Neuro-Fuzzy Based Clustering Approach for Content Based Image Retrieval Using 2D Wavelet Transform

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.8 78ABG8 G;<F GE4<A<A: 74G4 J<G; GJB F8GF B9 I86GBEF 'BF<G<I8 F4@C?8 F8G E8?8I4AG <@4:8F = { ,… } ⊂ 4A7 A8:4G<I8 F4@C?8 F8G ABA E8?8I4AG <@4:8F = { * , … * } ⊂ .8 4FF<:A 846; <ACHG I86GBE <A <AGB 4A *&+$ 4?:BE<G;@ GB 6E84G8 G;8 J8<:;G I86GBEF J;8E8 4F G;8 I86GBEF <A 4E8 HF87 <A G;8 #(- 4?:BE<G;@ 9BE 9HEG;8E 47=HFG@8AG B9 G;8 J8<:;G I86GBEF *&+$ 4?:BE<G;@ <F :<I8A 4F 9B??BJF 2 3 *G8C # ( # ;BBF8 G;8 EBBG AB78F { } 9EB@ G;8 4I4<?45?8 F8G B9 <ACHG I86GBEF { } <A 4 E4A7B@ @4AA8E *G8C ! #' # )4A7B@?L F8?86G 4 A8J 74G4 CB<AG ,4A7 9<A7 G;8 58FG @4G6;<A: A8HEBA 4G G<@8 FG8C 5L HF<A: G;8 @<A<@H@ 7<FG4A68 H7?<784A 6E<G8E<BA ∑ () , , ,….. *G8C # 9 ( ) − ≤ ( ),J;8E8 ( ) <F G;8 ;<8E4E6;L 9HA6G<BA HF87 GB 6BAGEB? G;8 ?8I8?F B9 GE88 G;8A 4FF<:A ( ) GB G;8 =G; 6?HFG8E 4A7 47=HFG G;8 FLA4CG<6 J8<:;G I86GBE 466BE7<A: GB G;8 E8<A9BE687 ?84EA<A: EH?8 ( + 1) = ( ) + ( ) ( ) − ( ) , .;8E8 ( )<F G;8 ?84EA<A: E4G8 J;<6; 786E84F8F @BABGBA<64??L J<G; G<@8 0 < ( ) < 1. ?F8 9BE@ 4 A8J FH5AB78 FG4EG<A: J<G; *G8C # $ # BAG<AH8 J<G; FG8C HAG<? AB 6;4A:8F <A G;8 984GHE8 @4C +;8 *&+$ 4?:BE<G;@ B5G4<AF 4 A8J F8G B9 6?HFG8E 68AGE8F <F = { , … . … },J;8E8 G;8 AH@58E B9 68AGE8F 6BAGEB??87 5L G;8 G;8 9HA6G<BA ( ) A G;8 8KC8E<@8AG ( ) J4F <A<G<4?<M87 5L G;8 ABE@ B9 G;8 GE4<A<A: I86GBEF <A 4A7 J4F E87H687 ?<A84E?L *G8C $"# ! # G G;<F FG4:8 G;8 A8:4G<I8 F4@C?8F <A 4E8 HF87 GB GHA8 G;8 786<F<BA 5BHA74E<8F B9 G;8 <A<G<4? F8G .8 8@C?BL G;8 4AG<E8<A9BE687 ?84EA<A: EH?8 <A G;8 #-( 4?:BE<G;@ GB C8E9BE@ G;<F BC8E4G<BA +;8 4?:BE<G;@ FG4EGF J<G; E4A7B@?L 6;BBF<A: <ACHG I86GBE * ( ) 9EB@ G;8 GE4<A<A: F8G { * } +;8A 6?4FF<9L * ( ) GB G;8 AB78 <9 * ( ) − < * ( )− , ≠ , 4CC?L G;8 4AG<E8<A9BE687 ?84EA<A: EH?8 GB G;8 6BEE8FCBA7<A: 6?HFG8E 68AG8EF 4F ( + 1) = ( ) − ( )[ * ( ) ( )] .;8E8 ( ) <F G;8 ?84EA<A: 6BAFG4AG J;<6; 786E84F8F @BABGBA<64??L J<G; G;8 AH@58E B9 <G8E4G<BAF 0 ≤ ( ) ≤ 1. 9G8E F8I8E4? GE4<A<A: 74G4 G;8 AB78 I86GBEF 6BAI8E:8 4A7 G;8 6?HFG8E @B7<9<64G<BA CEB68FF <F 6B@C?8G8 .8 4FF<:A 846; <ACHG I86GBE v 4F 68AGE8 B9 G;8 6BEE8FCBA7<A: 4HFF<4A >8EA8? <A G;8 A8GJBE> BE G;8

