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Linear Filtering – Part II Selim Aksoy Department of Computer Engineering Bilkent University

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CS 484, Spring 2010©2010, Selim Aksoy2 Fourier theory Jean Baptiste Joseph Fourier had a crazy idea: Any periodic function can be written as a weighted sum of sines and cosines of different frequencies (1807). Don’t believe it? Neither did Lagrange, Laplace, Poisson, … But it is true! Fourier series Even functions that are not periodic (but whose area under the curve is finite) can be expressed as the integral of sines and cosines multiplied by a weighing function. Fourier transform

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CS 484, Spring 2010©2010, Selim Aksoy3 Fourier theory The Fourier theory shows how most real functions can be represented in terms of a basis of sinusoids. The building block: A sin( ωx + Φ ) Add enough of them to get any signal you want. Adapted from Alexei Efros, CMU

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CS 484, Spring 2010©2010, Selim Aksoy4 Fourier transform

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CS 484, Spring 2010©2010, Selim Aksoy5 Fourier transform

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CS 484, Spring 2010©2010, Selim Aksoy6 Fourier transform

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CS 484, Spring 2010©2010, Selim Aksoy7 Fourier transform

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CS 484, Spring 2010©2010, Selim Aksoy8 Fourier transform

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CS 484, Spring 2010©2010, Selim Aksoy9 Fourier transform

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CS 484, Spring 2010©2010, Selim Aksoy10 Fourier transform Adapted from Alexei Efros, CMU

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CS 484, Spring 2010©2010, Selim Aksoy11 Fourier transform Adapted from Gonzales and Woods

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CS 484, Spring 2010©2010, Selim Aksoy12 Fourier transform Adapted from Gonzales and Woods

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CS 484, Spring 2010©2010, Selim Aksoy13 Fourier transform

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CS 484, Spring 2010©2010, Selim Aksoy14 Fourier transform

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CS 484, Spring 2010©2010, Selim Aksoy15 Fourier transform Adapted from Shapiro and Stockman

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CS 484, Spring 2010©2010, Selim Aksoy16 Fourier transform Example building patterns in a satellite image and their Fourier spectrum.

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CS 484, Spring 2010©2010, Selim Aksoy17 Convolution theorem

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CS 484, Spring 2010©2010, Selim Aksoy18 Frequency domain filtering Adapted from Shapiro and Stockman, and Gonzales and Woods

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CS 484, Spring 2010©2010, Selim Aksoy19 Frequency domain filtering Since the discrete Fourier transform is periodic, padding is needed in the implementation to avoid aliasing (see section 4.6 in the Gonzales-Woods book for implementation details).

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CS 484, Spring 2010©2010, Selim Aksoy20 Frequency domain filtering f(x,y) h(x,y) g(x,y) |F(u,v)| |H(u,v)| |G(u,v)| Adapted from Alexei Efros, CMU

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CS 484, Spring 2010©2010, Selim Aksoy21 Smoothing frequency domain filters Adapted from Gonzales and Woods

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CS 484, Spring 2010©2010, Selim Aksoy22 Smoothing frequency domain filters

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CS 484, Spring 2010©2010, Selim Aksoy23 Smoothing frequency domain filters The blurring and ringing caused by the ideal low- pass filter can be explained using the convolution theorem where the spatial representation of a filter is given below.

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CS 484, Spring 2010©2010, Selim Aksoy24 Smoothing frequency domain filters Adapted from Gonzales and Woods

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CS 484, Spring 2010©2010, Selim Aksoy25 Smoothing frequency domain filters

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CS 484, Spring 2010©2010, Selim Aksoy26 Sharpening frequency domain filters

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CS 484, Spring 2010©2010, Selim Aksoy27 Sharpening frequency domain filters Adapted from Gonzales and Woods

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CS 484, Spring 2010©2010, Selim Aksoy28 Sharpening frequency domain filters Adapted from Gonzales and Woods

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CS 484, Spring 2010©2010, Selim Aksoy29 Sharpening frequency domain filters Adapted from Gonzales and Woods

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CS 484, Spring 2010©2010, Selim Aksoy30 Template matching Correlation can also be used for matching. If we want to determine whether an image f contains a particular object, we let h be that object (also called a template) and compute the correlation between f and h. If there is a match, the correlation will be maximum at the location where h finds a correspondence in f. Preprocessing such as scaling and alignment is necessary in most practical applications.

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CS 484, Spring 2010©2010, Selim Aksoy31 Template matching Adapted from Gonzales and Woods

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CS 484, Spring 2010©2010, Selim Aksoy32 Template matching Face detection using template matching: face templates.

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CS 484, Spring 2010©2010, Selim Aksoy33 Template matching Face detection using template matching: detected faces.

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CS 484, Spring 2010©2010, Selim Aksoy34 Resizing images How can we generate a half-sized version of a large image? Adapted from Steve Seitz, U of Washington

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CS 484, Spring 2010©2010, Selim Aksoy35 Resizing images Throw away every other row and column to create a 1/2 size image (also called sub-sampling). 1/4 1/8 Adapted from Steve Seitz, U of Washington

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CS 484, Spring 2010©2010, Selim Aksoy36 Resizing images Does this look nice? 1/4 (2x zoom)1/8 (4x zoom)1/2 Adapted from Steve Seitz, U of Washington

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CS 484, Spring 2010©2010, Selim Aksoy37 Resizing images We cannot shrink an image by simply taking every k’th pixel. Solution: smooth the image, then sub-sample. Gaussian 1/4 Gaussian 1/8 Gaussian 1/2 Adapted from Steve Seitz, U of Washington

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CS 484, Spring 2010©2010, Selim Aksoy38 Resizing images Gaussian 1/4 (2x zoom) Gaussian 1/8 (4x zoom) Gaussian 1/2 Adapted from Steve Seitz, U of Washington

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CS 484, Spring 2010©2010, Selim Aksoy39 Sampling and aliasing Adapted from Steve Seitz, U of Washington

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CS 484, Spring 2010©2010, Selim Aksoy40 Sampling and aliasing Errors appear if we do not sample properly. Common phenomenon: High spatial frequency components of the image appear as low spatial frequency components. Examples: Wagon wheels rolling the wrong way in movies. Checkerboards misrepresented in ray tracing. Striped shirts look funny on color television.

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CS 484, Spring 2010©2010, Selim Aksoy41 Gaussian pyramids Adapted from Gonzales and Woods

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CS 484, Spring 2010©2010, Selim Aksoy42 Gaussian pyramids Adapted from Michael Black, Brown University

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CS 484, Spring 2010©2010, Selim Aksoy43 Gaussian pyramids Adapted from Michael Black, Brown University

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