.......2.2....3...4.4...6.5...24... 27 2....28 2.2...29 Y ( )...32 X ( )...35 2.3...37 2.4...43...43...44...44...45...46...48... 50 3....50...50...50...5...5 3.2...53 3.3...54 3.4...55... 58... 59 i
RGB 0 255 ii
Fuzzy C-Means Clustering RGB iii
[~3] [4] [5] [6,7] [8~] [2,3] [4~20] RGB (fuzzy theory) [2~3] (fuzzy relation)
.. ( ) 2
3
4
5
. 2. ( ) 3. 4.. 2. 3. 6
4. [32~35] 7
[44] 8
45 ( ) [44] [32] 9
. [45] 590 00 (733/00) (68/590) (P<0.0) 2. [46] 3. [47] 67 58 (94%) 9 (6%) 58 2 (3%) 63% 2% 0% 297 55 8% 28 7 7 2 4. [48] 98 44 20 44 5. [49] 7 5 0 ( 8 ) 5 0
2 00% 6. [50] 303 44.2%(34 ) 7. [5] 20 8. [52] 662 44.% 68.4% 72% 3/5 66% 2.7 68.73% 9. Friedberg C.K.[53] 20 ( 3 4.5 6 0 ).2 RGB HSI
B (0,0,) (0,,0) G (,0,0) R.2 RGB RGB RGB RGB.2 RGB [0,] RGB RGB RGB RGB HSI RGB 2
HSI HSI (I) (H) (S).3 HSI B R H P G I.3 HSI H P I S P RGB HSI ( R + G B) I = + 3 (-) 3 S =, (-2) [ min ( R, G B) ] ( R + G + B) [( R G) + ( R B) ] 2 2 ( R G) + ( R B)( G B) H = cos (-3) 2 [ ] 3
.3.4.5 (Histogram equalization) (gray- level transformation function) ( r) s = T (-4) r s r 4
[0,] (-4) (a) T(r)0r (b) 0r0r) n k P( r ) = 0 r k = 0,, L, L k n n L P(r k ) k n k r k n k n k k L -6) n j S = T ( rk ) = = P( rj ) k = 0,,, L j= 0 j= 0.4.6 5
.7 (thresholding).8 T T T.4 [2~3] 965 Zadeh (INFORMATION AND 6
CONTROL) FUZZY SET 0 35 50 34 0 30 0.235 0.6 42 6 6 (Fuzziness) (Randomness) (crisp set) (characteristic function) 0 (universal set) A X x X 7
f A ( x) = 0 x A x A (fuzzy set) (membership function) 0 X A f A X [0,] X={5,0,20,30,40,50,60,70,80}.2 ( ) 5 0 20 30 40 50 60 70 80 0 0..2.4.6.8 (fuzzy relation) (pattern classification) X (pattern set) (one-step fuzzy relation) f ( x, y ) n- f f ( x, x) = x X ( x y) = f ( y, x) x, y X, (n-step fuzzy relation) ( x y ) f n, 8
f n ( x, y ) = sup x, x 2, L, x n X L, f ( x min[ n, y )] f ( x, x ), f ( x, x 2 ), n = 2,3, L f n + ( x, y ) = = sup x, L, x n, x n X L, f x, L, sup L, f f n x n 2, ( x x ( x ( x, y ) n X min[ f, n n x n ), min[ f, y ), f ( x, x f ( x n ( x, x ),, y )] ), ( y, y )] 0 f ( x, y ) f 2 ( x, y ) L f n ( x, y ) f n + ( x, y ) L f ( x, y ) = lim fn ( x, y ) n λ [0,] (fuzzy binary relation) Rλ xr y f,. (reflexive)f ( x, x ) = λ ( x y) λ 2. (symmetric) f ( x, y ) = f ( y, x ) 3. (transitive) f ( x, z ) max min[ f ( x, y), f ( y, z )] Rλ (equivalence relation) y (threshold)λ R (class) R ( x, y ). R ' = R ( R o R ) (transitive closure) ( x y) λ R T, 2. R ' R R R ' = 9
3. R ' = R T R R o (composition) (membership matrix) M = P ], M = [ q ] R P o Q P [ i, k Q k, j = M = ] R [ r i, j [ r i j ] = [ p i, k ] o [ q k, r, j = max min( pi, k,, q k, k i, j j ] ) (product) min (sum) max.3 0.4.5.7.6.8.9 o.3.5.5.2 0.7 0.5.7.8.9 =.5.5.3.2.4.5.5.5.5.7.6 Fuzzy c-means Clustering Method (clustering) (Pattern Recognition) (clustering) (cluster) (cluster) (cluster center) X = x, x,..., x } { 2 n X (fuzzy pseudo partition) c- (fuzzy c-partition)x (fuzzy subset) (family) P = A, A,..., A } k Ν n c 0 < i= n A i ( x k ) = k= A ( x ) < n i k { 2 c X (vector)x k X 20
