Lecture-08 - Fall2012 Thursday,September13,2012 Outline...

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9/13/2012 1 Transforms of Images Prof. Ghassan AlRegib School of Electrical and Computer Engineering Georgia Institute of Technology ECE 6258 Digital Image Processing Fall 2012 Thursday, September 13, 2012 ECE 6258 Fall 2012 Outline DTFT vs. DFT Definition of DFT Properties of DFT Linear Convolution vs. Circular Convolution Ghassan AlRegib © Georgia Tech 2 ECE 6258 Fall 2012 Introduction We studied the Fourier Transform (D. T. F. T.), XXXX , but in practice computing the Fourier Transform is not possible Thus the DTFT is only an analytical tool In this part of the course, we study: —DFT —DCT Ghassan AlRegib © Georgia Tech 3 (,) F  ECE 6258 Fall 2012 2 D Discrete Fourier Transform Notes: The DFT is obtained by sampling the DTFT (on a rectangular lattice); The DFT is an impacted operator to transfer the image into the frequency domain and then perform various operations in that domain; The DFT is energy compactor; It helps with analyzing various processes; It is derived for a periodic sequence; Ghassan AlRegib © Georgia Tech 4
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9/13/2012 2 ECE 6258 Fall 2012 Discrete Fourier Transform (DFT) Let , be a 2 D sequence Let _____ Notice that or Ghassan AlRegib © Georgia Tech 5 [,] fmn M N 01 mn mM nN      [ , ] kl f mn f m kMn lN      Region of Support rectangularly periodic Horizontal period Vertical period M N 0 0 otherwise  0 mn mn otherwise  ECE 6258 Fall 2012 Discrete Fourier Transform (DFT) The 2 D discrete Fourier Series (DFS) gives us a way to write as a superposition of harmonically related complex sinusoids: Ghassan AlRegib © Georgia Tech 6  11 00 12 2 , [ , ]exp( ) MN Fkl j km j ln MN M N    Fourier Series Coefficient Periodic in both k , l and m , n ECE 6258 Fall 2012 Discrete Fourier Transform (DFT) The coefficient can be computed from using Ghassan AlRegib © Georgia Tech 7 22 [ , ] [ , ]exp( ) j km j ln ECE 6258 Fall 2012 Definition of DFT The 2 D DFT of a sequence with a finite region of support is: Notice that Ghassan AlRegib © Georgia Tech 8 [ ,] e x p ( ) j kM lN  2 [ , ] [ , ]exp( ) j MN M N , 0 1 0 otherwise  
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9/13/2012 3 ECE 6258 Fall 2012 Definition of DFT Recall , thus we can write: So far: The 2 D DFT is a Fourier Series representation for the periodic extension of ; The 2 D DTF consists of samples of the Fourier Transform; Ghassan AlRegib © Georgia Tech 9 11 2( ) 00 (,) [,] MN jm n mn Ff m n e        ( , ) kl Fkl F M N f mn ECE 6258 Fall 2012 Properties of the 2 D DFT Linearity When and have the support on the same region Note: If the two images don’t have support on the same region, we perform zero padding Ghassan AlRegib © Georgia Tech 10 af m n bg m n aF k l bG k l  gmn M N ECE 6258 Fall 2012 Circular Shift Example Circular Shift Ghassan AlRegib © Georgia Tech 11
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Lecture-08 - Fall2012 Thursday,September13,2012 Outline...

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