6 pagesInternational audienceThe aim of this paper is to propose a new method for building a fuzzy partition on multi-dimensional spaces. This method is based on a linear interpolation performed by means of Delaunay's triangulation of the multi-dimensional space. It is applied for creating fuzzy symbolic sensors which use multi-component measurements. Colour sensing is an interesting problem for applying this method. Indeed, humans have good control on this sensing but can not explain it simply. Fuzzy colour sensors can learn human perception of colours whithout explicit knowledge about the perception mechanisms

Benoit, Eric

Foulloy, Laurent

Mauris, Gilles

Hal - Université Grenoble Alpes

A fuzzy colour sensor

HAL Université de Savoie

A fuzzy colour sensorEric Benoit, Gilles Mauris, Laurent FoulloyTo cite this version:Eric Benoit, Gilles Mauris, Laurent Foulloy. A fuzzy colour sensor. 13th IMEKO worldcongress, 1994, Torino, Italy. IMEKO, pp.1015-1020, 1994, September 1994. <hal-00008819>HAL Id: hal-00008819https://hal.archives-ouvertes.fr/hal-00008819Submitted on 16 Sep 2005HAL 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.AbstractThe aim of this paper is to propose a new method for building a fuzzy partition on multi-dimensional spaces. Thismethod is based on a linear interpolation performed by means of Delaunay’s triangulation of the multi-dimensional space.It is applied for creating fuzzy symbolic sensors which use multi-component measurements. Colour sensing is aninteresting problem for applying this method. Indeed, humans have good control on this sensing but can not explain itsimply. Fuzzy colour sensors can learn human perception of colours whithout explicit knowledge about the perceptionmechanisms.Keywords: fuzzy sensor, fuzzy subsets, colour sensor, multisensorIntroductionIntroduced since several years ago, fuzzy symbolic sensors, also called fuzzy sensors, are intelligent sensors whichare able to receive, to produce and to handle fuzzy linguistic information [1], [2], [3]. A fuzzy linguistic information isrepresented by a fuzzy subset of symbols. This particularity allows them to be used as perceptive elements for symboliccontrollers like fuzzy controllers and fuzzy expert systems. Furthermore, this kind of sensor can own the most advancedfunctionalities of intelligent sensors, like auto-adaptation and learning, by using artificial intelligence techniques asproposed by Zingales [4].Most of the fuzzy sensors work on single-component measurements like distance, temperature and so on. Theygenerate fuzzy linguistic information by a process called fuzzy linguistic description, or simply fuzzy description. In orderto create fuzzy sensors working on multi-component measurements like colour or smell, two ways can be considered. Thefirst strategy consists in defining the fuzzy description by means of a rule based formalism that involves a fuzzydescription of every feature returned by the corresponding fuzzy sensor. It has been applied to a linguistic description ofcomfort [5]. The second strategy is based on an interpolation method in order to build fuzzy subsets on multi-dimensionalspaces.After having defined fuzzy sensors, the fuzzy description method based on multi-dimensional fuzzy sets ispresented. In the second section, we recall the principle of colour measurement, and we apply the method to the fuzzydescription of RGB components of colour and to fuzzy description of chrominance. Then a sensor which performs thefuzzy description of a coloured surface is described.1. FUZZY SENSORS1.1 DefinitionFuzzy symbolic sensors are based on the translation of information from a numerical representation to a linguisticone. One issue to provide a numeric-linguistic conversion is to associate a fuzzy set with each symbol in order to obtaina fuzzy measurement of physical quantities. The interest of the fuzzy subset approach is to take into account thecontinuous nature of the variables, inside the linguistic representation that is itself discrete.The fuzzy meaning is a mapping from a linguistic set S into the set of the fuzzy subsets of measurements [6]. Thefuzzy meaning of a symbol a is characterized by its membership function denoted µτ(a)(x). In a same manner, a fuzzydescription can be defined as a mapping from the measurement set into the set of the fuzzy subsets of symbols, so thefuzzy description is characterized by its membership function denoted µι(x)(a). The relation between the membershipfunctions of a fuzzy description and the corresponding fuzzy meaning, given by Eq. (1), states that if a symbol belongs tothe description of a measurement, then the measurement belongs to the meaning of the symbol.