(

.;8E8 σ <F 4 F@BBG;<A: C4E4@8G8E 789<A87 4F σ = δ@<A Vm, − Vi, W $ J<G; 58<A: 4A BI8E?4CC<A: 946GBE +;8 8FG<@4G87 9HA6G<BA BHGCHG K 9BE K <F G;8A :<I8A 4F X, V , Q

9 K

+ # % ,) # % +.&)" ' ) $ + )* &) /' ) $ %+* ACHGF

&HGCHGF

<778A ,A<GF

+E4<A<A: '4<EF

G8E4G<BAF

- &#&) +,) * ) +) - # @4:8 E8GE<8I4? FLFG8@F ;4I8 <@C?8@8AG87 BA8 BE @BE8 6B?BE 984GHE8F 9BE 8K4@C?8 7B@<A4AG 6B?BEF 6B?BE @B@8AGF 6B?BE F8GF 6B?BE 6B;8E8A68 I86GBE 6B?BE 6BEE8?B:E4@ 4A7 4HGB6BEE?B:E4@ +;8 :BB7 6B?BE FC468 9BE <@4:8 E8GE<8I4? FLFG8@ F;BH?7 CE8F8EI8 G;8 C8E68<I87 6B?BE 7<998E8A68F A BG;8E JBE7F G;8 AH@8E<64? H6?<784A 7<998E8A68 F;BH?7 4CCEBK<@4G8 G;8 ;H@4A C8E68<I87 7<998E8A682 3 ! % !" A BE78E GB HF8 4 :BB7 6B?BE FC468 9BE 4 FC86<9<6 4CC?<64G<BA 6B?BE 6BAI8EF<BA <F A88787 58GJ88A 6B?BE FC468F )87 E88A ?H8 ) H8 *4GHE4G<BA -4?H8 *- # % !" A <: &5G4<A45?8 *- 6B?BEF ?<8 J<G;<A 4 GE<4A:?8 J;BF8 I8EG<68F 4E8 789<A87 5L G;8 G;E88 CE<@4EL 6B?BEF <A ) FC4682 3

<:HE8 ) GB *- BAI8EF<BA

+;8 ;H8 B9 G;8 CB<AG ' <F G;8 @84FHE87 4A:?8 58GJ88A G;8 ?<A8 6BAA86G<A: ' GB G;8 GE<4A:?8 68AGE8 4A7 ?<A8 6BAA86G<A: ) CB<AG GB G;8 GE<4A:?8 68AGE8 +;8 F4GHE4G<BA B9 G;8 CB<AG ' <F G;8 7<FG4A68 58GJ88A ' 4A7 GE<4A:?8 68AGE8 +;8 I4?H8 <AG8AF<GL B9 G;8 CB<AG ' <F E8CE8F8AG87 4F ;8<:;G BA 4 ?<A8 C8EC8A7<6H?4E GB G;8 GE<4A:?8 4A7 C4FF<A: G;EBH:; <GF 68AGE8 +;8 :E8LF64?8 CB<AGF 4E8 F<GH4G87 <AGB G;8 F4@8 ?<A8 4A7 G;8 6BAI8EF<BA 9BE@H?4 <F 4F 9B??BJF (

=1−

) (

[(

=

)

(

)] )(

)

[min( , , )]

AG8EA4G<BA4? !BHEA4? BA *<:A4? 4A7 @4:8 'EB68FF<A: -B? %B !4A

V = (R + G + B)

