5 pagesInternational audienceThe colour, with the particularity to be defined simultaneously as a physical quantity and as a psychophysical quantity, is one of the concepts that can link hard sciences and behavioural sciences. From the viewpoint of behavioural sciences colours are basically measured with nominal scales, and in hard science colours are measured with interval scales. Our hypothesis is that the main relation that must be preserved during colour measurement is a metric. We suggest then that colours must be measured with metrical scales. The fuzzy metrical scale is preferred due to the possibility to define it like a nominal scale

Benoit, Eric

Hal - Université Grenoble Alpes

Measurement scale for colour perception

HAL Université de Savoie

Measurement scale for colour perceptionEric BenoitTo cite this version:Eric Benoit. Measurement scale for colour perception. XX IMEKO World Congress, Sep 2012,Busan, South Korea. pp.TC7-O-10, 2012. <hal-00763629>HAL Id: hal-00763629https://hal.archives-ouvertes.fr/hal-00763629Submitted on 11 Dec 2012HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Abstract – The colour, with the particularity to bedefined simultaneously as a physical quantity and as apsychophysical quantity, is one of the concepts that can linkhard sciences and behavioural sciences. From the viewpointof behavioural sciences colours are basically measured withnominal scales, and in hard science colours are measuredwith interval scales. Our hypothesis is that the main relationthat must be preserved during colour measurement is ametric. We suggest then that colours must be measured withmetrical scales. The fuzzy metrical scale is preferred due tothe possibility to define it like a nominal scale.Keywords: Measurement theory, Colour, Scale, Dis-tance, Fuzzy subsets theory.1. INTRODUCTIONSome quantities like colour, odour, software complexityare usually measured with inappropriate scales. Indeed, thetheories chosen to abstract such quantities usually define anaffine space to represent measurement values even if thischoice is not justified. For example, colours are representedin many different spaces like RGB, xyz, Luv, Lab, HSV andthe transformation from one to each other is not always anaffine transformation. We conclude from this situation thatthe empirical space of colours doesn’t hold an affinestructure and then cannot be represented by an affine space.Conversely, the metric, defined with psychophysicexperiments stays stable and is the most known relation oncolours. The basis hypothesis of this paper is that theempirical space of some quantities manifestations, moreparticularly the colour, can be represented by a non-affineabstract space that holds a metric.The determination of such metric depends on the theorythat is used to perform calculus reasoning and decision on anabstract world where quantities are represented by their wellknown quantity value. Let describe the full process [1]. Firstthe field of interest, i.e. the concrete world is identified. Thenconcrete objects and their properties are selected. A theory,made of entities, axioms and theorems, is chosen.Experiments are then performed to obtain observations ofquantity manifestations. The representations of themanifestations are named quantity values [VIM] and areexpressed into a space which structure depends on the theorychosen to infer conclusions. The choice of the theory iscrucial and depends on the goal of the experiment. In thecolour field, the goal can be to use the colour to identify thechemical components of a liquid. In this case the theory isdefined on the field of molecular physics and colourmanifestations are represented by spectral energydistributions. Each spectra are expressed into a n-dimensional vector space. If the goal is to check the qualityof a manufactured colour, then the experiment is based on atheory of colour vision and colours are expressed into ametric space. 2. COLOUR VISION REPRESENTATIONThis paper, will focus on psychophysical aspects ofcolours. This means that colour quantities are not consideredexclusively into the context of physics but also into thecontext of human perception. From a pure physicsconsideration, a colour of an electromagnetic flow is definedby its spectral power distribution (SPD). As for anydistribution, a general definition is never obtained due to thenecessity to define the spectral resolution. Indeed thequantity that represent a colour is a vector such lengthdepends on the chosen resolution, and the chosen range ofthe spectra. We can see that even with a given theory, thegoal of the measured quantity has a strong incidence on itsrepresentation. As an example, the InternationalCommission on Illumination (CIE) specified that for colourmeasurement of visible light the spectra range is from 360nmto 830nm with 1nm resolution. This institution gave also afirst approximation of human colour perception with thedefinition of the tristimulus values XYZ. The X, Y and Zvalues are obtained with 3 colorimetric observers x(O), y(O)and z(O) (see (1) and Fig. 1.) that approximate the spectralsensitivity of human photosensors [2]. (1)Where k is a constant, and  I(O) is the acquired SPD.X k I O( ) x O( )  Od380nm780nm³=XX IMEKO World Congress Metrology for Green GrowthSeptember 9-14, 2012, Busan, Republic of Korea MEASUREMENT SCALE FOR COLOUR PERCEPTIONEric BenoitLISTIC-Université de Savoie, Annecy, France, eric.benoit@univ-savoie.frFig. 