x k = [ x x 2... x ] R k k kp P v, v 2,..., vc P = { A, A2,..., Ac} c (cluster centers) v v,...,, 2 v c v i = n k = n [ A ( x k = i [ A ( x i k )] k m )] x m k ( i N c, m > R) Fuzzy c-means. t=0 (fuzzy pseudo partition) (0) P A,...,, A2 A c 2. v n k = i = n [ A ( x k= i i k [ A ( x )] k m )] x m k m P (t) ( t) c ( t ) ( t v ), v 2,..., vc 3. X x k 2 ( ) (i)x t > 0 i k v i N c A ( t+ ) i ( x k c ) = j= k x v k v v ( t) i ( t) j 2 2 m 2 ( ) (ii) t ( t x = 0 i I N I i A + ) ( x ) k v i ( t+ A ) ( x ) = i N I ( t+ A ) ( x ) = 0 i I i k ( t 4. + ) ( t p p ) ε t 2. m (fuzzier) m Fuzzy c-means Classical c-meansm (cluster center) X ε (t) P P ( t+) c i i k k 2
k k 2 k 3 k 5 5 x y k k 2 k 3 k 4 K 5 k 6 k 7 k 8 k 9 k 0 k k 2 k 3 k 4 K 5 x 0 0 0 2 3 4 5 5 5 6 6 6 y 0 2 4 2 3 2 2 2 2 3 0 2 4 5 A A 2 A A 2 k K 2 K 3 k 4 k 5 K 6 k 7 K 8 k 9 k 0 k k 2 k 3 k 4 K 5 A.85.85.85.85.85.85.85.85.85.85.85.85.85.85.85 A 2.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5 Fuzzy c-means k k 2 k 3 k 4 k 5 K 6 k 7 K 8 k 9 k 0 k k 2 k 3 k 4 K 5 A.99.99.99.47.0 0 0 0.0 0.0 A 2.0 0.0 0 0 0.0.53.99.99.99 A A 2 K 8 (3,2) A A 2 Fuzzy c-means 22
(pattern recognition) (decesion analysis) (fuzzy expert system) (fuzzy logic control FLC) (fuzzy associative memory) fuzzy model.9 (crisp) Fuzzy c-means (cluster relation) 23
24.5 () (2) (3).0 RGB Fuzzy c-means clustering Method.0
RGB 25
0 (uncertainty) (fuzzy theory) (fuzzy relation) (membership function) (fuzziness) 26
27 (Gray Card) 2. 2. RGB 2.2
2. 2.3 ( 5300K 300Lux) (Hue ) [0] 28
2.3 Nikon E2 (.3 pixels280*000) RGB CCD (Gray card) (Luminance) (saturation) [0] 2.2 ( 280*000 6bits bitmap ) 29
Y X 2.4 X Y Y X Y X X Y 30
Y ( ) Y (RGB ) Y (i) RGB R28~20G24~90B00~50 (ii) RGB R-G25~60G-B2~60R-B6 (iii) RGB R32 G50 B50 (iv) RGB R30~20G30~90B00~50 (v) RGB R-G20~60G-B20~60R-B20 (vi) RGB R32 G50 B50 (RGB ) ( ) ( ) Y Y Y ( ) ( ) Y ( ) ( ) X ( ) (RGB ) (RGB 3
Y (RGB ) (RGB ) 2.5 Y ( ) Y Y ( ) (RGB ) ( i )( ii ) ( ) ( ) (RGB ) ( iii ) (X 460~820Y 0~432) 5 5 0 0 0 0 5 5 0 0 0 0 32
2.6 (RGB ) ( i )( ii ) (RGB ( iii ) X 460~820Y 0~432 (RGB ) (RGB ) ( i )( ii )( iii ) Y Y 2.7 Y Y (X 460~780Y 248~770) 5 5 0 0 0 0 5 33
5 0 0 0 0 2.8 X 460~780Y 248~770 (RGB ) (RGB ) ( iv )( v )( vi ) Y Y Y X 600~724 Y 0~432 Y X 600~724 (RGB )Y Y Y 0~432 Y 248~770 34
2.9 Y X ( ) (X 264~579 Y ~ ) 5 5 2 2 5 5 5 5 5 2 2 5 5 5 2.0 X 264~579Y ~ (RGB )R35 G30 B30 ( ) (RGB ) R5 G5 B 35
5 X 2. X (X 742~085Y ~ ) 5 5 2 2 5 5 5 5 5 2 2 5 5 5 2.2 X 742~085Y ~ (RGB )R35 G30 B30 ( ) (RGB ) R5 G5 B 5 X 36
2.3 2.3 ( ) RGB 37
G G G B G B R G, R B (2-) R B G RB G R G G B + if R G, R B R G R G G B + G B + 38
(2-2) (2-2) (2-2) 80 < R + G + B < 480 (2-3) RGB 80 0 RGB 480 255 ( ) (2-3) (2-2) x M = (2-4) C ( x M ) V = C R G x = C (2-3) G B + 2.5 0 255 2.6 2 M-V M M+V 0 m 255 2.4 39
2.5 2.5 VLow=M-VVHigh=M+V (2-2) VLow VHigh (2-2) M (2-2) M VLow 0 VHigh 255 VLow VHigh (2-2) VLow VHigh 0 255 X V x VLow = m 0 X 0 V = 255 m x M X m 40 if if x M x > M (2-5) (2-6) R G x = m (2-5) VLow M G B + 0 m (2-6) M VHigh m 255 2.5
