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

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CS 484, Spring 2011©2011, 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). 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 2011©2011, 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 2011©2011, Selim Aksoy4 Fourier transform

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CS 484, Spring 2011©2011, Selim Aksoy5 Fourier transform

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CS 484, Spring 2011©2011, Selim Aksoy6 Fourier transform

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CS 484, Spring 2011©2011, Selim Aksoy7 Fourier transform

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CS 484, Spring 2011©2011, Selim Aksoy8 Fourier transform

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CS 484, Spring 2011©2011, Selim Aksoy9 Fourier transform

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CS 484, Spring 2011©2011, Selim Aksoy10 To get some sense of what basis elements look like, we plot a basis element --- or rather, its real part --- as a function of x,y for some fixed u, v. We get a function that is constant when (ux+vy) is constant. The magnitude of the vector (u, v) gives a frequency, and its direction gives an orientation. The function is a sinusoid with this frequency along the direction, and constant perpendicular to the direction. u v Adapted from Antonio Torralba

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Fourier transform CS 484, Spring 2011©2011, Selim Aksoy11 Adapted from Antonio Torralba Here u and v are larger than in the previous slide. u v

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Fourier transform CS 484, Spring 2011©2011, Selim Aksoy12 Adapted from Antonio Torralba And larger still... u v

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CS 484, Spring 2011©2011, Selim Aksoy13 Fourier transform Adapted from Alexei Efros, CMU

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CS 484, Spring 2011©2011, Selim Aksoy14 Fourier transform Adapted from Gonzales and Woods

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CS 484, Spring 2011©2011, Selim Aksoy15 Fourier transform Adapted from Gonzales and Woods

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CS 484, Spring 2011©2011, Selim Aksoy16 Fourier transform

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CS 484, Spring 2011©2011, Selim Aksoy17 Fourier transform

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CS 484, Spring 2011©2011, Selim Aksoy18 Adapted from Antonio Torralba Horizontal orientation Vertical orientation 45 deg. 0f max 0 fx in cycles/image Low spatial frequencies High spatial frequencies Log power spectrum How to interpret a Fourier spectrum:

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Fourier transform CS 484, Spring 2011©2011, Selim Aksoy19 Adapted from Antonio Torralba AB C 12 3

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CS 484, Spring 2011©2011, Selim Aksoy20 Fourier transform Adapted from Shapiro and Stockman

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CS 484, Spring 2011©2011, Selim Aksoy21 Fourier transform Example building patterns in a satellite image and their Fourier spectrum.

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CS 484, Spring 2011©2011, Selim Aksoy22 Convolution theorem

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CS 484, Spring 2011©2011, Selim Aksoy23 Frequency domain filtering Adapted from Shapiro and Stockman, and Gonzales and Woods

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CS 484, Spring 2011©2011, Selim Aksoy24 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 2011©2011, Selim Aksoy25 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 2011©2011, Selim Aksoy26 Smoothing frequency domain filters

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CS 484, Spring 2011©2011, Selim Aksoy27 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 2011©2011, Selim Aksoy28 Smoothing frequency domain filters

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CS 484, Spring 2011©2011, Selim Aksoy29 Sharpening frequency domain filters

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CS 484, Spring 2011©2011, Selim Aksoy30 Sharpening frequency domain filters Adapted from Gonzales and Woods

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CS 484, Spring 2011©2011, Selim Aksoy31 Sharpening frequency domain filters Adapted from Gonzales and Woods

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CS 484, Spring 2011©2011, Selim Aksoy32 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 2011©2011, Selim Aksoy33 Template matching Adapted from Gonzales and Woods

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CS 484, Spring 2011©2011, Selim Aksoy34 Template matching Face detection using template matching: face templates.

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CS 484, Spring 2011©2011, Selim Aksoy35 Template matching Face detection using template matching: detected faces.

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CS 484, Spring 2011©2011, Selim Aksoy36 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 2011©2011, Selim Aksoy37 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 2011©2011, Selim Aksoy38 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 2011©2011, Selim Aksoy39 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 2011©2011, Selim Aksoy40 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 2011©2011, Selim Aksoy41 Sampling and aliasing Adapted from Steve Seitz, U of Washington

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CS 484, Spring 2011©2011, Selim Aksoy42 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 2011©2011, Selim Aksoy43 Gaussian pyramids Adapted from Gonzales and Woods

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CS 484, Spring 2011©2011, Selim Aksoy44 Gaussian pyramids Adapted from Michael Black, Brown University

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CS 484, Spring 2011©2011, Selim Aksoy45 Gaussian pyramids Adapted from Michael Black, Brown University

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