eee508_Transforms_Part2

eee508_Transforms_Part2 - Transforms • Transforms that...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 03/11/2012.

Page1 / 26

eee508_Transforms_Part2 - Transforms • Transforms that...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online