial colour matching i i structure, a s-preclass P is called a s-class in hiS; s if 1 s S; s Hi :¼ [ # ðÞH (4) there is no other -preclass Q in hisuch that P Q. H H¼fgH22 jhi2H So, s-classes are maximal s-preclasses in hiS; s .It means that they cannot be extended without ceasing to be will be called the component hue class providing it is not s s -preclasses. More specifically, given a -class K, for each empty. [ x 62 K there is y K that does not partially match x (i.e., All colour stimuli in a component hue class, H , par- s s s i y x). Every -preclass is shown to be in at least one - tially match each other in hue because they share the class, and every element in S belongs to some s-class.16 component hue h . The set of component hue classes in a It follows that for any x; y 2 S the existence of a s-class i simple hue structure proves to be a minimal set of s- containing both x and y is necessary and sufficient for x classes inducing the hue partial matching s. and y to partially match each other. This is summarised in the following proposition.16 Proposition 3 Let hS; H;#i be a simple hue structure, where h ; ...; h . Then there exist m component Proposition 1 Let S be the colour stimulus set, and H¼fg1 m hue classes H ,...,H that make a unique basis in the hiS; s a partial colour matching structure on it. Two colour 1 m partial hue-matching structure hiS; s associated with stimuli x; y 2 S are partially match each other, i.e., x s y,if S; ;# . and only if there exists a s-class containing both x and y. h H i Proof As S; ;# is simple, for each i W21 (h ) = It follows from Proposition 1 that given a partial colour h H i i Ø, thus Hi = Ø. Each Hi is a s-class. Indeed, consider a matching structure hiS; s , the partial colour matching s 2 2 s-class K containing W 1(h ). Each element in W 1(h ) must can be defined in terms of s-classes as in Definition 1. i i partially match in hue any element in K. It follows that K Specifically, let K be a set of all s-classes. Let us define a ( Hi.AsK is a s-class, we conclude that K ¼ Hi. Let us H ... s yThe preimage, W21(h), of the hue function W at hue h comprises all show now that ¼ {H1, ,Hm} is a basis. If x y then the colour stimuli x such that W(x) ¼ h. there is a component hue hi [W(x) \W(y), thus x, y [

Volume 00, Number 0, Month 2011 3 Hi. Conversely, if x, y [ Hi then hi [W(x) \W(y), thus x Note that, in general, the hue structure hiS; H; t associ- s y. Let K is another basis in hS; H;#i. There must be a ated with the m-chromatic partial colour matching structure 21 class K in K which contain W (hi) . It can be only Hi. hiS; s might differ from the hue structure hS; H;#i that Therefore, K includes H. However, the latter is a basis induces hiS; s . However, they coincide (up to permutation itself. Hence, K coincides with H. of the component hues) when hS; H;#i is simple. There- So, the set of component hue classes in a simple hue struc- fore, a simple hue structure and a m-chromatic partial col- ture is the only basis of the partial hue-matching structure our matching structure associated with it are just the two associated with this hue structure. The component hue classes sides of the same coin. I shall use ‘‘m-chromatic structure’’ completely characterise the simple hue structure. All the other as a common term for both. As the chromaticity classes s-classes in it (as in the example below) can be ignored. play the key role, I shall use the notation hiS; H1; ...; Hm Example 1 Let hS; H¼fh ; h ; h g;#i be a simple for a m-chromatic structure with the chromaticity classes 1 2 3 ... complete hue structure. Then, the colour stimulus set H1, ,Hm. Proposition 4 justifies the following definition. È ÀÈ ÉÁ ÀÈ ÉÁ ÀÈ ÉÁ #1 ; #1 ; #1 ; Definition 9 Let hiS; H ; ...; H be a m-chromatic h1 h2 [ h1 h3 [ h2 ÀÈh3 ÉÁ 1 m 1 structure. Consider two complementary subsets of chro- [ # h1; h2; h3 g maticity classes: ,..., and ,..., . A subset Hk1 Hkp Hkpþ1 Hkm À Á È is a s-class but it is not a component hue class. ; ...; S ... C Hk1 Hkp ¼ x 2 x 2 Hk1 \ \ Hkp The