.;8E8 4A7 4E8 <AG8:8EF H8 GB G;8 BEG;BABE@4? CEBC8EGL G;8 J4I8?8G 6B899<6<8AGF B9 4 F<:A4? ( ) 64A 58 84F<?L 6B@CHG87 I<4 X ( ) ( ) , ∫

# % !" BAI8EF<BA 9EB@ *- FC468 GB ) FC468 <F @BE8 6B@C?8K 4A7 :<I8A GB G;8 A4GHE8 B9 G;8 ;H8 <A9BE@4G<BA J8 J<?? ;4I8 4 7<998E8AG 9BE@H?4 9BE 846; F86GBE B9 G;8 6B?BE GE<4A:?8 2 3 )87 E88A F86GBE 9BE B Y B r= 1+ (

) = (1 − ) =1−( + )

E88A ?H8 F86GBE 9BE 120 < ≤ 240 = (1 − ) = 1+ ( =1−( + )

)

?H8 )87 F86GBE 9BE B Y B =1−( + ) = (1 − ) =

1+

(

)

,

4A7 G;8 FLAG;8F<F 9BE@H?4 ( )=∑ , , , ( ) 64A 58 HF87 GB E86BI8E 6B899<6<8AGF

)

<: &E:<A4? <@4:8 <: &A8 ?8I8? 786B@CBF<G<BA +B 6BAFGEH6G G;8 @BG;8E J4I8?8G ( ) J8 @4L 9<EFG 78G8E@<A8 4 F64?<A: 9HA6G<BA ( ),J;<6; F4G<F9<8F G;8 GJB F64?8 7<998E8A68 8DH4G<BA ( ) = √2 ∑ ( ) (2 − ) +;8A G;8 J4I8?8G >8EA8? ( ) <F E8?4G87 GB G;8 F64?<A: 9HA6G<BA I<4 ( ) = √2 ∑ ( ) (2 − ) J;8E8 ( ) = (−1) (1 − ) +;8 6B899<6<8AGF ( ) <A ;4I8 GB F8I8E4? 6BA7<G<BAF 9BE G;8 F8G B9 54F<F J4I8?8G 9HA6G<BAF <A GB 58 HA<DH8 BEG;BABE@4? 4A7 ;4I8 4 68EG4<A 78:E88 B9 E8:H?4E<GL +;8 6B899<6<8AGF ( ) 4A7 ( ) C?4L 4 I8EL 6EH6<4? EB?8 <A 4 :<I8A 7<F6E8G8 J4I8?8G GE4AF9BE@ +B C8E9BE@ G;8 J4I8?8G GE4AF9BE@ 7B8F ABG E8DH<E8 G;8 8KC?<6<G 9BE@F B9 ( )4A7 ( ) 5HG BA?L 78C8A7F BA ( )4A7g(k) BAF<78E 4 ! ?8I8? J4I8?8G 786B@CBF<G<BA J;<6; 64A 58 JE<GG8A 4F ( )=∑ , , ( ) =∑ ( , , ( )+∑ , , ( )) .;8E8 6B899<6<8AGF , 4E8 :<I8A 4A7 6B899<6<8AGF , 4A7 , 4G F64?8 + 1 4E8 E8?4G87 GB G;8 6B899<6<8AGF , 4G F64?8 I<4 ( −2 ) , = ∑ , ( −2 ) , = ∑ , .;8E8 0 ≤ ≤ +;HF 4A7 CEBI<78F 4 E86HEF<I8 4?:BE<G;@ 9BE J4I8?8G 786B@CBF<G<BA G;EBH:; ( ) 4A7 9BE 4 ?BJ E8FB?HG<BA ( ) 4A7 G;8 6B899<6<8AGF , 6B@CBA8AG , ( ) L HF<A: 4 F<@<?4E 4CCEB46; J8 64A 78E<I8 4 E86HEF<I8 4?:BE<G;@ 9BE 9HA6G<BA FLAG;8F<F 54F87 BA <GF J4I8?8G 6B899<6<8AGF , , 0 ≤ ≤ 4A7 , ( −2 )+∑ ( − 2 ) , , , = ∑