1. colour-matching functions x(O), y(O), z(O)This representation of colour into a 3D space is justifiedby the fact that 2 colours represented with the same XYZtristimuli are considered as equal by any human. Its then easyto deduce that this colour measurement is based on a nominalscale, i.e. on a scale that links an equivalence relation onempirical quantities with an equality on measured values. The colorimetric spaces are derived from the XYZtristimuli by the use of bijective transformations. Thisconfirms that colour measurement scales are equivalentwhen considered as nominal scales: bijections are admissibletransformations for nominal scales. For a given colourclassification, colour histogram methods do not depend onthe colorimetric space and they use the classification as anominal scale. Most other methods need a metric on thecolorimetric space or at least a similarity relation betweencolours [3]. In the first case, a metrical scale, i.e. a scale thatpreserves a metric, is needed. With the main property ofpreservation of a similarity relation, a fuzzy nominal scale isa good candidate for the second case [4][5]. Some methodsuse colorimetric spaces as affine spaces [6]. These lastapproaches are questionable when the different colorimetricspaces must be considered as equivalent representationspaces of the same quantity. From formal point of view, a method might not dependson the choice of a representation space that actually dependson the choice of scale. In this paper, we promote thehypothesis that the number of different colorimetric spacesproposed shows that the representation space of a scale forcolour measurement may have a metric but not necessary bean affine space. The preservation of distance is then thegeneric relation of such scale known as metrical scales. Theconsequence is that all signal processing need to be definedon the basis of a distance.3. COLOUR REPRESENTATION BY LEXICAL FUZZY SUBSETSFuzzy nominal scales where introduced in order toformalize an application to the measurement process of amechanism of description of a quantity by a fuzzy subset ofsymbols [7]. With these scales, values in the representationspace are fuzzy subsets of symbols also called in this paperlexical fuzzy subsets (LFS). The measurement is split into ameasurement from the set of manifestations to a numericalspace X, then a mapping D called fuzzy description or simplydescription translates a numerical scalar or vector into alexical fuzzy subset. In the following example, amanifestation is represented by a scalar itself described by aLFS defined by its membership function P on a lexical set S= {a,b,c,d}.Fig. 2. Example of a fuzzy description mapping a scalar into a lexical fuzzy subset (LFS).In this paper, we restrict our study to the fuzzy nominalscales that respect:(2)Such scales define a fuzzy equivalence relation betweenLFSs like for example the simplest one (3). (3)This relation also known as similarity relation is arepresentation of a relation between manifestations. Itrespects the reflexivity condition, expressed by (2), and aweak version of the transitivity condition. In our case P~ isTL-transitive:(4)where(5)Using fuzzy scales for colour measurement is justified bythe fact that the existence of a similarity relation betweencolours even if such relation is not clearly known. On thebasis of a lexical set S = {red, blue, yellow, green, ...} a fuzzynominal scale defines the meaning of each symbol with afuzzy subset of manifestations, actually a fuzzy subset on acolorimetric space. The first step to define such scale is to define the lexicalset. A usual set will be S = {green, yellow, red, purple, blue,cyan, black, white}, the 8 colours of the RGB cube and of itsaffine transformations. In the paper we restrict to a chromaticplane then white, black and intermediate colours arerepresented with the same value. We choose the symbol012400 500 600 700 800O:wavelength (nm)xyza b c d017.625DPD(7.625)scalar lexical fuzzy subsetA D X  PA s( )s S¦ 1=A B D X  P ~ A B( )  min PA s( ) PB s( ) s S¦=TL P~ x y( ) P~ y z( )  P~ x z( )dTL x y  max 0 x y 1–+ =neutral for this value. We propose also to use differentsymbols to represent real colours and colours that define theextremes of the chromatic plane. Finally we add the colourorange to the set in order to have a lexical set morerepresentative