following proposition shows how the hue equiva- É and x 62 H for q ¼ p þ 1; ...; m lence classes of a hue structure can be expressed in terms kq of its hue component classes. will be called a chromatic class in S unless it is empty. A Proposition 4 Let hS; H;#i be a hue structure, and chromatic class C(Hi) will be called unitary (of the degree {H ,...,H } the set of its hue component classes. A non- 1 m of chromaticity 1), a chromatic class C(Hi , Hi ) binary empty subset C S is a component hue equivalence 1 2 (of the degree of chromaticity 2), a chromatic class C(Hi , class if and only if there are hue component classes 1 Hi , Hi ) tertiary (of the degree of chromaticity 3), and so ... S 2 3 Hk1, ,Hkp such that C consists of all such x in that x on. If for some p m there is a chromatic class with the belong to these and only to these hue component classes degree of chromaticity p, but there is no chromatic class ... (written C(Hk1, ,Hkp)). with the degree of chromaticity more than p, then we say It follows that S is partitioned into subsets that the degree of chromaticity for hiS; H1; ...; Hm is p. C(H ,...,H ) each of which is assigned the same set of k1 kp It follows from Proposition 4 that, given a m-chromatic component hues. structure hiS; H1; ...; Hm , its chromatic classes are the As the existence of a unique basis is an important prop- hue equivalence classes of the hue structure erty of the partial hue-matching structure associated with hiS; ðH1; ...; HmÞ; t associated with hiS; H1; ...; Hm . a hue structure, we will distinguish partial colour match- Definition 10 Let hiS; H1; ...; H be a m-chromatic ing structures with this property as a special type. m structure. A chromaticity class Hi will be called singular S; s Definition 8 A partial colour matching structure hi if Hi ¼ C(Hi). If there is only one singular chromaticity will be called m-chromatic if there is a unique finite basis class it will be called neutral. H ... s ... ¼ {H1, ,Hm} in it. The -classes H1, ,Hm will be Now assume that we observe a subject in experiment on called chromaticity classes. partial colour matching. Let us also assume that the observer A simple hue structure with m component hues responses are based on an unknown hue structure hS; H;#i, uniquely determines a m-chromatic partial colour match- and we want to identify this structure. The experiment ing structure. As the component hue classes make a basis, allows for evaluating the partial colour matching structure S; s S; ;# they constitute the chromaticity classes in this partial hiassociated wiht h H i. If there exists a unique ba- H ... S; s ... colour matching structure. Vice versa, a m-chromatic sis ¼ {H1, ,Hm}inhiits elements H1, ,Hm will be chromaticity classes, and the hue structure S; H; t partial colour matching structure determines a simple hue hi associated with hiS; s can be determined. If for each chro- structure with m component hues. Indeed, let hiS; s be a maticity class there is a unitary chromatic class the underly- partial colour matching structure, and H ¼ {H1,...,Hm} H ing hue structure hS; H;#i amounts to hiS; H; t , the basis its unique basis. Define a function t: S ! 2 assigning H comprising its component hue classes. to each x 2 S all the chromatic classes containing x. When at least for one chromaticity class, H , a subset Formally, hiS; H; t is a hue structure, the chromaticity i C(H ) is empty (i.e., there is no unique hue corresponding classes {H ,...,H } playing a role of component hues. i 1 m to this chromaticity class), one has to conclude that the As for each x; y 2 S x s y , t (x) \ t (y) = Ø, { underlying hue structure is not simple. In this case, one hiS; s is a partial hue-matching structure associated cannot suggest that the recovered in experiment hue struc- with hiS; H; t . Likewise, let us call hiS; H; t the hue { structure associated with the m-chromatic partial colour Or the observer responses did not rest upon any chromatic structure matching structure hiS; s . at all. This case will be dealt with in Discussion.

