Multi-valued Fuzzy Spaces for Color Representation

Transcription

1 Multi-valued Fuzzy Saces for Color eresentation Vasile Patrascu Tarom Information Technology, ucharest, omania bstract. This aer rooses two comlementary color systems: red-greenblue-white-blac and cyan-magenta-yellow-blac-white. oth systems belong to the five-valued category and they reresent some articular case of neutrosohic information reresentation. The roosed multi-valued fuzzy saces are obtained by constructing fuzzy artitions in the unit cube. In the structure of these five-valued reresentations, the negation, the union and the intersection oerators were defined. Next, using the roosed multi-valued reresentation in the framewor of fuzzy clustering algorithm, it results some color image clustering rocedure. Keywords: fuzzy color sace, five-valued reresentation, intuitionistic fuzzy sets, neutrosohic set. 1 Introduction color image generally contains tens of thousands of colors. Therefore, most color image rocessing alications first need to aly a color reduction method before erforming further sohisticated analysis oerations such as segmentation. The use of color clustering algorithm could be a good alternative for color reduction method construction. In the framewor of color clustering rocedure, we are faced with two color comarison subject. We want to now how similar or how different two colors are. In order to do this comarison, we need to have a good coordinate system for color reresentation and also, we need to define an efficient color similarity measure in the considered system. The color sace is a three-dimensional one and because of that for a uniue descrition there are necessary only three arameters. mong of the most imortant color systems there are the following:, HSV, HSI, HSL, Luv, Lab, I1II3. This aer resents two systems for color reresentation called rgbw resectively cmyw and they belong to the multi-valued color reresentation [14]. The resented systems are obtained by constructing a five-valued fuzzy artition of the unit cube. The sum of the arameters r,g,b,w, and c,m,y,,w verifies the condition of artition of unity and we can aly some similarities related to this roerty. Thus, one obtains new formulas for color similarity/dissimilarity. The aer has the following structure: Section resents the construction modality for obtaining of the fivevalued color reresentation rgbw, the inverse transform from the rgbw color system to one, and the definition in the framewor of the roosed color reresentation for the negation, the union and the intersection oerators. Section 3 resents the using. Laurent et al. (Eds.): IPMU 014, Part II, CCIS 443, , 014. Sringer International Publishing Switzerland 014

2 Multi-valued Fuzzy Saces for Color eresentation 175 of the roosed multi-valued reresentation in the framewor of the -means clustering algorithm. The resentation is accomanied with some exerimental results. Finally, Section 4 outlines some conclusions. The Construction of a Five-Valued Color eresentation For reresenting colors, several color saces can be defined. color sace is a definition of a coordinate system where each color is reresented by a single vector. The most commonly used color sace is [6]. It is based on a Cartesian coordinate system, where each color consists of three comonents corresonding to the rimary colors red, green and blue. Other color saces are also used in the image rocessing area: linear combination of (similar to I1II3 [9]), color saces based on human color terms lie hue, saturation and luminosity (similar to HIS [4], HSV [18], HSL [8]), or ercetually uniform color saces (similar to Lab [5], Luv [3]..1 The Fuzzy Color Sace rgbw We will construct this new reresentation starting from the (red, green, blue) color system. We will suose that the three arameters tae value in the interval [0,1]. We will define the maximum V, the minimum v, the hue H, the luminosity L [1] and the saturation S [10]: V = max(,, ), v = min(,, ) (1) ( ) 3 + H = atan, () V L = 1 + V v (3) ( V v) S = (4) 1+ v V 0.5 Firstly, we will define a fuzzy artition with two sets: the fuzzy set of chromatic colors and the fuzzy set of achromatic colors. These two fuzzy sets will be defined by the following two membershi functions: We obtained the first fuzzy artition for the color sace: ρ C = S (5) ρ =1 S (6) ρ ρ =1 (7) C +