of the human feeling. A possible lexical set isS = {full_green, full_orange, full_yellow, full_red,full_purple, full_blue, full_cyan, neutral, green, orange,yellow, red, purple, blue, cyan}.The meaning are define by a piece wise interpolationbased on a triangulation of the chromatic plane. First eachsymbol is associated with a chromatic coordinate that ischaracteristic to the symbol. Then the plane is split intotriangles such that vertices are characteristic coordinates.The meaning of a symbol is a fuzzy subset whichmembership function is equal to 1 for the characteristiccoordinate of the symbol and equal to 0 on the othercharacteristic coordinates, and interpolated on the triangles.The next figure shows the triangulation used to define themeaning of the lexical set on the ab chromatic plane. Fig. 3. Triangulation that defines the fuzzy meaning of S = {full_green, full_orange, full_yellow, full_red, full_purple, full_blue, full_cyan, neutral, green, orange, yellow, red, purple, blue, cyan}. Letters a,b,c,.. replace full_green, full_orange, full_yellow, ...Building a metrical scale from a fuzzy scale needs todefine a distance d on the lexical set and to define a distanced’ between LFSs that verifies:. • the singleton coincidence: d’({a},{b}) = d(a,b)• the continuity property.• the precision property that imposes that the distancebetween two LFSs must be a positive real number.• the consistency property that is usually verified by dis-tances on crisp subsetsThe transportation distance, denoted dtp, verifies all theseproperties [9][10]. It is computed as solution for a masstransportation problem [10] where the masses aremembership degrees, sources and destinations are items ofthe lexical set and the unit cost from a source to a destinationis given by the distance d on S. This distance can be definedrelatively to the goal of the measurement process orrelatively to the application. Another solution is to use thetriangulation to compute a distance on the basis of theadjacency of the symbols provided by the graph ofcharacteristic coordinates.4. ADAPTATION OF THE SCALEThe crucial point of the scale definition is the location ofthe characteristic coordinates. The coordinates given in Fig.3. are defined for a general use and are usually not usable forspecific cases. For example, Van Gogh painting usually notfit with this generic scale.Fig. 4. Van Gogh painting as a context for colour measurement100100-100-100abfedcbaghijklmnoFig. 5. Colour histogram of the painting in comparison with the characteristic coordinates of each symbol.The proposal of this paper is to perform a Fuzzy C-Meansclustering (FCM) to adapt the generic knowledge given bythe initial characteristic coordinates. The idea is to fit eachcharacteristic point with the center of its closest cluster. Thedifficulty is that the FCM algorithm is based on theminimization of an objective function based on thecomputation of the Euclidean distance between samples andcluster centers and cannot be directly used. Indeed, as seenbefore, nothing can justify that the colorimetric space, or thespace of LFSs, holds an Euclidean metric. In the original FCM algorithm a set of clusters is firstdefined. Each cluster is randomly defined by a fuzzy subsetof samples. At each iteration, the FCM algorithm computesthe center of each cluster. Then the membership degree ofeach sample to each cluster is re-evaluated relatively to itsproximity to the associated cluster center. In our approach, we propose several adaptations to thisprocess.Let M be a set of samples in X.In the initial state, Each cluster Cs is identified by asymbol s and is defined by the fuzzy subset of X derivedfrom the fuzzy description.(6)where(7)As the fuzzy equivalence relation that characterizes thescale defines a distance for short range LFSs, the eq. (6) canbe simplified into:(8)The main difference with the standard FCM is theinclusion of a basic knowledge at the initial step of thealgorithm. This knowledge can be considered as an averageknowledge about the representation of colours.At each iteration, the cluster center is simply computed asthe gravity center of the cluster. (9)The scale is then transformed such that the center of thecluster Cs become the characteristic point of the symbol s.As for the original algorithm the iterations stop whenchanges reach a termination criterion.The D parameter, must be defined into [0,1]. It representsthe inertia of the learning process. If D = 1, each iteratedcluster includes only its characteristic point as uniquesample. The characteristic points never move during thealgorithm. If D = 0, each iterated cluster can include newsamples far from the characteristic point.The next figure shows the triangulation after theadaptation of the scale with this method. As the coloursfull_green, full_orange, full_yellow, full_red, full_purple,full_blue, full_cyan, neutral are synthetic