ece542-5 - 1 Digital Image Processing Digital Image...

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Unformatted text preview: 1 Digital Image Processing Digital Image Processing ECE542-5 Digital Image Processing Digital Image Processing 4- Image Enhancement in the Frequency Domain - I Dr. Z. Aliyazicioglu Cal Poly Pomona Electrical & Computer Engineering 1 Cal Poly Pomona Electrical & Computer Engineering Electrical & Computer Engineering Office 9-143 The Continues Time Fourier Transform and inverse FT The Fourier transform, F(u), of a single variable continuous function f(x) 2 ( ) ( ) j ux F u f x e dx ( ) ( ) F u f x e dx The inverse Fourier transform of given F(u) 2 ( ) ( ) j ux f x F u e du Fourier Transform Pair The Fourier transform, F(u,v), of two variables continuous function f(x,y) 2 ( ) ( ) ( ) j ux vy F u v f x y e dx dy ECE542 - 5 2 Cal Poly Pomona Substitute and yield ( , ) ( , ) F u v f x y e dx dy The inverse Fourier transform of given F(u,v) 2 ( ) ( , ) ( , ) j ux vy f x y F u v e du dv 2 The One dimensional Discrete Fourier Transform and inverse FT The Discrete Fourier transform (DFT) of a discrete 2 function f(x), x=0,1,,M-1 (DFT) of a f(x) in trigonometric form 2 1 1 ( ) ( ) , for 0,1,2,..., 1 ux M j M x F u f x e u M M The inverse Discrete Fourier Transform (IDFT) of given F(u) 2 1 ( ) ( ) , 0,1,2,..., 1 ux M j M u f x F u e x M ECE542 - 5 3 Cal Poly Pomona 1 1 2 2 ( ) ( ) sin , for 0,1,2,..., 1 M x ux ux F u f x Cos j u M M M M ( ) ( ) ( ), 0,1,2, , 1 F u R u jI u u M The One dimensional DFT and inverse FT DFT F(u) in polar coordinate ( ) ( ) ( ) , for 0,1,2,..., 1 j u F u F u e u M where Magnitude of spectrum of FT 2 2 ( ) ( ) ( ) F u R u I u 1 ( ) ( ) tan ( ) I u u R u where Phase angle or Phase Spectrum 2 2 2 ( ) ( ) ( ) ( ) P u F u R u I u Power spectrum Let x denote the first point in the sequence and x denotes ECE542 - 5 4 Cal Poly Pomona Let x denote the first point in the sequence and x denotes fixed interval, the sample x is means ( ) ( ) f x f x x x 1 u M x Similarly ( ) ( ) F u F u u 3 Example: A discrete function of M points A 1 M 1024 A=1, M=1024, and K=8 f(x) multiplied by (-1) x before Transform Its Fourier Transform A discrete function with ECE542 - 5 5 Cal Poly Pomona function with twice the number of nonzero points Its Fourier Transform The two-Dimensional DFT and Its Inverse The Discrete Fourier transform of a function f(x, y) of size M x N 1 1 2 1 ( , ) ( , ) ux uy M N j M N F u v f x y e where u=0,1,2,,M-1 and v=0,1,2,,N-1 ( , ) ( , ) x y F u v f x y e MN The inverse Discrete Fourier Transform of given F(u,v) 1 1 2 ( , ) ( , ) ux uy...
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This note was uploaded on 10/24/2010 for the course ECE 542 taught by Professor Zeki during the Spring '10 term at Cal Poly Pomona.

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ece542-5 - 1 Digital Image Processing Digital Image...

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