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)
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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 www.tfh-berlin.de/~schwenk/hobby/fourier/Welcome.html 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
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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
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7 Smoothing Frequency Domain Filters Smoothing is achieved in the frequency
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