4 COLOR research and application 12}, C(H3) ¼ {5G5/10} and C(H4) ¼ {5R4/14}. Notably, these were exactly the papers that the observer verbally categorised as unique hues (after the experiment). There- fore, C(H1) represents the papers that share the compo- nent hue, which was verbalised as ‘‘yellow,’’ C(H2) ‘‘blue,’’ C(H3) ‘‘green’’ and C(H4) ‘‘red,’’ in an agreement with the Munsell notation. The rest of four chromatic classes contain papers with binary hues: C(H1, H3)—yel- low-green, C(H1, H4)—yellow-red, C(H2, H3)—blue-green and C(H2, H4)—blue-red. The chromaticity classes C(H1) and C(H2) do not overlap. Nor do the chromaticity classes C(H3) and C(H4). This is in line with an idea, enunciated by Hering, that blue and yellow, and red and green hues make perceptually op- ponent pairs. Generally, this observer yielded results, which were in accord with the classical view on human FIG. 1. Averaged and rounded matrix of observer 1 responses. White checks denote those pairs of Munsell papers trichromatic colour vision (e.g., Refs. 13 and 17). which were judged as having common hue components. The second observer brought about different results (Fig. 3). As can be seen in Fig. 4, there are six chromatic- ity classes: H ¼ {10YR7/14 to, 10GY6/12}, H ¼ ture hiS; H; t is that used by the observer. However, one 1 2 {10G5/10 to 5PB5/12}, H ¼ {10Y8.5/12 to 5B5/10}, H can assert that H is the minimal set of component hues 3 4 ¼ {5P4/12 to 10R5/16}, H ¼ {5PB5/12 to 10RP5/14} resulting in such a partial colour matching that was 5 and H ¼ {10R5/16 to 10YR7/14}. Six unitary chromatic obtained in experiment. 6 classes have been found: C(H1) ¼ {5Y8.5/14}, C(H2) ¼ {10B5/12}, C(H3) ¼ {5G5/10}, C(H4) ¼ {5R4/14}, EXPERIMENT C(H5) ¼ {10PB4/12} and C(H6) ¼ {5YR6/14}. There- fore, the first four chromaticity classes, H1 to H4, seem to An experiment on partial hue-matching was conducted be ‘‘classical’’ yellow, blue, green and red respectively. with three inexperienced trichromatic human observers.§ The component hue common to the papers in the fifth They were presented with a series of 20 Munsell papers chromaticity class, H5, was verbally named by the ob- (in the Munsell notation: 5BG6/10, 10BG5/10, 5B5/10, server afterwards as purple, and in the sixth, H6, as or- 10B5/12, 5PB5/12, 10PB4/12, 5P4/12, 10P4/12, 5RP5/12, ange. 10RP5/14, 5R4/14, 10R5/16, 5YR6/14, 10YR7/14, 5Y8.5/ The third observer (Fig. 5) produced a pattern of 14, 10Y8.5/12, 5GY7/12, 10GY6/12, 5G5/10 and 10G5/ results, which is rather different. Such a matrix of 10) with the task to pick out those which hue has some- thing in common with that of a given Munsell paper (test). Each of the 20 papers was used as a test for five times. An averaged, symmetrized and rounded matrix of results for one observer is presented in Fig. 1. White entries mark those pairs that partially matched each other. Further details of experimental procedures and analysis of results can be found elsewhere12 (Logvinenko and Beattie, submitted for publication). Four chromaticity classes have been found: H1 ¼ {10R5/16 to 10GY6/12}, H2 ¼ {10G5/10 to 10RP5/14}, H3 ¼ {10Y8.5/12 to 5B5/10} and H4 ¼ {5PB5/12 to 10YR7/14} (Fig. 2) by using a special computer program. The stimulus set proves to be partitioned into eight chro- matic classes—four unitary and four binary. Each of the four unitary chromatic classes happens to contain just one Munsell paper: C(H1) ¼ {5Y8.5/14}, C(H2) ¼ {10B5/

§A series of experiments with more subjects have been carried out using the partial hue-matching technique12 (Logvinenko and Beattie, sub- mitted for publication). Presented here are three prototypical patterns of results observed. Most of trichromatic observers yielded the results simi- lar to observer 1. Although rather rare, the data produced by observers 2 and 3 are of great theoretical importance, showing possible limitations of FIG. 2. Chromaticity classes (marked with arcs) as the method. derived from the response matrix in Fig. 1.