- + /+,) * $ ) +) - # +8KGHE8 <F 4ABG;8E <@CBEG4AG CEBC8EGL B9 <@4:8F -4E<BHF G8KGHE8 E8CE8F8AG4G<BAF ;4I8 588A <AI8FG<:4G87 <A C4GG8EA E86B:A<G<BA 4A7 6B@CHG8E I<F<BA 4F<64??L G8KGHE8 E8CE8F8AG4G<BA @8G;B7F 64A 58 6?4FF<9<87 <AGB GJB 64G8:BE<8F FGEH6GHE4? 4A7 FG4G<FG<64? *GEH6GHE4? @8G;B7F <A6?H7<A: @BEC;B?B:<64? BC8E4GBE 4A7 47=468A6L :E4C; 78F6E<58 G8KGHE8 5L <78AG<9L<A: FGEH6GHE4? CE<@<G<I8F 4A7 G;8<E C?468@8AG EH?8F +;8L G8A7 GB 58 @BFG 89986G<I8 J;8A 4CC?<87 GB G8KGHE8F G;4G 4E8 I8EL E8:H?4E *G4G<FG<64? @8G;B7F <A6?H7<A: BHE<8E CBJ8E FC86GE4 6B B66HEE8A68 @4GE<68F F;<9G <AI4E<4AG CE<A6<C4? 6B@CBA8AG 4A4?LF<F *' +4@HE4 984GHE82 3 JB?7 786B@CBF<G<BA $4E>BI E4A7B@ 9<8?7 9E46G4? @B78? 4A7 @H?G<E8FB?HG<BA 9<?G8E<A: G86;A<DH8F FH6; 4F 45BE2 3 4A7 J4I8?8G GE4AF9BE@2 3 2 3 2 3 *<A68 <@4:8 <F GLC<64??L 4 GJB 7<@8AF<BA4? F<:A4? 4 8DH<I4?8AG B9 G;8 .+ <F C8E9BE@87 +;<F <F 46;<8I87 5L 9<EFG 4CC?L<A: G;8 # 4A7 9<?G8EF GB G;8 ?<A8F B9 F4@C?8F EBJ 5L EBJ G;8A E89<?G8E<A: G;8 BHGCHG GB G;8 6B?H@AF 5L G;8 F4@8 9<?G8EF F G;8 E8FH?G G;8 <@4:8 F;BJA <A <: <F 7<I<787 <AGB FH554A7F ## # # 4A7 4F 78C<6G87 <A <: +;8 ## FH5 54A7 6BAG4<AF G;8 ?BJ C4FF <A9BE@4G<BA B9 ;BE<MBAG4? I8EG<64? 4A7 7<4:BA4? BE<8AG4G<BA +;8 ## FH5 54A7 CEBI<78F 4 ;4?9 F<M87 I8EF<BA B9 <ACHG <@4:8 J;<6; 64A 58 GE4AF9BE@87 4:4<A GB ;4I8 @BE8 ?8I8?F B9 E8FB?HG<BA 8A8E4??L 4A <@4:8 <F C4EG<G<BA87 <AGB # E8FB?HG<BA ?8I8?F 5L 4CC?L<A: G;8 .+ # G<@8F2 3 2 3 L J4I8?8G GE4AF9BE@ J8 @84A G;8 786B@CBF<G<BA B9 4A <@4:8 J<G; 94@<?L B9 E84? BEG;B:BA4? 54F8F , ( ) o5G4<A87 G;EBH:; GE4AF?4G<BA 4A7 7<?4G<BA B9 4 >8EA8? 9HA6G<BA ( ) >ABJA 4F @BG;8E J4I8?8G (2 − ) , ( ) =2

G <F 6BAI8A<8AG GB I<8J G;8 786B@CBF<G<BA 4F C4FF<A: 4 F<:A4? , G;EBH:; 4 C4<E B9 9<?G8EF 4A7 J<G; <@CH?F8 E8FCBAF8F ( ) 4A7 ( ) 4A7 7BJA F4@C?<A: G;8 9<?G8E87 F<:A4?F 5L GJB 7EBCC<A: 8I8EL BG;8E F4@C?8 J;8E8 ( ) 4A7 ( ) 4E8 789<A87 4F

