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Department of Computer Engineering

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

2 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 CS 484, Fall 2017 ©2017, Selim Aksoy

3 Fourier transform CS 484, Fall 2017 ©2017, Selim Aksoy

4 Fourier transform CS 484, Fall 2017 ©2017, Selim Aksoy

5 An illustration By Lucas V. Barbosa - Own work, Public Domain, CS 484, Spring 2017 Bilkent University

6 Fourier transform CS 484, Fall 2017 ©2017, Selim Aksoy

7 Fourier transform CS 484, Fall 2017 ©2017, Selim Aksoy

8 Fourier transform CS 484, Fall 2017 ©2017, Selim Aksoy

9 Fourier transform CS 484, Fall 2017 ©2017, Selim Aksoy

10 Fourier transform 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 CS 484, Fall 2017 ©2017, Selim Aksoy

11 Fourier transform Here u and v are larger than in the previous slide.
Adapted from Antonio Torralba CS 484, Fall 2017 ©2017, Selim Aksoy

12 Fourier transform And larger still... v u
Adapted from Antonio Torralba CS 484, Fall 2017 ©2017, Selim Aksoy

13 Fourier transform Adapted from Alexei Efros, CMU CS 484, Fall 2017
©2017, Selim Aksoy

14 Fourier transform Adapted from Gonzales and Woods CS 484, Fall 2017
©2017, Selim Aksoy

15 Ringing artifact revisited
Adapted from Gonzales and Woods CS 484, Spring 2017 Bilkent University

16 Fourier transform - matlab
A=1; K=10; M=100; t=[ones(1,K)*A zeros(1,M-K)]; subplot(3,1,1); bar(t); ylim([0 2*A]) subplot(3,1,2); bar(abs(fftshift(fft(t)))); ylim([0 A*K+1]) % Matlab uses DFT formulation without normalization by M. subplot(3,1,3); bar(real(fftshift(fft(t)))); ylim([-A*K+1 A*K+1]) Adapted from Gonzales and Woods CS 484, Spring 2017 Bilkent University

17 Fourier transform Adapted from Gonzales and Woods CS 484, Fall 2017
©2017, Selim Aksoy

18 Fourier transform CS 484, Fall 2017 ©2017, Selim Aksoy

19 Fourier transform CS 484, Fall 2017 ©2017, Selim Aksoy

20 Fourier transform How to interpret a Fourier spectrum: fmax
Horizontal orientation Vertical orientation 45 deg. fmax fx in cycles/image Low spatial frequencies High spatial frequencies Log power spectrum Adapted from Antonio Torralba CS 484, Fall 2017 ©2017, Selim Aksoy

21 Fourier transform C A B 1 2 3 Adapted from Antonio Torralba
CS 484, Fall 2017 ©2017, Selim Aksoy

22 Fourier transform Adapted from Shapiro and Stockman CS 484, Fall 2017
©2017, Selim Aksoy

23 Convolution theorem CS 484, Fall 2017 ©2017, Selim Aksoy

24 Frequency domain filtering
Adapted from Shapiro and Stockman, and Gonzales and Woods CS 484, Fall 2017 ©2017, Selim Aksoy

25 Frequency domain filtering
CS 484, Fall 2017 ©2017, Selim Aksoy

26 Frequency domain filtering
f(x,y) h(x,y) g(x,y) |F(u,v)| |H(u,v)| |G(u,v)| CS 484, Fall 2017 ©2017, Selim Aksoy Adapted from Alexei Efros, CMU

27 Smoothing frequency domain filters
CS 484, Fall 2017 ©2017, Selim Aksoy

28 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. CS 484, Fall 2017 ©2017, Selim Aksoy

29 Sharpening frequency domain filters
CS 484, Fall 2017 ©2017, Selim Aksoy

30 Sharpening frequency domain filters
Adapted from Gonzales and Woods CS 484, Fall 2017 ©2017, Selim Aksoy

31 Sharpening frequency domain filters
Adapted from Gonzales and Woods CS 484, Fall 2017 ©2017, Selim Aksoy

32 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. CS 484, Fall 2017 ©2017, Selim Aksoy

33 Template matching Adapted from Gonzales and Woods CS 484, Fall 2017
©2017, Selim Aksoy

34 Face detection using template matching: face templates.
CS 484, Fall 2017 ©2017, Selim Aksoy

35 Face detection using template matching: detected faces.
CS 484, Fall 2017 ©2017, Selim Aksoy

36 Template matching Where is Waldo?
CS 484, Fall 2017 ©2017, Selim Aksoy

37 Resizing images How can we generate a half-sized version of a large image? Adapted from Steve Seitz, U of Washington CS 484, Fall 2017 ©2017, Selim Aksoy

38 Resizing images 1/8 1/4 Throw away every other row and column to create a 1/2 size image (also called sub-sampling). Adapted from Steve Seitz, U of Washington CS 484, Fall 2017 ©2017, Selim Aksoy

39 Resizing images 1/2 1/4 (2x zoom) 1/8 (4x zoom) Does this look nice?
Adapted from Steve Seitz, U of Washington CS 484, Fall 2017 ©2017, Selim Aksoy

40 Resizing images We cannot shrink an image by simply taking every k’th pixel. Solution: smooth the image, then sub-sample. Gaussian 1/8 Gaussian 1/4 Gaussian 1/2 Adapted from Steve Seitz, U of Washington CS 484, Fall 2017 ©2017, Selim Aksoy

41 Resizing images Gaussian 1/2 Gaussian 1/4 (2x zoom)
Adapted from Steve Seitz, U of Washington CS 484, Fall 2017 ©2017, Selim Aksoy

42 Sampling and aliasing Adapted from Steve Seitz, U of Washington
CS 484, Fall 2017 ©2017, Selim Aksoy

43 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. CS 484, Fall 2017 ©2017, Selim Aksoy

44 Sampling and aliasing CS 484, Fall 2017 ©2017, Selim Aksoy
Adapted from Ali Farhadi

45 Gaussian pyramids Adapted from Gonzales and Woods CS 484, Fall 2017
©2017, Selim Aksoy

46 Gaussian pyramids Adapted from Michael Black, Brown University
CS 484, Fall 2017 ©2017, Selim Aksoy

47 Gaussian pyramids Adapted from Michael Black, Brown University
CS 484, Fall 2017 ©2017, Selim Aksoy

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