Volume 00, Number 0, Month 2011 5 FIG. 3. Averaged and rounded matrix of observer 2 FIG. 5. Averaged and rounded matrix of observer 3 responses. responses. responses could be expected if the observer substituted (unique) hues. If the two sets coincide (i.e., each compo- the task of hue-matching with that of similarity judg- nent hue is at the same time a unitary hue), then one can ments. Indeed, let us assume that the observer yielded the assert that the derived component hues are the component yes-response when the similarity between a pair of Mun- hues really used by observer in experiment. sell papers exceeded some criterion value. In this case, When the number of the derived unitary hues is fewer than the response matrix would be a narrow white band around that of the derived component hues, there is a possibility that the main diagonal, similar to what is observed in Fig. 5. observer might have used more component hues. However, these additional hues did not affect the observer responses. Removing these does not change the response matrix, thus DISCUSSION the partial hue-matching relation. For example, using pink as a component hue along with red and white component hues The partial hue-matching technique implies the observer results in the same partial hue-matching relation as that ability (i) to decompose any hue into a number of compo- induced by using only red and white. Therefore, the deriva- nent hues and (ii) to decide if there is a common compo- tion of component hues from the response matrix accom- nent hue for a pair of hues. When this is the case the par- plishes also removing redundant component hues (i.e., those tial hue-matching technique permits to derive the minimal which do not affect the observer response matrix). set of component hues compatible with the results of the The results of this experiment show that in spite of the experiment on partial hue-matching and the set of unitary instruction to make a judgement on the basis of hue content, not all observers could base their judgements solely on component hues. Indeed, an important part of instruction was that the strength of the component hue should not mat- ter. Based on hue content, slightly yellowish red and slightly reddish blue should partially match each other although the dissimilarity between these two colours is quite large. However observer 3 had a strong tendency to judge such pairs (i.e., with a large dissimilarity) as ‘‘not partially matched’’. In other words, this observer was prone to replacing the task of hue judgement with one of colour similarity judgement. This might happen when observers are not experienced enough in evaluating the chromatic content of their colour sensations. Indeed, at the first glance, it is hard to spot that, for instance, Munsell papers 5PB and 10YR have a common component hue (so small is the common shade of red in these Munsell papers). After some training, how- ever, most trichromatic observers are rather confident in judging these Munsell papers as a partial match. Still, an objective criterion to test whether observers rest their judgements upon hue content is needed. FIG. 4. Chromaticity classes as derived from the Had one have such a criteria it could be applied not response matrix in Fig. 3. only to the results of observer 3 but observer 2 as well.

6 COLOR research and application In a sense the observer 2 results are even more controver- chromatic structures, and when six component hues give sial. Although it is more or less clear that observer 3 rise to the same hue structure. A distinction between these failed to follow to make a judgement of component hues, two cases can be produced only within a broader theoreti- observer 2 produced a matrix that can be interpreted as a cal context. graph of the partial colour matching relation s with a I believe that the colours within each component hue unique basis consisting of six chromaticity classes. More- class should lend themselves to ordering with respect to over, for each chromaticity class, there is a unitary chro- the strength of the component hue constituting the class matic class. Strictly speaking, one has to conclude that s (referred to as chromatic order). In fact, a chromatic is induced by a simple chromatic structure with six order is implicitly assumed by widely used hue scaling component hues. techniques which intend to describe colour in terms of Yet, there is an alternative interpretation. One can sug- the relative amount of component hues contained in this gest that observer 2 rested his judgements not upon com- colour.19 The preliminary study showed that observers ponent hues but upon colour categories. In other words, were capable to sort out the Munsell papers within the observer 2 could have answered ‘‘yes’’ when the papers chromaticity classes with respect to the strength of the belonged to the same