MKM_501E_Sunum_6inciHafta_2D_FT

MKM_501E_Sunum_6inciHafta_2D_FT - The Fourier Transform...

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The Fourier Transform Jean Baptiste Joseph Fourier
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0 50 100 150 200 250 0 500 1000 1500 2000 2500 0 0 3000 3500 Original histogram Image Operations in Different Domains 1) 2) 3) Frequency (Fourier) domain - Histogram stretching, equalization, specification, etc. .. - Average filter, median filter, gradient, laplacian, etc… Original image Gradient magnitude 2 2 y x f f f + = f Blurry Image Laplacian + = Sharpened Image Noisy image (Salt & Pepper noise) 3 X 3 Average 5 X 5 Average 7 X 7 Average Median
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= 3 sin(x) A + 1 sin(3x) B A+B + 0.8 sin(5x) C A+B+C + 0.4 sin(7x) D A+B+C+D A sum of sines and cosines sin(x) A
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Higher frequencies due to sharp image variations (e.g., edges, noise, etc.)
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The Continuous Fourier Transform = -∞ = + u iux du x f π 2 F(u)e ) ( ) 2 sin( ) 2 cos 2 ux i ux ( iux + = e
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Complex Numbers Real Imaginary Z=(a,b) a b | Z θ ) (conjugate (phase) spectrum) (Fourier vector) unit (a θ i i e Z ib a Z Z a b tg b a Z i e i i - - = - = = = + = + = - = - = * 1 2 2 2 ) / ( sin cos ) 1 ( 1 i e Z ib a Z i Z Z = + = + = ) Im( ) Re(
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x ( 29 ux π 2 cos The wavelength is 1/u .
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This note was uploaded on 12/04/2011 for the course TRADE 1056596 taught by Professor Ayladinç during the Spring '11 term at Istanbul Bilgi University.

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MKM_501E_Sunum_6inciHafta_2D_FT - The Fourier Transform...

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