lecture6_2D_DFT

# Lecture6_2D_DFT - DFT Domain Image Filtering Yao Wang Polytechnic Institute of NYU Brooklyn NY 11201 With contribution from Zhu Liu Onur Guleryuz

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DFT Domain Image Filtering Yao Wang Polytechnic Institute of NYU, Brooklyn, NY 11201 With contribution from Zhu Liu, Onur Guleryuz, and Gonzalez/Woods, Digital Image Processing, 2ed

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Lecture Outline 1D discrete Fourier transform (DFT) D discrete Fo rier transform (DFT) 2D discrete Fourier transform (DFT) Fast Fourier transform (FFT) DFT domain filtering D unitary transform 1D unitary transform 2D unitary transform Yao Wang, NYU-Poly EL5123: DFT and unitary transform 2
Discrete Fourier Transform (DFT): TFT for Finite Duration Signals DTFT for Finite Duration Signals : becomes ansform Fourier tr : 1 ,..., 1 , 0 for defined only is signal the If N n ) 1 , 0 ( , ) 2 exp( ) ( ' 1 0 f fn j n f (f) F N n : (DFT) transform Forward : yields rescaling and , 1 1 0 at ' Sampling ,...,N- , k/N, k f (f) F 1 ,..., 1 , 0 , ) 2 exp( ) ( 1 ' ) ( 1 0 N k n N k j n f N ) N k ( F k F N n 1 ,..., 1 , 0 , ) 2 exp( ) ( 1 : (IDFT) transform Inverse 1 N n n k j k F f(n) N Yao Wang, NYU-Poly EL5123: DFT and unitary transform 3 0 N N k

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Property of DFT (1) Periodicity . modulo represents )) (( where . 0 ), )) ((( ) ( N k N k or k k F k F N p N ) ) ( 2 exp( ) ( 1 ) ( 1 0 n N mN k j n f N mN k F N n Proof A A A F(k) ) ( ) 2 2 exp( ) ( 1 1 0 k F mn j n N k j n f N N n B C D E F G B C D E F G B C D E F G 0N - 1 N Yao Wang, NYU-Poly EL5123: DFT and unitary transform 4 low low high Note: Highest frequency is at k=[N/2]. k=0,1, N-1 represent low frequency.
Property of DFT (2) Translation { ( ). )) ((( } / 2 exp{ ) ( )} / ( 2 exp{ ) ( ) )) ((( 0 0 N o N o k k F N n k j n f N kn j k F n n f Special case N is even, k 0 = N/2. ( p{ n N ). )) 2 ((( ) 1 )( ( } exp{ ) ( N k F n f n j n f Shifting the frequency up by N/2 Yao Wang, NYU-Poly EL5123: DFT and unitary transform 5

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Property of DFT (3) Conjugate symmetry for real sequences | ) ( | | ) ( | ) ( ) ( ) ( * * or k N F k F k N F k F k F . |, ) 2 ( | | ) 2 ( | even is N when k N F k N F 0 1234567 0 8 Yao Wang, NYU-Poly EL5123: DFT and unitary transform 6 N=8 N=9
2D Discrete Fourier Transform Definition ssuming f(m n) m = 0 1 M n=0 1 Assuming f(m, n), m = 0, 1, …, M-1, n = 0, 1, …, N-1, is a finite length 2D sequence 1 11 ) ( 2  MN ln km j . 1 ,..., 1 , 0 , 1 ,..., 1 , 0 , ) , ( 1 ) , ( ; 1 ,..., 1 , 0 , 1 ,..., 1 , 0 , ) , ( ) , ( ) ( 2 00   N n M m e l k F n m f N l M k e n m f MN l k F N ln M km j mn N M Comparing to DTFT MN kl l k nv mu j  ) ( 2 N v M u e n m f v u F     , , ) , ( ) , ( 2 / 12 / 1 ) ( 2 udv nv mu j Yao Wang, NYU-Poly EL5123: DFT and unitary transform 7 2 / / 1 ) , ( ) , ( dudv e v u F n m f

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Property of 2D DFT (1) Periodicity ( r r . 0 , 0 ), )) (( , )) ((( ) , ( N l or l M k or k l k F l k F N M -M Low Frequency A C B D A C B D -M/2 High requency A B A B 0 (0,0) (0,N-1) Frequency C D C D M M/2 (M-1,0) (M-1,N-1) Displayed rea using Yao Wang, NYU-Poly EL5123: DFT and unitary transform 8 0N -N -N/2 N/2 k l area using fftshift
Property of 2D DFT (2) Translation )}, / / ( 2 exp{ ) , ( ) )) (( , )) ((( 0 0 0 0 N M N ln M km j l k F n n m m f Special Case: M,N=even, k 0 =M/2, l 0 =N/2 ). )) (( , )) ((( )} / / ( 2 exp{ ) , ( 0 0 0 0 N M l l k k F N n l M m k j n m f ). )) 2 (( , )) 2 ((( ) 1 )( , ( )} ( exp{ ) , ( ) ( N M n m N l

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## This note was uploaded on 01/22/2012 for the course EL 5123 taught by Professor Yaowang during the Fall '07 term at NYU Poly.

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Lecture6_2D_DFT - DFT Domain Image Filtering Yao Wang Polytechnic Institute of NYU Brooklyn NY 11201 With contribution from Zhu Liu Onur Guleryuz

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