category. In this case, the results component hue constituting the class.12 The results are reveal six colour categories rather than six unique hues. consistent with the existence of chromatic order with More specifically, assume that there are m colour cate- the properties of weak order. It follows that the features gories. Formally, it means that there is a cover C of S, that observers used in partial hue-matching are one- that is, a family of subsets {C1,...,Cm}ofS whose union dimensional variables (continua), most likely, compo- is the whole set S. Each Ci is a collection of colour nent hues. If an observer failed to weakly order the stimuli, which belong to the i-th colour category. Let us elements of chromaticity classes, it would testify for define a partial colour matching s as follows. For every that the partial colour matching revealed in experiment x; y 2 S assume that x s y if and only if there is such a is based on colour categories rather than component colour category Ci that x; y 2 C. The partial colour match- hues. Indeed, a colour category based on a binary hue ing s will be referred to as that generated by the colour (e.g., orange or violet) can hardly be weakly ordered categories C. because it is not clear what the strength of a binary hue It must be said that, generally, there is no simple rela- might mean. Thus, a necessary condition for a chroma- tionship between the colour categories C and the s-classes ticity class to represent a component hue is, formally of the generated tolerance s. For an arbitrary set of colour speaking, the existence of a chromatic order on it. categories a colour category may happen even not to be a The definitive features of a weak order are its totality s-class of the generated tolerance. Yet, it turns out that if and transitivity.20 Totality of the chromatic order means the generated tolerance s induces a m-chromatic structure, that any two colours x and y are chromatically compara- then each chromaticity class contains some colour cate- ble (i.e., either x has the component-hue strength not gory (Ref. 16 Theorem 3.7). However, the colour catego- weaker than y or y not weaker than x). Transitivity means ries can not be restored from the s-classes of the gener- that for any colour stimuli x,y, and z,ifx has the compo- ated tolerance. nent-hue strength not weaker than y, and y not weaker Still, the set of chromaticity classes derived from the than z, then the component-hue strength of x is not response matrix can be treated as a set of colour catego- weaker than that of z. Although totality does not require ries compatible with the response matrix (i.e., generating special testing (observers never refuse to compare any the matrix). If they make a basis then it is a minimal set colours), a preliminary experiment showed that transitivity of colour categories compatible with the response matrix. failures occurred from trial to trial. Hence, a proper statis- If this basis is unique then it is a unique minimal set of tical test should be conducted to decide whether these colour categories compatible with the response matrix. In failures are inconsistent with the transitivity hypothesis. this latter case, one can say that irrespective of the colour For this purpose, one can use a special test developed on categories that the observer really used in experiment purpose to test transitivity statistically.21 (they can be rather redundant), the derived set of colour It follows from the fact that colours in a chromaticity categories can be used to model the observer responses. class can be ordered that a chromaticity class can be inter- An analogous situation takes place with applying the sig- preted as a fuzzy subset.22 Moreover, a set of fuzzy chro- nal detection theory to psychophysics.18 I do not believe maticity classes defines a fuzzy tolerance on the set of col- that observers really evaluate the likelihood ratio. Yet, I ours. Therefore, one can use the existing algorithms of assume that they behave as if they do so. deriving fuzzy tolerance classes23 to evaluate fuzzy chro- An important question then arises. How can one differ- maticity classes. The advantage of this might be that entiate between the chromaticity classes representing observers will not have to make a (crisp) decision about component hues and those representing colour categories? belongingness of colours to the same chromaticity class. For instance, observer 2 provides a particular case when Instead, they will have to decide for which of two pairs of the set of colour categories and the basis of generated tol- colour the likelihood of the belongingness to the same chro- erance coincide. There seems to be no difference between maticity class is higher. Preliminary observations show that the case when six colour categories give rise to a six- this task is easier and more natural for observers.

