# Lecture_11.pdf - Lecture Outline CSE 4227 Digital Image...

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1 CSE 4227: Digital Image Processing Lecture 11 Fourier Transform & Filtering Dr S Rahman Professor, CSE, AUST CSE | AUST Lecture Outline ° Overview of Fourier Transform – Jean Baptiste Joseph Fourier – The Fourier series & the Fourier transform – Image Processing in the frequency domain Image smoothing Image sharpening 2

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2 Jean Baptiste Joseph Fourier Fourier was born in Auxerre, France in 1768 – Most famous for his work “ La Théorie Analitique de la Chaleur” published in 1822 – Translated into English in 1878: The Analytic Theory of Heat” Nobody paid much attention when the work was first published One of the most important mathematical theories in modern engineering The Big Idea = Any function that periodically repeats itself can be expressed as a sum of sines and cosines of different frequencies each multiplied by a different coefficient – a Fourier series Images taken from Gonzalez & Woods, Digital Image Processing (2002)
3 The Big Idea (cont…) Notice how we get closer and closer to the original function as we add more and more frequencies Taken from The Big Idea (cont…) Frequency domain signal processing example in Excel

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4 The Discrete Fourier Transform (DFT) The Discrete Fourier Transform of f(x, y) , for x = 0, 1, 2… M -1 and y = 0,1,2… N -1, denoted by F(u, v), is given by the equation: for u = 0, 1, 2… M -1 and v = 0, 1, 2… N -1. ∑∑ - = - = + - = 1 0 1 0 ) / / ( 2 ) , ( ) , ( M x N y N vy M ux j e y x f v u F π DFT & Images The DFT of a two dimensional image can be visualised by showing the spectrum of the images component frequencies Images taken from Gonzalez & Woods, Digital Image Processing (2002) DFT
5 DFT & Images (cont…) Images taken from Gonzalez & Woods, Digital Image Processing (2002) DFT Scanning electron microscope image of an integrated circuit magnified ~2500 times Fourier spectrum of the image The Inverse DFT It is really important to note that the Fourier transform is completely reversible The inverse DFT is given by: for x = 0, 1, 2… M -1 and y = 0, 1, 2… N -1 ∑∑ - = - = + = 1 0 1 0 ) / / ( 2 ) , ( 1 ) , ( M u N v N vy M ux j e v u F MN y x f π

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6 The DFT and Image Processing To filter an image in the frequency domain: 1. Compute F(u,v) the DFT of the image 2. Multiply F(u,v) by a filter function H(u,v) 3. Compute the inverse DFT of the result Images taken from Gonzalez & Woods, Digital Image Processing (2002) Some Basic Frequency Domain Filters Images taken from Gonzalez & Woods, Digital Image Processing (2002) Low Pass Filter High Pass Filter
7 Smoothing Frequency Domain Filters Smoothing is achieved in the frequency

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