New HSL Distance Based Colour Clustering Algorithm

Transcription

1 The 4th Midwest Artificial Itelligece ad Cogitive Scieces Coferece (MAICS 03 pp 85-9 New Albay Idiaa USA April New HSL Distace Based Colour Clusterig Algorithm Vasile Patrascu Departemet of Iformatics Techology Tarom Compay Calea Bucurestilor 4F Bucharest Romaia Abstract I this paper we defie a distace for the HSL colour system Next the proposed distace is used for a fuzzy colour clusterig algorithm costructio The preseted algorithm is related to the well-kow fuzzy c-meas algorithm Fially the clusterig algorithm is used as colour reductio method The obtaied experimetal results are preseted to demostrate the effectiveess of our approach Itroductio A colour image geerally cotais tes of thousads of colours Therefore most colour image processig applicatios first eed to apply a colour reductio method before performig further sophisticated aalysis operatios such as segmetatio The usig of colour clusterig algorithm could be a good alterative for colour reductio method costructio I the framework of colour clusterig procedure we are faced with two colour compariso subject We wat to kow how similar or how differet two colours are I order to do this compariso we eed to have a good coordiate system for colour represetatio ad also we eed to defie a efficiet iter-colour distace measure i the cosidered system I this paper we will cosider the particular case of colour represetatio by the perceptual system HSL (hue saturatio ad lumiosity (Smith 978 The obtaied degree of similarity or dissimilarity is depedet o the used iter-colour distace (Carro 995 (Lim et al 990 (Sarifuddi et al 005 (Trivedi et al 986 I the HSL colour compariso procedure the three colour compoets have ot the same importace Thus the most importat is the hue the saturatio is the ext ad fially the lumiosity is less importat Because of this reaso we ca defie some colour clusterig methods that use the bi-dimesioal space HS O this way it is eglected the third compoet the lumiosity This method is very good for those images that are quite saturated I the same time this method has the disadvatage of supplyig erroeous clusters for the less saturated colours because it does ot take ito accout the achromatic compoet the lumiosity O the other side whe the lumiosity is take ito accout i the distace formula there are arisig situatios whe two colours are placed i differet clusters because of the differece betwee their lumiosities despite of their strog similarity i the chromatic space HS I this paper we propose a distace that miimizes the two disadvatages metioed above This distace is costructed as a quasi-liear combiatio of the three stadard distaces of H S L scalar compoets Next this distace is used for a fuzzy colour clusterig costructio I the followig sectios the paper is thus orgaized: sectio presets the particular form of the system HSL used i this paper; sectio 3 presets the ew colour distace; sectio 4 presets the colour clusterig algorithm based o the proposed distace ad related to the fuzzy c-meas algorithm; sectio 5 presets some experimetal results while sectio 6 outlies the coclusios The Perceptual Colour System HSL The most part of the colour images are represeted by the RGB colour system However this space presets the followig two importat limitatios: the difficulty to determie colour features like the presece or the absece of a give colour ad the iability of the Euclidea distace to correctly capture colour differeces i the RGB space Startig from the RGB system there were defied other systems for colour represetatio Oe of them is the HSL system where H is the hue S is the saturatio ad L is the lumiosity Colour space HSL is also commoly used i image processig As opposed to RGB system HSL is cosidered as atural represetatio colour system The HSL system belogs to the perceptual system category because it is very close to the huma colour perceptio I the HSL system colour is decomposed accordig to physiological criteria like hue saturatio ad lumiosity Hue refers to the pure spectrum colours ad correspods to domiat colour as perceived by huma Saturatio correspods to the relative purity or the quatity of white light that is mixed with hue while lumiosity refers to the amout of light i a colour (Gozales et al 007 A great 85