µι(x) (a) =  µτ(a) (x) (1)If the set of the fuzzy meanings forms a fuzzy partition of E then we have a fuzzy nominal scale [7], which is anextension to fuzzy subsets of the concept of nominal scale [8]. The meanings associated with a fuzzy nominal scale verifyEq. (2). According to Eq. (1), the fuzzy description of any measurement verifies Eq. (2).:      Proc. of the XIII IMEKO world congress, Torino, Italy, sep. 1994, pp.:      1015--1020.A fuzzy colour sensorE. Benoit, G. Mauris, L. FoulloyLAMII/CESALPLaboratoire d’Automatique et de MicroInformatique IndustrielleUniversité de Savoie, BP 806, 74016 Annecy Cedex, France(2)(3)A simple example for a fuzzy telemeter is provided in Fig. 1. The fuzzy description of the distance x = 12 cm isobtained from the fuzzy meaning of the symbols in S = {Close, Medium, Far}.Fig. 1 Fuzzy description of the measurement x = 12 cm.1.2 Multi-component fuzzy sensorsWhen a sensor uses several transducers, the measurement is a vector of numerical values. Then, the meaning of eachsymbol must be a fuzzy set defined in a multi-dimensional volume. Furthermore the set of meanings has to be a fuzzypartition of the measurements set.In this paper, we consider an initial knowledge about measurements. This knowledge is materialized by the meaningof symbols on a small subset V of the measurements set E. Then we have to extend the meaning of each symbol on all themeasurements set. In order to obtain a fuzzy partition, the measurements set is cut into n-simplexes. A n-simplex in a n-dimensional space is a polyhedra with n+1 vertices. For example, a 2-simplex is a triangle and a 3-simplex is atetrahedron. Then meanings are defined on each n-simplex such that Eq. (2) is verified.First, the measurements set is partitioned into n-simplexes using Delaunay’s triangulation. The points used toperform the triangulation are the elements of the set V. As any triangulation can be used to cut the measurements set, itmust be chosen according to constraints associated to the measurements [9]. The Delaunay triangulation is preferred whenno constraint can be found.The membership function of the meaning of each symbol is defined on each n-simplex by a multi-linearinterpolation. We suppose the restriction on a n-simplex of the membership function of the meaning of a symbol s is:µτ(s)(v) = µτ(s)(x1, x2, ..., xn) = a1x1+a2x2+ ... +anxn+an+1 (4)The value of this function is known for the n+1 vertices of the n-simplex, therefore the n+1 factors ai can becalculated.(5)The ith vertex of the n-simplex is denoted vi, and its jth component is denoted xij.Once this process is performed on each n-simplex for each symbol, we obtain a fuzzy nominal scale defined on E.This scale is an extension on E of the fuzzy nominal scale on V.With this method, the knowledge needed for the configuration of the sensor is very compact. It can be acquired bymeans of a system called a teacher which can be a human or an expert system. During the configuration phase, the teacherand the sensor analyse the same phenomenon, and the teacher gives its description to the sensor. The sensor increases itsknowledge with its measurement associated with the teacher description. Then it can build the fuzzy nominal scale. Itshould be noted that the description given by the teacher must be a fuzzy subset of symbols which verify Eq. (2). If theteacher is a human then its description is generally a crisp set which contains only one symbol.x E∈( )∀ µτ a( ) x( )a S∈∑, 1=µι x( ) a( )a S∈∑ 1=τ(Close) τ(Medium) τ(Far)x (cm)151210µ01Close MediumFarES µ ι(12)01A M 1– B= Mx11... x1n1... ... ... …xn 11+... xn 1+ n1= Aa1…an 1+= Bµτ s( ) v1( )µτ s( ) v2( )...µτ s( ) vn 1+( )=2. FUZZY COLOUR SENSOR2.1 Colour measurementHuman beings perceive electromagnetic light beams by four types of photochemical transducers: three cones for theday vision and one rod for the night vision. The photometric sensors give back information in relation to the receivedenergy. The sensation of colour is generated by the different spectral sensitivities of the cones. The “blue” cones detectsshort wavelength, whereas the “green” and “red” cones detect respectively medium and long wavelengths.The artificial sensing (for example the video camera) is based on three photometric transducers that recreate theeffects of red, green and blue cones. Their respective sensitivity are denoted R(λ), G(λ), B(λ) where λ is the wavelength.The responses of the detectors are given in Eq. (6) where s(λ) denotes the spectrum of the light to be analysed. When theresponses are normalized between 0 and 1, the colour space is simply a unit cube called the RGB cube.