Volume 00, Number 0, Month 2011 7 CONCLUSION 7. Van Brakel J, Saunders BAC. On the existence of a fixed number of unique opponent hues. In: Dickinson CM, Murray IJ, Carden D, edi- It is generally believed that all hues are composed of tors. John Dalton’s Colour Vision Legacy. London: Taylor and Fran- some finite number of component hues. However, because cis; 1997, p 399–402. of the lack of an operational definition of component hue, 8. Kuehni RG. Variability in unique hue selection: A surprising phe- nomenon. Color Res Appl 2004;29:158–162. there is no consensus on the nomenclature of the compo- 9. Kuehni RG, Schwarz A. Color Ordered: A Survey of Color Order nent hues. For example, Brentano (Ref. 24, p 352) and Systems from Antiquity to the Present. New York: Oxford Univer- Holt (Ref. 25, p 175–176) believed that green was com- sity Press; 2008. posed of blue and yellow, which is at odds with current 10. Hard A, Sivik L. NCS—: A Swedish standard views. This controversy can be resolved by using the par- for color notation. Color Res Appl 1981;6:129–138. 11. Munsell AH. Atlas of the Munsell Color System. Malden, MA: tial hue-matching technique, which implies the observers’ Wadsworth-Holland; 1915. ability to judge whether two colours share a common hue. 12. Logvinenko AD, Beattie LL. Partial colour matching: A new method It is proved that if for each component hue there exists a to evaluate colour appearance and derive unique hues. In: ‘‘CGIV – hue consisting of only this component hue, then the com- Third European Conference on Colour in Graphics, Imaging, and ponent hues can be uniquely derived from the observer Vision, Leeds, UK’’. The Society for Imaging Science and Technol- ogy (CD-ROM), 2006. p 262–265. response matrix. Furthermore, it is shown that the same 13. Hurvich LM, Jameson D. An opponent-process theory of color technique can be used to determine colour categories. vision. Psychol Rev 1957;64:384–404. More specifically, if observers indicate whether two col- 14. Zeeman EC. The Topology of the brain and visual perception. In: ours belong to at least one colour category, then one can Fort MK, editor. The topology of 3-Manifolds, Englewood Cliffs, NJ: Prentice Hall, 1962. p 240–256. derive the set of minimal colour categories that are com- 15. Shreider YA. Tolerance spaces. Kibernetika 1970;6:124–128. patible with the observer matrix response. 16. Schreider JA. Equality Resemblance and Order. Moscow: Mir; 1975. 17. De Valois RL, De Valois KK. A multi-stage color model. Vision Res 1993;33:1053–1065. 18. Green DM, Swets JA. Signal Detection Theory and Psychophysics. New York: Wiley; 1966. 1. Valberg A. Unique hues: An old problem for a new generation. 19. Abramov I, Gordon J. Seeing unique hues. J Opt Soc Am A Vision Res 2001;41:1645–1657. 1994;22:2143–2153. 2. Indow T. Psychologically unique hues in aperture and surface . 20. Krantz DH, Luce RD, Suppes P, Tversky A, editors. Foundations of Die Farbe 1987;34:253–260. Measurement, Vol. 1. Additive and Polynomial Representations. 3. Hering E. Outlines of a Theory of the Light Sense. (Hurvich LM, New York and London: Academic Press; 1971. Jameson D, Trans.). Cambridge, MA: Harvard University Press; 21. Logvinenko AD, Byth W, Vityaev EE. In search of an elusive hard 1964 (Original work published 1920). threshold: A test of observer’s ability to order sub-threshold stimuli. 4. Wyszecki G, Stiles WS. Color Science: Concepts and Methods, Vision Res 2004;44:287–296. Quantitative Data and Formulae, 2nd edition. New York: Wiley; 22. Zadeh LA. Fuzzy sets. Inf Control 1965;8:338–353. 1982. P 487. 23. Das M, Chakraborty MK, Ghoshal TK. Fuzzy tolerance relation, 5. Quinn PC, Rosano J, Wooten B. Evidence that brown is not an ele- fuzzy tolerance space and basis. Fuzzy Sets Syst 1998;97:361–369. mental color. Percept Psychophys 1988;43:156–164. 24. Nida-Rumelin M, Suarez J. Reddish green: A challenge for modal 6. Mollon JD, Jordan G. On the nature of unique hues. In: Dickinson claims about phenomenal structure. Philos Phenomenol Res CM, Murray IJ, Carden D, editors, John Dalton’s Colour Vision 2009;LXXVIII:346–391. Legacy, London: Taylor and Francis; 1997. p 381–392. 25. Boring EG. Mind and mechanism. Am J Physiol 1946;59:173–192.

8 COLOR research and application