3 176 V. Patrascu The arameter ρ C is related to the color chromaticity while ρ is related to the color achromaticity. Next in the framewor of the chromatic colors, we will define the reddish, bluish and greenish color sets by the following formulae: S min( γ ) + min( γ ) r = γ σ S min( γ ) + min( γ ) g = γ σ S min( γ ) + min( γ ) b = γ σ (8) (9) (10) where γ = cos(h ), π π γ = cos H = cos H + and 3 3 σ = max( γ ) min( γ ) There exists the following euality: r + g + b = ρ (11) C fter that, in the framewor of the achromatic colors, we define two subsets: one related to the white color and the other related to the blac color: There exists the following euality: From (7), (11) and (14) it results the subseuent formula: w = ρ L (1) = ρ ( 1 L) (13) w+ = ρ (14) r + g + b + w + = 1 (15) We obtained a five-valued fuzzy artition of unity and in the same time we obtained a five-valued color reresentation having the following five comonents: r (red), g (green), b (blue), w (white) and (blac). We must observe that among the three chromatic comonents r, g, and b at least one of them is zero, exlicitly min( r, g, b) = 0.

4 Multi-valued Fuzzy Saces for Color eresentation 177. The Inverse Transform from rgbw to In this section, we will resent the comuting formulas for the comonents having as rimary information the rgbw comonents. Firstly, we will comute the HSL comonents and then the ones. Thus for the comuting of luminosity L, we will use the achromatic comonents w,. w L = (16) w + For the comuting of saturation S and hue H, we will use the chromatic comonents r,g,b. S = r + g + b (17) where 3( ω ω ) ω + ω H = atan, ω (18) ω = r + min( r, b g) (19) + ω = g + min( g, b r) (0) + ω = b + min( b, r g) (1) + For the comonents, we have the following formulae: ω = ( V v) + v () S ω = ( V v) + v (3) S ω = ( V v) + v (4) S The arameters V and v can be determined solving the system of euations (3) and (4) and taing into account (16) and (17)..3 Negation, Union and Intersection for rgbw Sace In the following, we consider the negation of color Q = (,, ), namely Q = ( C, M, Y ) = (1,1,1 ). Using (), (8), (9), (10), (1) and (13) it results:

5 178 V. Patrascu r g b S max( γ ) + max( γ ) c = γ σ = S max( γ ) + max( γ ) m = γ σ = S max( γ ) + max( γ ) y = γ σ = (5) (6) (7) w = = ρ ( 1 L) (8) = w = ρ L (9) The five comonents defined by (5), (6), (7), (8) and (9) verify the condition of artition of unity, namely: and in addition: c + m + y + + w = 1 (30) c + m + y = ρ fter the negation oeration, we obtained a new five-valued fuzzy artition of unity and in the same time we obtained a five-valued color reresentation having the following comonents: c (cyan), m (magenta), y (yellow), (blac) and w (white). We must observe that among the three chromatic comonents c, m and y at least one of them is zero, exlicitly, C min ( c, m, y) = 0 There exist the following euivalent relations between these two comlementary systems: m + y min( c, m) + min( c, y) r = min( y, m) (31) c + y min( c, m) + min( m, y) g = min( y, c) (3) m + c min( c, y) + min( m, y) b = min( c, m) (33) g + b min( r, g) + min( r, b) c = min( g, b) (34)

6 Multi-valued Fuzzy Saces for Color eresentation 179 r + b min( r, g) + min( g, b) m = min( r, b) (35) r + g min( b, r) + min( b, g) y = min( r, g) (36) We must highlight that the air ed-cyan (, C) defines a fuzzy set [19] for the reddish color and it verifies the condition of fuzzy sets, namely + C = 1. The air red-cyan ( r, c) defines an tanassov s intuitionistic fuzzy set [1] for the reddish colors and it verifies the condition r + c 1. Similarly, the air blue-yellow ( b, y) defines an tanassov s intuitionistic fuzzy set for bluish colors, the air greenmagenta ( g, m) defines an tanassov s intuitionistic fuzzy set for greenish colors while the air ( w, ) defines an tanassov s intuitionistic fuzzy set for the white color. Thus for the color Q = (0.3,0.5,0.8), one obtains the fuzzy set = 0. 3 and C = 0.7, while for intuitionistic fuzzy descrition one obtains r = 0, c = The intuitionistic descrition is better than fuzzy descrition because the color Q is a bluish one and then reddish membershi degree must be zero. More than that for the white color Q = (1,1,1 ) one obtains for the fuzzy set descrition = 1 and C = 0 while for intuitionistic descrition one obtains r = 0, c = 0. gain, the intuitionistic descrition is better than the fuzzy one. Thus, the fuzzy descrition is identically with that of the red color while in the framewor of intuitionist fuzzy descrition, the intuitionistic index is 1. This value is a correct value for an achromatic color lie the white color. Taing into account the neutrosohic theory roosed by Smarandache [15], [16], [17] we can consider that the vector ( r, g, b, w, ) rovides a neutrosohic set for reddish colors. From this oint of view, r reresents the membershi function, b, g reresent two non-membershi functions while w, reresent two neutralities functions. For this neutrosohic set, we define the union and intersection. The Union For any two colors = ( r, g, b, w, ) and = ( r, g, b, w, ) we define the union by formulae: r = max( r, r ) (37) g = min( g, g ) (38) b = min( b, b ) (39) w = max(( w + r, w + r ) max( r, r ) (40) = min( + g + b, + g + b ) min( g, g ) min( b, b ) (41)

