IEG4160_Part5 - IEG 4160: Image and Video Processing....

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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
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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
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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
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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
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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 + −∞ −∞ = ∫ ∫
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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 = +
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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
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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 + −∞ −∞ = ∫ ∫ ...
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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
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IEG4160_Part5 - IEG 4160: Image and Video Processing....

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