Fuzzy Set Theoretical Approach to the Tone Triangular System
Total Page:16
File Type:pdf, Size:1020Kb
JOURNAL OF COMPUTERS, VOL. 6, NO. 11, NOVEMBER 2011 2345 Fuzzy Set Theoretical Approach to the Tone Triangular System Naotoshi Sugano Tamagawa University, Tokyo, Japan sugano@eng.tamagawa.ac.jp Abstract—The present study considers a fuzzy color system gravity of the attribute information of vague colors. This in which three input fuzzy sets are constructed on the tone fuzzy set theoretical approach is useful for vague color triangle. This system can process a fuzzy input to a tone information processing, color identification, and similar triangular system and output to a color on the RGB applications. triangular system. Three input fuzzy sets (not black, white, and light) are applied to the tone triangle relationship. By treating three attributes of chromaticness, whiteness, and II. METHODS blackness on the tone triangle, a target color can be easily A. Color Triangle and Additive Color Mixture obtained as the center of gravity of the output fuzzy set. In Additive color mixing occurs when two or three beams the present paper, the differences between fuzzy inputs and inference outputs are described, and the relationship of differently colored light combine. It has been found between inference outputs for crisp inputs and for fuzzy that mixing just three additive primary colors, red, green, inputs on the RGB triangular system are shown by the and blue, can produce the majority of colors. In general, a input-output characteristics between chromaticness, color vector can be described by certain quantities as a whiteness, and blackness as the inputs and redness (as one scalar and a direction. These quantities are referred to as of the outputs). the tri-stimulus values, R for the red component, G for the green component, and B for the blue component, and are Index Terms—fuzzy set theoretical approach, tone triangle, given as follows: color triangle, RGB triangle, additive color mixture, fuzzy set of triangular pyramid, conical fuzzy input, tone-hue C = R + G + B (1) mapping, fuzzy rules, vague color, equilateral triangle This is referred to as the RGB color model (Fig. 1), which allows colors to be represented by a planar diagram. The RGB color model can be used to draw the I. INTRODUCTION red, green, and blue components (R, G, B) on the axis of Using the additive color mixing reported in recent color space (including to a color triangle), as shown in studies [6], [7], the relationship between input fuzzy sets Fig. 1. The coordinates (r, g, b) on the color triangle can on the RGB triangle and fuzzy inputs of conical specify various colors. The coordinates correspond to the membership functions was examined. The color triangle amounts of R, G, and B that make up the color. The (plane) represents the hue and saturation of a color. The coordinates specifying the center of the color triangle six fundamental colors and white can be represented on represent the case in which the three primary colors are the same color triangle (See Fig. 3b). Vague colors on the mixed in equal proportion and indicate the color white. color triangle and chromaticity diagram were clarified. Such representations are referred to as chromaticity However, a technique for obtaining expressions of the diagrams. A chromaticity diagram represents hue and tone triangle in the RGB system using the fuzzy set saturation, but not lightness [12]. On the color triangle theoretical method and the additive color mixture has not (the dotted area in Fig. 1) [13], the percentages of redness, been reported. In the present study, the relationship greenness, and blueness, where the total of the three between two or three input fuzzy sets on the tone triangle attributes is equivalent to 100%, specify a color. In order (See Fig. 3a) and fuzzy inputs of conical membership to indicate only the direction of a color vector, i.e., the functions is examined. The six fundamental colors and chromaticity, the redness r, greenness g, and blueness b white can be represented on the RGB triangle (which is are obtained as follows: R similar to the color triangle in Fig. 3b). Vague colors on r = (2) the tone triangle are clarified. Such a system will help us R + G + B to determine the average color value as the center of G g = (3) R + G + B Corresponding author: N. Sugano is with Department of Intelligent B b = (4) Information Systems, Tamagawa University, 6-1-1 Tamagawagakuen, R + G + B Machida, Tokyo194-8610, Japan. r + g + b = 1 (5) © 2011 ACADEMY PUBLISHER 2346 JOURNAL OF COMPUTERS, VOL. 6, NO. 11, NOVEMBER 2011 In other words, the direction is shown as the ratio of tri-stimulus values R, G, and B. The total of these ratios is a W equal to 1, as shown in Eq. (5). In Fig. 1, at red R, the components are (R, G, B) = (1, 0, 0) and the coordinates are (r, g, b) = (1, 0, 0). At red, light green, blue, the components are (R, G, B) = (r, g, b). At yellow, for instance, the components are (R, G, B) = (1, 1, 0) and the coordinates are (r, g, b) = (0.5, 0.5, 0). Colors on three squares: WMRY, WYGCy, and WCyBM in Fig. gray dull C 2b (color space) corresponds to those on three diamonds: WMRY, WYGCy, and WCyBM in Fig. 3b (color triangle). Thus, most of the colors on part of the surface dark of a color space can be displayed in the color triangle. See Appendix C (Table 1). W S M R R Y (1, 0, 0) Y (R, G, B) b G = (1, 1, 0) Cy (r, g, b) B = (0.5, 0.5, 0) B Y (0, 0, 1) G G (0, 1, 0) S (0, 0, 0) Cy W R Figure 1. A color triangle and additive color mixture. The origin is black S. M a W M B Y R Figure 3. (a) A tone triangle. A point in the plane of the triangular Cy system represents the lightness and saturation of a color. S is black. C is the maximum chromaticness of each hue. (b) A color triangle. A point B in the plane of the triangular system represents the hue and saturation of a color. Cy is cyan. G B. Tone Triangle and Color Triangle Designs S In RGB color space (Fig. 2), the tone triangle and the color triangle are considered. In the color space, for b W instance, when the hue of a color is red R, the tone M triangle and the color triangle are fixed as shown in Fig. 2a and b. The intersection between the tone triangle and Y the color triangle corresponds to line W-R in Fig. 3b (not R in Fig. 2). That is, a coordinate on the tone triangle Cy (CWS) is mapped to the line W-R on the RGB triangle. The color triangle (which covers the colors on part of the B surface of the color space) and the RGB triangle (which covers all of the colors of the color space) designed as the G output in this fuzzy system are obviously different. In Fig. 3a, C is the maximum chromaticness (No.66: red R in Fig. S 4a) as a hue, but Cy in Fig. 3b is cyan, midway between blue and green (No.6: cyan Cy in F ig. 4b). Figure 2. (a) A tone triangle and (b) a color triangle in the same color The color triangle in Fig. 2b is an equilateral triangle, if space. S is black. Cy is cyan. the hue is red R, the tone triangle is a right triangle. When the hue of a color is orange, i.e., midway between yellow and red, the tone triangle is nearly an isosceles triangle. © 2011 ACADEMY PUBLISHER JOURNAL OF COMPUTERS, VOL. 6, NO. 11, NOVEMBER 2011 2347 By shaping another triangle into an equilateral triangle, it and S, respectively. Dark (or deep), light (or pale), and is possible to normalize the equilateral coordinates in the dull are modifiers. fuzzy system. The tone triangle and color triangle in Fig. 4 correspond to the schematic diagrams shown in Fig. 3. The color a 11 names or modifiers in Fig. 4a are No.1: black, No.6: gray, 21 No.11: white, No.46: dark (or deep), No.51: light (or 10 30 20. 38 pale), and No.66: C, maximum chromaticness [3] (e.g., 9 29 45 vivid red). The color names in Fig. 4b are No.1: blue, 19 37 51 No.6: cyan, No.11: green, No.51: yellow, No.66: red, 8 28 44 56 No.46: magenta, and No.104: white. 18 36 50 60 . 7 27 43 55 63 a1 Whiteness b1 Greenness c1 Redness w 17 35 49 59 65 Aj 1 go Ok 1 ro Unknown 6 26 42 54 62 66 go’ ro’ S w’ In αk B αk G O ’ 16 34 48 58 64 Blackness k R W 0 G 0 5 25 41 53 61 U’ V’ uko V’ 15 33 47 57 c’ ro’ uko’ s’ c ro s bo bo’ 4 24 40 52 C R B 14 32 46 Chromaticness 3 23 39 13 31 a2 Whiteness b2 Greenness c2 Redness 2 22 w g Aj 1 o Ok 1 ro 12 Unknown go’ ro’ S w’ In αk B αk G 1 Ok’ Blackness R W 0 G 0 U’ V’ uk V’ o c’ ro’ uko’ s’ c ro 35 s bo bo’ 105 C R B 11 102 107 b Chromaticness 21 104 42 Figure 5. Fuzzy system using the membership function of input fuzzy 10 30 101 106 sets Aj with (a) conical fuzzy input In on the tone triangle, (b) output 20 38 103 crisp sets Ok on the RGB triangle, and (c) a color coordinate on the 9 29 45 34 graphic plane in each trace.