7 180 V. Patrascu The Intersection For any two colors = ( r, g, b, w, ), = ( r, g, b, w, ) we comute the intersection using two stes. Firstly, we comute the intersection for the color negations by formulae: c = max( c, c ) (4) m = min( m, m ) (43) y = min( y, y ) (44) = max(( + c, + c ) max( c, c ) (45) w = min( w + m + y, w + m + y ) min( m, m ) min( y, y ) (46) Secondly, having g and b. c, m, y and using (31),(3) and (33) we comute r, The results of union and intersection verify the condition of artition of unity. lso, the union and intersection are associative, commutative and verify the De Morgan roerties. Similarly, we can define these two oerations for blue, green, yellow, magenta or cyan colors. More than that, we can construct five-valued neutrosohic set for any color hue but this construction will not be subject of this aer. 3 Color Clustering in the rgbw Color Sace For any two colors = r, g, b, w, ), = r, g, b, w, ) we comute the hattacharyya similarity []: ( ( and its dissimilarity: F (, ) = r r + g g + b b + w w + (47) D(, ) = 1 F(, ) (48) Using the dissimilarity defined by (48) in the framewor of -means algorithm, one obtains a color clustering algorithm. The algorithm -means [7], [11] is one of the simlest algorithms that solve the clustering roblem. The rocedure classifies a given data set through a certain number of clusters fixed a riori. The main idea is to define centroids, one for each cluster. The next ste is to tae each oint belonging to the data set and associate it to the nearest centroid. fter that, the cluster centroids are recalculated and new centroids are obtained. Then, a new binding has to

8 Multi-valued Fuzzy Saces for Color eresentation 181 done between the same data set oints and the nearest new centroid. loo has been generated. This algorithm aims at minimizing an objective function, in this case a suared error function. The objective function is defined by: J = j= 1 n i= 1 D ( x, c ) (49) ( j) i j ( j) j where D ( x i, c j ) is a chosen dissimilarity measure between a data oint x i and the cluster center c j. The function J reresents an indicator of the dissimilarity of the n data oints from their resective cluster centers. Using in (49) the dissimilarity (48), we obtained the exerimental results shown in figures 1 and. For the image bird shown in figure 1, only in the case (i) for the rgbw system the orange color was searated. For the image bird, the uniform green bacground was well searated for the HIS, HSV, Lab, Luv and rgbw color systems. For the image flower shown in figure, the orange color was searated in the case (h) for the Lab system and in the case (i) for the rgbw system. For the image flower, the uniform gray bacground was well searated using the Lab, Luv and rgbw color systems. a) Original b) HSI c) HSL d) HSV e) I1II3 f) Lab g) Luv h) i) rgbw Fig. 1. The image bird

9 18 V. Patrascu a) Original d) HSI g) HSL b) HSV e) I1II3 h) Lab c) Luv f) i) rgbw Fig.. The image flower 4 Conclusions Two comlementary fuzzy color saces, rgbw and cmyw which are useful in the color image analysis are introduced. The semantic of the five values defining a color in the rgbw sace is the amount of red, green, blue, white and blac necessary to rovide the color. The transformation from to rgbw or cmyw turns out to be very simle. The similarity/dissimilarity formula using the five arameters r,g,b,w, is introduced and also the negation, the union and intersection oerators were defined. The hue and saturation can be retrieved from the chromatic comonents red, green and blue while the luminosity can be retrieved from the achromatic comonents white and blac. Exerimental results verify the efficiency of rgbw fuzzy color sace for color clustering.

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