2 The 4th Midwest Artificial Itelligece ad Cogitive Scieces Coferece (MAICS 03 pp 85-9 New Albay Idiaa USA April advatage of HSL system over the RGB lies i its capacity to recogize the presece of colours i a give image There exist may formulae for H S L compoets calculatio For the hue H the most part of the defiitios are closed to the followig formula that uses the ata fuctio (ECMA-6 0: B G R B G H ata ( 6 ad H ( ] For the lumiosity calculatio i this paper the followig formula will be used (Patrascu 0: M L M m ( where M max( R G B ad m mi( R G B Regardig to the saturatio calculatio may variats are defied by the distace betwee max( R G B ad mi( R G B I this paper we will use for saturatio calculatio the followig formula (Patrascu 009: ( M m S (3 M 05 m 05 We suppose R G B [0 ] ad it results that M m S L[0] The New HSL Colour Distace I the Cartesia coordiate system ( R 3 x y z for two vectors v ( x y z v ( x y z oe defies the Euclidea distace by: D E ( v v ( x x ( y y ( z z I order to obtai the variat of Euclidea distace for the cylidrical coordiate system we use the followig substitutio: x cos( y si( z l It results: D E ( v v 4 d ( d ( d ( l l were: d ( si d ( ( d ( l l ( l l The coordiate system HSL is a cylidrical oe ad for two colours Q ( H S L ad Q ( H S L the Euclidea distace becomes (Gozales et al 007: D E ( Q Q 4SS d ( H H d ( S S d ( L L The colour Euclidea distace D E has three terms: the distace betwee hues the distace betwee saturatios ad the distace betwee lumiosities The distace betwee hues is multiplied by a factor that depeds o the colour saturatios This factor has a multiplicative structure Thus whe the saturatio values icrease the hues distace ifluece icreases i framework of distace D E (4 Whe the saturatio values decrease the hue distace ifluece decreases From here the idea to multiply the lumiosity distace with a similar factor comes up This factor will have the followig behaviour: whe the saturatio values icrease the lumiosity distace ifluece decreases ad whe the saturatio values decrease the lumiosity distace ifluece icreases We ca geeralize (4 usig two real ad positive parameters ad by the followig: D P ( Q Q d ( H H d ( L L d ( S S The parameter will be related to the colour chromaticity while the parameter will be related to the colour achromaticity Before the costructio of parameters ad we defie the idex of chromaticity c ad the idex of achromaticity a havig the followig properties: idex of chromaticity c :[0] [0] (i c ( 0 0 (ii c ( (iii S S [0 ] if S S the c( S c( S We will use for the chromaticity idex the followig particular fuctio: (5 86

3 The 4th Midwest Artificial Itelligece ad Cogitive Scieces Coferece (MAICS 03 pp 85-9 New Albay Idiaa USA April idex of achromaticity a :[0] [0] (i a ( 0 (ii a ( 0 c( S S (6 (iii S S [0 ] if S S the a( S a( S We will use for the achromaticity idex the followig particular fuctio: a( S S (7 The two parameters ad will have a multiplicative structure Thus will be a product of chromaticity idexes while will be a product of achromaticity idexes It results: c S c( (8 ( S a S a( (9 ( S Usig (6 (7 (8 ad (9 for the two multipliers ad it results the followig particular forms: The membership fuctio w ij is calculated usig formula (3 where the distace D P is calculated with formula ( i [ ] j [ k] wij The fuctios uity amely: k DP Qi q j m DP Qi qm m j (3 wij verify the coditio of the partitio of i [ ] w i wi wik The cluster ceter compoets h j s j l j are calculated with formulae (4 (5 ad (6 j [ k] wij si i h j ata wij i H i wij cosh i i wij i (4 S S (0 with Si The distace (5 becomes: D P ( Q Q ( S ( S S S ( S S ( L H si L H ( S S The Colour Clusterig Algorithm ( Let there be colours Q Q Q that must be clustered ito k sets Each cluster j is characterized by the membership coefficiets w j w j w j for the cosidered colours ad the cluster ceter defied by the colour q j h j s j l j For colour clusterig we will costruct a algorithm that is similar to the fuzzy c-meas algorithm (Bezdek 98 j [ k] wij Ai Li i l j (5 wij Ai i with Ai Si j [ k] wij Si i s j (6 wij i where is a fuzzificatio-defuzzificatio parameter ad also (5 87