colours defined bya norm, they are not supposed to be changed during thelearning process. So D = 0 for these colours. Fig. 6. Characteristic coordinates after the adaptation of the scale with the FCM like clustering (D = 1 for a to h, D = 0.5 for i to o).100100-100-100abfedcbaghijklmnoPCsx( ) ED 1 d s^ ` D x  – 2=ED u( ) u if u Dt0 else        ©§=PCsx( ) ED P~ s^ ` D x   2= ED PD x  s( ) 2=csx PCsx( )x M¦ PCsx( )x M¦----------------------------------=100100-100-100abfedcbaghijklmnoDISCUSSIONThe information of colour has the property on one side tobe typically a psychophysic information, on the other side tobe acquired with accurate measuring instrument and then tobe accurately represented on a numerical space. Between thephysical world, from where the colour entities are issued,and the abstract human mental world, where they are definedand represented, is the sensitive world that is a partialperception of the physical world. Colour entities, like theorange colour for example, cannot be considered as concretephysical entities of the concrete physical world. they areissued from a concrete entity, a spectra of a electromagneticflow, and appear into the sensitive world. The difficulty ofcolour measurement is that the human sensitive world, i.e.the human perception of the concrete world, differs from theinstrumental sensitive world, i.e. the instrumentedperception of the world. The consequence is that the abstractworlds used to represent these sensitive worlds also differs.In particular, the structure of the colorimetric spaces of theinstrumental abstract world are richer than the structure ofthe colorimetric spaces of the abstract human mental world. Indeed, each colorimetric space of the instrumentalabstract world holds a metric. Usually, but not necessary, anEuclidean metric. So it’s legitimate to consider that thecolorimetric space in instrumental sensitive world also holdsthis metric. A colorimetric space of the abstract humanmental world is a metrizable space, but the associated metricis not defined. This fits with the general knowledge that adistance between colour exists but cannot be preciselydefined. Within this context, the space of lexical fuzzysubsets gives an alternative to usual numerical colorimetricspaces. Indeed, this space is a metrizable space, and themetric depends on the goal of the colour process and not onthe sensitive world. Furthermore, the distance is based on afuzzy scale that can be adapted to colour process throw alearning algorithm.CONCLUSIONColour measurement does not lead to a unique theory andneeds a scale for each application, or more precisely for eachconcept. We proposed in this paper to use scales thatpreserve a similarity relation or scales preserving a metric.The fuzzy scales, with the expression of measurement valueson a non affine space give a good solution for colourmeasurement. The counter part is the necessity to adapt thescale according to a context or a colour process. This papergave an algorithm to perform such adaptation. Thisadaptation can be compared with a calibration process wherethe calibration standards are colour entities. Finally that thecolorimetric space associated to a fuzzy scale has a structurecloser to the human representation than classicalcolorimetric spaces.REFERENCES[1] L. Finkelstein, “Representation by symbol systems as anextension of the concept of measurement,” Kybernetes, Vol. 4,pp. 215-223, 1975.[2] H. Pauli, "Proposed extension of the CIE recommendation on“Uniform color spaces, color difference equations, and metriccolor terms”," J. Opt. Soc. Am. 66, 866-867 (1976) [3] M. Stricker, M. Orengo, Similarity of color images, Proc. SPIE2420, 381 (1995)[4] E. Benoit, L. Foulloy,  “Towards fuzzy nominal scales”,Measurement, ISSN 0263-2241, Vol. 34, No. 1 (Fundamental ofMeasurement), pp. 49-55, 2003. [5] E. Benoit, L. Foulloy, G. Mauris,  “Fuzzy approaches formeasurement”, Handbook of Measuring Systems Design,ISBN: 0-470-02143-8, Vol. 1, Eds. Peter Sydenham andRichard Thorn, John Wiley & Sons, pp. 60-67, 2005.[6] T.A. Ell, Hypercomplex Color Affine Filters, ICIP 2007, IEEEInternational Conference on Image Processing, 249-252 (2007)[7] Zadeh L.A., Quantitative fuzzy semantics, InformationSciences, Vol. 3, pp. 159-176, 1971.[8] T. Allevard, E. Benoit, L. Foulloy,  “Dynamic gesturerecognition using signal processing based on fuzzy nominalscales”, Measurement, No. 38, pp. 303-312, 2005.[9] T. Allevard, E. Benoit, L. Foulloy,  “The transportation distancefor fuzzy descriptions of measurement”, Metrology andMeasurement Systems, Vol. XIV, No. 1/2007, pp. 25-35, 2007.[10]S.T. Rachev, L. Rüschendorf, Mass transportation problems,Springer-Verlag, New-York, 1998.

https://hal.archives-ouvertes.fr/hal-00763629

Measurement scale for colour perception

Abstract

Similar works

Full text

Available Versions

Hal - Université Grenoble Alpes

HAL Université de Savoie