P & <=F<C

( ) 9EB@ <GF J4I8?8G

AG8EA4G<BA4? !BHEA4? BA *<:A4? 4A7 @4:8 'EB68FF<A: -B? %B !4A

?HFG8EF F;BJA <A <:

&HE FLFG8@ ;4F 588A G8FG87 J<G; G;8 4I8E4:8 E864?? E4G8 -)) 4A7 CE86<F<BA @84FHE8F2 3HF87 G;8 74G454F8 J<G; E84? JBE?7 <@4:8F ∩ Precision =

( ) = (− ), ( ) = (− ). +;8 C4<E B9 9<?G8EF 4A7 6BEE8FCBA7 GB G;8 ;4?9 54A7 ?BJ C4FF 4A7 ;<:; C4FF 9<?G8EF E8FC86G<I8?L 4A7 4E8 64??87 G;8 DH47E4GHE8 @<EEBE 9<?G8EF <A G;8 F<:A4? CEB68FF<A: ?<G8E4GHE8 2 3 +;8 E86BAFGEH6G<BA CEB687HE8 <F <@C?8@8AG87 5L HC F4@C?<A: G;8 FH5 F<:A4?F , 4A7 , 4A7 9<?G8E<A: J<G; ( ) 4A7 ( ) E8FC86G<I8?L 4A7 477<A: G;8F8 GJB 9<?G8E87 F<:A4?F GB:8G;8E ,FH4??L G;8 F<:A4? 786B@CBF<G<BA F6;8@8 <F C8E9BE@87 E86HEF<I8?L GB G;8 BHGCHG B9 ?BJC4FF 9<?G8E ( ) +;8 J4I8?8G C46>8G 54F<F 9HA6G<BAF { }, 64A 58 :8A8E4G87 9EB@ 4 :<I8A 9HA6G<BA 4F 9B??BJF ( ) = √2 ∑ ( ) (2 − ) ( ) = √2 ∑

( )

(2 − )

AVRR =

%H@58E B9 E8GE<8I87 <@4:8F %H@58E B9 E8?8I4AG <@4:8F

.;8E8 G;8 9HA6G<BA ( ) 64A 58 <78AG<9<87 J<G; G;8 F64?<A: 9HA6G<BA 4A7 J<G; G;8 @BG;8E J4I8?8G +;8A G;8 J4I8?8G C46>8G 54F8F 64A 58 789<A87 GB 58 G;8 6B??86G<BA B9 BEG;BABE@4? 54F8F 6B@CBF87 B9 9HA6G<BAF B9 G;8 9BE@ (2 − ) J;8E8 . 46; 8?8@8AG <F 78G8E@<A87 5L 4 FH5F8G B9 G;8 <A7<68F 4 F64?<A: C4E4@8G8E 4 ?B64?<M4G<BA C4E4@8G8E 4A7 4A BF6<??4G<BA C4E4@8G8E +;8 J4I8?8G BE J4I8?8G C46>8G 54F<F 9HA6G<BAF 64A 58 8KCE8FF87 5L G8AFBE CEB7H6G B9 GJB J4I8?8G BE J4I8?8G C46>8G 54F<F 9HA6G<BAF 4?BA: G;8 ;BE<MBAG4? 4A7 I8EG<64? 7<E86G<BAF +;8 6BEE8FCBA7<A: 9<?G8E 6B899<6<8AGF 64A 58 8KCE8FF87 4F ( , ) = ( ) ( ), ( , ) = ( ) ( ), ( , ) = ( ) ( ),

( , )= ( ) ()

∩

<:HE8 8AE8E4? ) E8GE<8I4?

