IEG4160_Part5

IEG4160_Part5 - IEG 4160 Image and Video Processing...

This preview shows pages 1–10. Sign up to view the full content.

Fourier Transform IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu Background 2D Fourier transform Spatial frequency Properties of Fourier transform Fourier transform and LSI systems 2D Sampling theorem Discrete Fourier Transform Circular/cyclic convolution Circular/cycle correlation

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

View Full Document
Page 2 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 5. Fourier Transform Background Fourier published his book “ The Analytic Theory of Heat ” in 1822 New revolutionary concept ± Any function that periodically repeated itself can be expressed as the sum of sines and/or cosines of different frequencies , each multiplied by a different coefficient. T
Page 3 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 5. Fourier Transform Background Two forms ± Fourier series ± Fourier transform Important characteristic ± A function, expressed in either a Fourier series or transform, can be reconstructed completely Digital computation and FFT algorithm appearing in 1950s revolutionized signal processing

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

View Full Document
Page 4 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 5. Fourier Transform Fourier transform 1D case = summation of weighted cosines and sines of different frequencies 2 () j ut F u f te d t π −∞ = 2 ( ) ju t f tF u e d u −∞ = 2 1 , cos(2 ) sin(2 ) t j eu t j ut =− = + ft
Page 5 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 5. Fourier Transform Fourier transform 2D case = sum of weighted cosines and sines of different spatial frequencies 2( ) (,) (, ) ju x v y Fuv f x y ed x d y π −+ −∞ −∞ = ∫ ∫ ) c o s 2 () s i n 2 x v y eu x v yj ux vy + = ++ + fx y ) x v y f x y F u v e dudv + −∞ −∞ = ∫ ∫

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

View Full Document
Page 6 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 5. Fourier Transform Spatial frequency What is spatial frequencies? (,) [ c o s 2( ) s i n ) ] f x y Fuv u x v yj ux vy dudv ππ −∞ −∞ =+ + + ∫ ∫ When 0, 1, 2,. ..... , cos2 ( ) 1 (maximal) ux vy n nu x v y π + = ± + = Sinusoidal image y x , where and keep unchanged ux vy v + = 1/ v 1/ u 22 1 T uv = + the spatial frequency is 1/ Tu v = +
Page 7 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 5. Fourier Transform Spatial frequency These lines cross x -axis u cycles/unit and y -axis v cycles/unit u and v : spatial frequencies along x -and y -axis, respectively Sinusoidal image y x 1/ v 1/ u 22 1 T uv = + the spatial frequency is: 1/ Tu v = + α

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

View Full Document
Page 8 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 5. Fourier Transform Spatial frequency When u and v vary, T and of the pattern vary too Interpretation: f ( x, y ) can be considered to be a linear combination of elementary periodic patterns of the form α j2 π (ux+vy) e 2( ) (, ) (,) ju x v y f x y F u v e dudv π + −∞ −∞ = ∫ ∫ ...
Page 9 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 5. Fourier Transform Spatial frequency F ( u,v ): weighting factor measuring the relative contribution of the elementary pattern ( u , v ): frequency domain F ( u,v ): frequency component of the transform |F ( u,v )|: frequency spectrum of the transform

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/13/2009 for the course CS 5150 taught by Professor Xulei during the Spring '09 term at University of Central Arkansas.

Page1 / 46

IEG4160_Part5 - IEG 4160 Image and Video Processing...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online