(2-4) M (2-2) M M m 69 (2-2).5 (2-2).5 M m HC LC ratio = HC LC m 2. M m 4
ratio M ratio m 0~0.24 78 0.89~.05 39 0.25~0.33 86.06~.3 44 0.34~0.42 95.4~.2 48 0.43~0.50 05.22~.32 52 0.5~0.60 5.33~.50 58 0.6~0.80 25.5~.70 63 0.8~0.88 35.70 67 ratio m m m 05 20 2.6 2.6 42
2.4. 0( ) 2. 255( ) 255( ) 0( ) 43
2.7 (flood fill) 2.8 0( ) 44
(flood fill) 255( ) 0( ) 2.9 X Y 0 5 45
2.20 2.2 ( ) 46
2.22 ( ) 2.23 ( ) 47
2.24 ( ) x, ) x, ) x, ) y y y 2 3 ( y = ax 2 = ax = ax 2 2 2 3 + bx + bx + bx 3 ( 2 y2 2 + c + c + c ( 3 y 3 abc 2 c y = ax + bx + 2.25 48
2.26 RGB 49
3. (i)rgb R88~20G72~2B56~04 (ii)rgb R-G0~20G-B4~6R-B8~32 (i)(ii) 255( ) 0( ) 3. 9 9 255( ) 0( ) 50
3.2 (flood fill) 3.3 0( 5
52 ) (flood fill) 255( ) 0( ) 3.4
3.5 3.2 53
3.3.4 Fuzzy c-means Fuzzy c-means 83.77 % 83.77 % 0.99 % 94.76 % 5.24 % 00 % 72.75 % 72.75 % 20.90 % 93.65 % 6.35 % 00 % 84.96 % 84.96 % 2.75 % 97.7 % 2.29 % 00 % 0/5 3.90 % 3.90 % /5 7.0 % 20.9 % 2/5 50.90 % 7.8 % 3/5 26.80 % 98.6 % 4/5.39 % 00 % 3/5 (83.77, 54
72.75, 84.96, 7.8) 3/5 (94.76, 00, 97.7, 98.6) Fuzzy c-means 3.4 55
56
57
RGB 58
[] 995:-525 224-347 [2] 993:2-5 [3] [4] [5] [6] 993():58 [7] 600 986:-2 [8] 989 2:47 [9] 996 2 [2] APSC 993:0-65-69 [3] 993:06-23265-272 [4] Rafael C. Gonzalez and Richard E. Woods, Digital Image 59
Processing, Addison-Wesley Publishing Company, USA, 992. [5] William K. Pratt, Digital Image Processing, Second Edition, A Wiley-Interscience Publication, USA, 99. [6] Anil K. Jain, Fundamentals of Digital Image Processing, Prentice-Hall International, Inc., USA, 989. [7] Kenneth R. Castleman, Digital Image Processing, Prentice-Hall International, Inc., USA, 996. [8] Czerwinski R., and Jones D., and O'Brien Jr W., Line and Boundary Detection in Speckle Images, IEEE Trans. Imag. Proc., vol. 7, No. 2, 998 [9] Djuric P., Fwu J., Line and Boundary Detection in Speckle Images, IEEE Trans. Imag. Proc., vol. 6, No., 997 [20] Sharma G., Trussell H., Digital Color Imaging, IEEE Trans. Imag. Proc., vol. 6, No. 7, 997 [2] George J. Klir and Tina A. Folger, Fuzzy Sets, Uncertainty, and Information, New Jersey, 988. [22] Shinichi Tamura, Seihaku Higuchi, and Kokichi Tanaka, Pattern Classification Based on Fuzzy Relations, IEEE Trans. Syst., Man, Cyber., Vol., No., pp. 6-66, Jan. 97. [23] L. A. Zadeh, Fuzzy algorithms, Inform. Contr., Vol. 2, pp. 99-02, Feb. 968. [24] L. A. Zadeh, Fuzzy sets and systems, Proc. 965 Symp. On Syst. Theory, pp. 29-39. [25] Fuzzy 993:-66 [26] Bart Kosko 60
995:2-54 [27] 997:6-~8-32 [28] L. A. Zadeh, Fuzzy sets, Inform. Contr., vol. 8, pp. 338-353, June 965. [29] R. Bellman, R. Kalaba, and L. Zadeh, Abstraction and pattern classification, J. Math. Anal. Appl., Vol. 3, pp. -7, Jan. 966. [30] Krzysztof J. Cios, Inho Shin and Lucy S. Goodenday, Using Fuzzy Sets to Diagnose Coronary Artery Stenosis, IEEE Computer, pp. 57-63, Mar. 99. [3] Yannis A. Tolias, and Stavros M. Panas, A Fuzzy Vessel Tracking Algorithm for Retinal Images Based on Fuzzy Clustering, IEEE Trans. Med. Imag., Vol. 7, No. 2, April 998 6
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