4freq - 4. Image Enhancement in the Frequency Domain...

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1 4. Image Enhancement in the Frequency Domain Chapter 4 (Image Enhancement in the Frequency Domain)
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2 Fourier Transform
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3 Fourier Transform Fourier Transform of a real function is generally complex ) ( ) ( tan ) ( ) ( ) ( 1 , 2 sin 2 cos ] 2 exp[ ] 2 exp[ ) ( ) ( )) ( ( ) ( ) ( ) ( ] 2 exp[ ) ( ) ( )) ( ( 2 2 2 1 2 1 2 2 1 ) ( (u) I (u) R u F u F F(u) R(u) I(u) Φ (u) u I u R u F j ux j ux ux j du ux j u F x f u F F e u F u jI u R dx ux j x f u F x f F - u j Spectrum Fourier Spectrum Power phase Ex. A f(x) X 0 x 0 0 ( ) e x p [ 2] [ e x p [ ] 2 sin( )exp[ ] sin( ) () X X A F uA j u x d x j u x ju A uX j uX u uX Fu AX uX  
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4 Convolution: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( u G u F x g x f u G u F x g x f d x g f x g x f Correlation: 2 () () ( )( ) ()() () p ow e r s p e c t r um fx gx f gx d F uGu fx g x F uG u f x f x F uFu Fu     Some Properties of Fourier Transform
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5 2-D Fourier Transform (,) 1 1 22 2 1 2 (( ,) ) ( , ) ( e x p [ 2( ) ] ( , ) , ) ) ( ( , ) e x p [ 2 ( ) ] (,) , ,t a n , , ,( , ) ( , ) j uv - Ffxy Fuv fxy j u x v yd x d y Fuve FF u v f x y F u v j u x v y d u d v R uv I uv I(u v) Φ (u v) R(u v) F(u v) F u v F u v            ,, R(uv ) I(uv )
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6 Discrete Fourier Transform (DFT) 1-D DFT pair for a signal sequence x(n) of length N: 1 0 2 1 0 2 . 1 ,..., 0 , ) ( ) ( 1 ,..., 0 , ) ( 1 ) ( N k N nk j N n N nk j N n k X n x N k n x N k X e e  1 0 2 1 1 0 ) ( 2 2 1 2 1 1 0 2 1 1 0 ) ( 2 2 1 2 2 1 12 2 2 1 1 2 2 1 1 1 ,..., 0 , , ) , ( ) , ( , 1 ,..., 0 , , ) , ( 1 ) , ( NN N j N j k N n n k k n k n k k X n n x n N k k n k n k n n n x k k X e e N N 2 1 N 1 • Sometimes appears in inverse, or appears in each side . • 2-D DFT pair for a signal sequence x(n 1 ,n 2 ) of area NxN: 11 2 00 1 (0,0) ( , ) ( , ) nn X xn n n N   DC component
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7 Example: 1-D DFT If f(n) = { 2, 3, 4, 4}: 3 0 0 3 2/ 4 0 /2 3 11 (0) ( ) [2 3 4 4] 3.25 44 1 (1) ( ) 4 1 3 4 4 ] 0.5 0.25 4 (2) (3) n jn n jj j Ff n e n e ee e j F Fj
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4freq - 4. Image Enhancement in the Frequency Domain...

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