eee508_Transforms_Part2

# eee508_Transforms_Part2 - Transforms • Transforms that...

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Unformatted text preview: Transforms • Transforms that are commonly used are separable ¾ Examples: Two-dimensional DFT, DCT, DST, Hadamard ¾ We can then use 1-D transforms in computing the 2D separable transforms separable transforms: 9 Take 1-D transform of the rows => X rows (K 1 ,K 2 ) 9 Take 1-D transform of columns of X rows (K 1 ,K 2 ) => X(K 1 ,K 2 ) EEE 508 1 The 2D Discrete Fourier Transform (2D DFT) • The DFT consists of samples of the DTFT at ¡ ¢ 2 1 , Z Z X 2 1 ¡ ¢ 2 1 , , ; 2 1 1 1 1 1 1 £ K N K N K K ¡ S S Z ¤ periodicity in space (or time) domain (and in frequency domain) ¡ ¢ 1 , , ; 2 2 2 2 2 2 2 £ N K N K K ¡ S Z ¤ sequence x ( n 1 , n 2 ) extended periodically with horizontal period of N 1 and vertical period of N 2 x ( n 1 , n 2 + N 2 ) = x ( n 1 , n 2 ) x ( n 1 + N 1 , n 2 ) = x ( n 1 , n 2 ) But, we restrict ourselves to the main period to recover the original signal EEE 508 1 2D DFT . ¡ ¢ ¡ ¢ 2 1 DFT 2 1 , , K K X n n x o m ¡ ¢ ¡ ¢ 1 ; , , 1 1 2 2 2 1 2 1 1 1 2 2 2 2 2 1 1 1 £ d d ¦ ¦ £ £ £ £ i i N n N n n N K j n N K j N K e e n n x K K X S S ¡ ¢ ¡ ¢ 1 ; , 1 , 1 1 2 2 2 1 2 1 2 1 1 1 2 2 2 2 2 1 1 1 £ d d ¦ ¦ £ £ i i N K N K n N K j n N K j N n e e K K X N N n n x S S EEE 508 1 2D DFT • Unitary DFT (Symmetric Form): ¡ ¢ ¡ ¢ 1 ; , 1 , 1 1 2 2 2 1 2 1 2 1 1 1 2 2 2 2 2 1 1 1 £ d d ¦ ¦ £ £ £ £ i i N n N n n N K j n N K j N K e e n n x N N K K X S S ¡ ¢ ¡ ¢ 1 ; , 1 , 1 1 2 2 2 1 2 1 2 1 1 1 2 2 2 2 2 1 1 1 £ d d ¦ ¦ £ £ i i N K N K n N K j n N K j N n e e K K X N N n n x S S ¡ ¢ 2 2 2 1 2 1 2 2 2 1 1 1 2 1 1 , e e N N K K n N K j n N K j n n ) £ £ S S ¡ ¢ ¡ ¢ 2 1 2 1 K K n n ) ) ¤ separable ¡ ¢ 2 , 1 ; 1 2 ) £ i e K i i i n N K j i n S where EEE 508 1 N i i 2D DFT ¾ Note: ¡ ¢ 1 ; 2 , 2 , 1 1 2 1 £ d d ¸ · ¨ § N K K K X K K X S S ¡ ¢ 1 ; , , 2 2 2 1 2 1 £ d d ¸ ¹ ¨ © N K N N X K K X c samples of period between [0 2 S of DTFT DTFT periodic of period 2 S ¾ 2D DFT of an image: [0,2 S ) of DTFT N 1 u N 2 2 S ( N 2 ) u 10 5 2 S ( N 2 ) S 2D DFT S magnitude EEE 508 1 2 S ( N 1 ) S 2D DFT ¾ Note: ¡ ¢ 1 ; 2 , 2 , 1 1 2 1 £ d d ¸ · ¨ § N K K K X K K X S S ¡ ¢ 1 ; , , 2 2 2 1 2 1 £ d d ¸ ¹ ¨ © N K N N X K K X c samples of period between [0 2 S of DTFT DTFT periodic of period 2 S ¾ 2D DFT of an image: [0,2 S ) of DTFT N 1 u N 2 2 S ( N 2 ) u 10 5 2 S ( N 2 ) S 2D DFT S magnitude 2 S 2 S EEE 508 1 2 S ( N 1 ) S 2D DFT ¾ For convenience, use the shorthand notation a N j a N N j N e W e W S S 2 2 ¡ ¡ ¢ ¾ 2D DFT Separable ¢ can use only 1D DFTs £ ¤ £ ¤ K X n x l £ ¤ £ ¤ 1 , , ; 1 1 ¡ ¦ ¡ N K W n x N K X N n Kn N ¡ £ ¤ £ ¤ 1 , , ; 1 1 ¡ ¦ ¡ ¡ N n W K X N n x N K Kn N ¡ EEE 508 1 2D DFT • Matrix Representation: where F i are N i u N i unitary matrices and...
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## This document was uploaded on 03/11/2012.

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eee508_Transforms_Part2 - Transforms • Transforms that...

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