4 The 4th Midwest Artificial Itelligece ad Cogitive Scieces Coferece (MAICS 03 pp 85-9 New Albay Idiaa USA April Experimetal Results We have applied this method to images: red-gree (figure a flower (figure a house (figure 3a parrots (figure 4a ad bird (figure 5a The clustered images obtaied usig the ew distace ca be see i figures b b 3b 4b 5b ad those obtaied usig the Euclidea distace ca be see i figures c c 3c 4c 5c Figure a shows a sythetic image Usig the proposed distace it was obtai the 3-colour represetatio show i figure b while figure c shows the image obtaied usig the Euclidea distace I the first case the black regio has a small area while i the secod case it has a large oe This fact proves the ifluece of achromatic parameter that was used i the proposed distace I the case of image flower usig the Euclidea distace it was obtaied two clusters for backgroud These two clusters have the same hue but the lumiosities are differet (figure c Usig the proposed distace oe obtaied oly oe cluster for the etire backgroud (figure b For image house usig the Euclidea distace the white ad bright blue colours were ot separated (figure 3c I the case of image parrots the yellow ad red colours were ot separated (figure 4c ad the clusterig is differet from that obtaied usig the proposed distace (figure 4b For the image bird usig the Euclidea distace the orage ad grey colours were ot separated (figure 5c The experimetal results illustrate that our distace performs well compared to Euclidea distace I the same time the obtaied results show the helpfuless of usig this ovel iter-colour distace measure i colour clusterig algorithms Coclusios I this paper oe presets a ehacemet of fuzzy c-meas algorithm for the particular case of colour clusterig It was used the perceptual colour system HSL for colour represetatio The mai step is represeted by defiitio of a ew distace i the HSL colour space I this costructio there were used two multipliers that make the balace betwee the hue weight ad lumiosity weight i the framework of this three-term colour distace We ca coclude that the ew iter-colour distace D P defied by ( the two multipliers defied by (8 ad defied by (9 the particular form of idex of chromaticity c defied by (6 ad achromaticity a defied by (7 costruct a ew ad useful framework for colour clusterig procedures The obtaied experimetal results were compared with those obtaied by usig the Euclidea distace This compariso shows the efficiecy of the proposed clusterig algorithm usig for the colour reductio method (a (b (c Figure : The image red-gree (a ad its 3-colour represetatio based o proposed HSL distace (b ad based o HSL Euclidea distace (c 88

5 The 4th Midwest Artificial Itelligece ad Cogitive Scieces Coferece (MAICS 03 pp 85-9 New Albay Idiaa USA April (a (a (b (b (c Figure : The image flower (a ad its 3-colour represetatio based o proposed HSL distace (b ad based o HSL Euclidea distace (c (c Figure 3: The image house (a ad its 6-colour represetatio based o proposed HSL distace (b ad based o HSL Euclidea distace (c 89

6 The 4th Midwest Artificial Itelligece ad Cogitive Scieces Coferece (MAICS 03 pp 85-9 New Albay Idiaa USA April (a (a (b (b (c (c Figure 4: The image parrots (a ad its 6-colour represetatio based o proposed HSL distace (b ad based o HSL Euclidea distace (c Figure 5: The image bird (a ad its 4-colour represetatio based o proposed HSL distace (b ad based o HSL Euclidea distace (c 90

7 The 4th Midwest Artificial Itelligece ad Cogitive Scieces Coferece (MAICS 03 pp 85-9 New Albay Idiaa USA April Refereces Bezdek J C 98 Patter Recogitio with Fuzzy Objective Fuctios New York: Pleum Press Carro A T 995 Segmetatio d'images couleur das la base Teite Lumiace Saturatio: approche umérique et symbolique PhD Thesis Uiversité de Savoie ECMA-6 0 ECMAScript Laguage Specificatio wwwecma-iteratioalorg/publicatios/stadards/ecma- 6htm Gozalez R C; Woods R E 007 Digital Image Processig Pretice Hall Lim Y W S; Lee U 990 O the colour image segmetatio algorithm based o the thresholdig ad the fuzzy c-meas techiques Patter Recogitio volume 3 o Patrascu V 009 New fuzzy color clusterig algorithm based o hsl similarity I Proceedigs of the Joit 009 Iteratioal Fuzzy Systems Associatio World Cogress (IFSA 009 Lisbo Portugal Patrascu V 0 Fuzzy Membership Fuctio Costructio Based o Multi-Valued Evaluatio i vol Ucertaity Modelig i Kowledge Egieerig ad Decisio Makig Proceedigs of the 0 th Iteratioal FLINS Coferece (FLINS 0 Istabul Turkey World Scietific Press Smith A R 978 Colour Gamut Trasform Pairs Computer Graphics ( Sarifuddi A M; Missaoui R 005 A ew perceptually uiform colour space with associated colour similarity measure for cotet based image ad video retrieval I Proceedigs of Multimedia Iformatio Retrieval Workshop 3-7 Trivedi A M; Bezdek J C 986 Low-level segmetatio of aerial images with fuzzy clusterig IEEE Tras O Systems Ma ad Cyberetics volume 6 o