(6)When decribing a colour in the common language, the luminosity is usually separated. For example we say “paleblue” or “dark red”. In order to introduce this knowledge into the sensor, a non linear mapping is applied to the RGB cube.This mapping has been chosen in order to obtain a colour information which do not depend on the luminosity. Thisinformation (C’1, C’2) is called the chrominance.(7)Fig. 2 The RGB cube and its transformation.2.2 Fuzzy description of chrominanceA first way to build a fuzzy description of colours is to describe separately the luminance and the chrominance. Themeasurements set of the luminance is monodimensional, and its description is not developed in this paper. Themeasurements set of the chrominance is two-dimensional. It is called the chrominance plane of colours and it is used asthe measurements set. Initially, seven colour are used for the configuration of the sensor:S={Neutral, Red, Magenta, Blue, Cyan, Green, Yellow} (8)ι(0,0) = {Neutral}, ι(-1/2, /2) = {Blue}, ι(1/2, /2) = {Magenta}, ι(-1/2, - /2) = {Green}, ι(1/2, - /2) ={Yellow}, ι(1,0) = {Red}, ι(-1, 0) = {Cyan}. (9)r s λ( )R λ( ) λdλ1λ2∫=v s λ( )V λ( ) λdλ1λ2∫=b s λ( )B λ( ) λdλ1λ2∫=withλ1 400nm=λ2 700nm=  C'1C'21 12--–12--–0 32------–32------rY'----vY'----bY'----= Y' max r v b, ,( )=Y’C’1C’2WhiteMagentaCyanGreenYellowBlueRedBlackBlackWhiteRedGreenBlueCyanYellowMagentaRBG1111-1132------32------–luminancechrominance3 3 3 3Fig. 3 Delaunay’s triangulation of the chrominance plane and the fuzzy meaning of Red.Experiments show that only one or two characteristic colours are needed to perform the configuration of the sensor.The small number of configuration points can be explained by the choice of the mapping used to obtain the chrominanceplane. Indeed, if two colours differ only by their darkness then they have the same coordinates in the chrominance plane.The following figure shows the result of the configuration performed to adapt the sensor to return a description of coloursclose to the human perception in a classification task of coloured objects. We have here four types of coloured objetc: red,yellow, green and blue objects. Each class is represented by a characteristic colour which is used for the configuration. Ingeneral, any number of class and any number of colour per class can be used to perform the configuration. The onlycondition is to define one symbol per class.Fig. 4 shows Delaunay’s triangulation when learning four new characteristical points, one red, one yellow, one greenand one blue. A new interpolation is built which leads to a new nominal scale. The symbols have a new fuzzy meaning,see for example the fuzzy meaning of Red given in Fig. 4.Fig. 4 New Delaunay’s triangulation and new fuzzy meaning of Red.2.3 Fuzzy description of RGB componentsNow, the RGB colour space is used as the measurements set of the fuzzy sensor. The characteristic points used forthe triangulation are the vertices of the cube. Since the description of each vertex of the RGB cube is known, the initiallinguistic set can be defined by Eq. (10).ι(0,0,0) = {Black}, ι(1,0,0) = {Red}, ι(0,1,0) = {Green}, ι(0,0,1)= {Blue}, ι(1,1,0) = {Yellow},ι(1,0,1) = {Magenta}, ι(0,1,1) = {Cyan}, ι(1,1,1) = {White} (10)S = {White, Black, Red, Magenta, Blue, Cyan, Green, Yellow} (11)Now, the triangulation can be performed, and fuzzy meanings of symbols can be calculated. If we try to use thisinitial configuration to describe a real colour, the result is generaly far from the desired one. For example the descriptionof a red colour is ι(0.639, 0.224, 0.326)={0.09/Red, 0.36/Black, 0.22/Yellow, 0.33/Magenta}. It means that the structureof the real subset of Red colours is more complex than a single tetrahedron. Furthermore, this initial configuration basedRedMagentaBlueCyanGreenYellowNeutralmembership function ofthe meaning of Red10µ τE(Red)a bC’1C’2RedRed MagentaBlueBlueCyanGreen GreenYellowYellowNeutral10µ τE(Red)a bC’1C’2membership function ofthe meaning of Redon a theoretical knowledge about colour measurements is not close to the reality. In order to define a configuration closerto the real perception, we provide to the sensor new measurements and their description. Then the sensor uses this newknowledge to perform a new triangulation, and build the new fuzzy meanings of symbols. After this learning phase, thenew fuzzy description of the previous red colour is ι(0.639, 0.224, 0.326)={0.89/Red, 0.03/Blue, 0.04/Yellow, 0.04/Magenta} which is closer to the reality because the grade of membership of Red is now 0.89.Experiments show that for several basis colours, the learning need three or four characteristic points for one symbol.Contrary to humans, the sensor makes an important difference between pale and dark colours. For example, a correctlearning of Red, includes pale red, red and dark red.Fig. 