.;8E8 G;8 9<EFG 4A7 F86BA7 FH5F6E<CGF <A

4A7 78ABG8F G;8 ?BJ C4FF 4A7 ;<:;C4FF 9<?G8E<A: 6;4E46G8E<FG<6F <A G;8 4A7 7<E86G<BAF E8FC86G<I8?L2 3 - ) *,#+* % * ,** &% +B 8I4?H4G8 G;8 C8E9BE@4A68 B9 BHE <@4:8 E8GE<8I4? 6?HFG8E<A: 4?:BE<G;@ G;8 CEBCBF87 9E4@8JBE> <F G8FG87 J<G; 4 :8A8E4? CHECBF8 <@4:8 74G454F8 B9 45BHG <@4:8F 846; F<M8 × 4CCEBK<@4G8?L 64G8:BE<8F 9EB@ &) # A BHE FLFG8@ J8 ;4I8 F8?86G87 64G8:BE<8F FH6; 4F EBF8F 5HF8F 8?8C;4AGF ?<BAF 4A7 FHA9?BJ8EF 8G6 +;8 FLFG8@ FHCCBEGF 5BG; <@4:8 E8GE<8I4? 4F J8?? 4F 6?HFG8E<A: BE 846; 8KC8E<@8AG J8 9<EFG F8?86G87 BA8 <@4:8 4F 4 DH8EL <@4:8 E4A7B@?L 9EB@ G;8 74G454F8 4A7 G;8A B5G4<A87 G;8 E8GE<8I87 <@4:8F +;8A J8 F8?86G87 BA8 BE @BE8 <@4:8F 4F 6?HFG8EF 9EB@ G;8 E8GE<8I87 <@4:8F +;8 F8?86G87 <@4:8F 6?HFG8EF J8E8 HF87 4F DH8EL <@4:8F 9BE A8KG FG8C GB E8GE<8I8 G;8 C4EG<6H?4E 6?HFG8EF +;8 J<78 E4A:8 B9 DH8E<8F J;<6; 6BH?7 58 FH5@<GG87 4F 4 E8FH?G B9 7<998E8AG DH8EL <@4:8F 4?FB CBFF<5?8 +;8 F<@H?4G<BA E8FH?GF B9 G;8 <@4:8 E8GE<8I4? <F F;BJA <A <: )8GE<8I4? B9 ?HFG8EF F;BJA <A <: )8GE<8I4? B9 ?HFG8EF F;BJA <A <: )8GE<8I4? B9 ?HFG8EF F;BJA <A <: )8GE<8I4? B9 ?HFG8EF F;BJA <A <: )8GE<8I4? B9 P & <=F<C

<:HE8 ?HFG8E E8GE<8I4?

<:HE8 ?HFG8E E8GE<8I4?

AG8EA4G<BA4? !BHEA4? BA *<:A4? 4A7 @4:8 'EB68FF<A: -B? %B !4A

+ # - ) ) ## ) + &) - ) &,* '')& * CCEB46;

%% ) $ HMML 'EBCBF87

'E86<F<BA %H@58E B9 DH8EL <@4:8F

+45?8 CEBI<78F 4 78G4<?87 6B@C4E<FBA B9 I8E4:8 E864?? E4G8 B5G4<A87 9BE G;8 GBC 9<I8 DH8E<8F 6BAF<78E87 HF<A:

<:HE8 ?HFG8E E8GE<8I4?

! "

%% ) $ HMML 4A7 CEBCBF87 4CCEB46; 'E86<F<BA C8E9BE@4A68 <@CEBI87 HCGB 6B@C4E87 GB %% 4A7 HCGB 6B@C4E87 GB ) $ 4A7 HCGB 6B@C4E87 GB HMML 4CCEB46; BA DH8EL <@4:8 AB 4A7 4?@BFG CE86<F<BA 9BE 4?? BG;8E DH8EL <@4:8F BE G;8 GBC 9<I8 DH8E<8F 6B@CHG87 4A7 G45H?4G87 <A +45?8 4A7 BG;8E DH8EL <@4:8F F;BJA <A <:

<:HE8 ?HFG8E E8GE<8I4?

$''&

! "

$# "

% ! ! ! #

<:HE8 'E86<F<BA 9BE 7<998E8AG DH8EL <@4:8F

<:HE8

?HFG8E E8GE<8I4? + # ') * &% &) - ) &,* '')& * CCEB46; %% ) $ HMML 'EBCBF87

I8E4:8 E864?? E4G8 %H@58E B9 DH8EL <@4:8F

$''& $# "

! "

<:HE8

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