5 This figure shows the two possible triangulations of the RGB cube. The choice of the initial triangulation isnot critical: it will be totally modified during the configuration phase.3. ApplicationThe previous methods were applied to implement a fuzzy sensor which performs the fuzzy description of thecoloured surfaces. The measurement is based on the analysis of the light reflected by an illuminated surface. The reflectionphenomenon can be modelized by the combination of two cases: the diffuse reflection which is predominant for matsurfaces, and the specular reflection which is predominant for polished surfaces. In general, the light coming from thespecular reflection depends only on the colour of the emitted light while the colour of the surface can be acquired byanalysing the diffuse light.The sensor has three components: a light source, a sensing unit, and a measurement head (Fig. 6). The measurementhead is designed to illuminate the surface and to integrate the light coming from the diffuse reflection. The spatialintegration reduces the effect of surface texture. The light source is a thermic one. It was chosen to have an illuminationclose to the sun light.The sensing unit includes three transducers, a signal conditioner, and a computation device which performs thesignal processing, the fuzzy descriptions and the learning. The transducers are phototransistors with optical filters. Thecomputational device is based on a 80C196KB micro-controller, with 32Kbytes of RAM and an I2C interface for networkcommunication. The computational device performs the configuration process and generates the fuzzy description of theanalysed surface. The I2C network interface is used for the communication with the fuzzy control system, fuzzy actuators,or other fuzzy sensors.Fig. 6 Description of the fuzzy colour sensor.BlackWhiteRedGreenBlueCyanYellowMagentaRBGBlackWhiteRedGreenBlueCyanYellowMagentaRBGLight sourceoptical fibermeasurementheadcoloured surface rgb filters transducersprocessingNeutralRedGreenBlueCyanMagentaYellowµ10(80C196)deviceI2CConclusionMulti-component fuzzy sensors can be used in many domains where the measurement depends on several sources.Due to their configuration capabilities, they are especially adapted for complex sensing especially those perfomed byhumans. Indeed, the sensor does not need explicit numerical model of these mechanisms to be configurated. Only a smallset of measurements associated with their fuzzy or crisp description is needed.Fuzzy colour sensors return a fuzzy representation of the measure,therefore they can be used in a rule based fuzzycontrol system. This ability is interesting when colours represent significant information to control the process. A simpleexample is the control of the colour of painting oils.The second type of applications is classification. In this case, the measurements used for the sensor configuration arecharacteristic elements of classes to identify. After the configuration, the sensor is able to return at which grade ameasurement is member of a class.The method applied to build multi-dimensional fuzzy sets can be used in many applications. For example, patternrecognition is an interesting domain to apply this method.References[1] Foulloy, L., Mauris, G., An ultrasonic fuzzy sensor; Proc. of  Int. Conf. on Robot Vision and Sensory Control,Zürich, Switzerland, February 1988, pp. 161-170.[2] Benoit, E., Foulloy, L., Symbolic sensors; Proc. the International Symposium on Artificial Intelligence BasedMeasurement and Control, Kyoto, Japan, September 1991.[3] Mauris G., Benoit E., Foulloy L., Fuzzy symbolic sensor : from concept to applications; Measurement, to appear.[4] Zingales, G., Narduzzi, C., The role of artificial intelligence in measurement; 8th International Symposium on Arti-ficial Intelligence based Measurement and Control (AIMaC’91), september 1991, Ritsumeikan University, Kyoto,Japan. pp. 3-12.[5] Benoit E., Mauris G., Foulloy L., Fuzzy sensors aggregation: Application to comfort measurement; 5th Int. Conf.IPMU, July 4-8, 1994, Paris, France.[6] Zadeh L.A., Quantitative fuzzy semantics; Information Sciences, Vol. 3, 1971, pp. 159-176[7] Benoit, E., Capteurs symboliques et capteurs flous: un nouveau pas vers l’intelligence; Thèse de doctorat de l’Uni-versité de Grenoble I, Janvier 1993.[8] Finkelstein, L., Leaning, M. S., A review of the fundamental concepts of measurement; Measurement, Vol 2, No 1,pp. 25-34, 1984[9] Posenau, M.-A. K., Approaches to high aspect ratio triangulations; Techreport of NASA Langley Research Center,Hampton, Virginia, 1993.

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A fuzzy colour sensor

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