CSE 185 Introduction to Computer Vision

1. CSE 185 Introduction to Computer Vision Image Filtering: Frequency Domain

2. Image filtering • Fourier transform and frequency domain • Frequency view of filtering • Hybrid images • Sampling • Today’s lecture covers material in 3.4 • Reading: Chapters 3 • Some slides from James Hays, David Hoeim, Steve Seitz, …

3. Gaussian filter Gaussian Box filter

4. Hybrid Images • Filter one image with low-pass filter and the other with high-pass filter, and then superimpose them • A. Oliva, A. Torralba, P.G. Schyns, “Hybrid Images,” SIGGRAPH 2006

5. Why do we get different, distance-dependent interpretations of hybrid images? ? Slide: Hoiem

6. Why does a lower resolution image still make sense to us? What do we lose?

7. Thinking in terms of frequency

8. Jean Baptiste Joseph Fourier ...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour. had crazy idea (1807): Any univariate function can be rewritten as a weighted sum of sines and cosines of different frequencies. • Don’t believe it? • Neither did Lagrange, Laplace, Poisson and other big wigs • Not translated into English until 1878! • But it’s (mostly) true! • called Fourier Series • there are some subtle restrictions Laplace Legendre Lagrange

9. A sum of sines Our building block: Add enough of them to get any signal g(x) you want!

10. Frequency spectra • example : g(t) = sin(2πf t) + (1/3)sin(2π(3f) t) = + Slides: Efros

11. Frequency spectra

12. Frequency spectra = + =

13. Frequency spectra = + =

14. Frequency spectra = + =

15. Frequency spectra = + =

16. Frequency spectra = + =

17. Frequency spectra =

18. Example: Music • We think of music in terms of frequencies at different magnitudes Slide: Hoiem

19. Fourier analysis in images Intensity Image Fourier Image

20. Signals can be composed + = More: http://www.cs.unm.edu/~brayer/vision/fourier.html

21. Fourier transform • Fourier transform stores the magnitude and phase at each frequency • Magnitude encodes how much signal there is at a particular frequency • Phase encodes spatial information (indirectly) • For mathematical convenience, this is often notated in terms of real and complex numbers Amplitude: Phase:

22. The Fourier transform of the convolution of two functions is the product of their Fourier transforms Convolution in spatial domain is equivalent to multiplication in frequency domain! The convolution theorem

23. Fourier transform (discrete case) Inverse Fourier transform: u, v : the transform or frequency variables x, y : the spatial or image variables 2D FFT Euler’s formula

24. 2D FFT Sinusoid with frequency = 1 and its FFT

25. 2D FFT Sinusoid with frequency = 3 and its FFT

26. 2D FFT Sinusoid with frequency = 5 and its FFT

27. 2D FFT Sinusoid with frequency = 10 and its FFT

28. 2D FFT Sinusoid with frequency = 15 and its FFT

29. 2D FFT Sinusoid with varying frequency and their FFT

30. Rotation Sinusoid rotated at 30 degrees and its FFT

31. 2D FFT Sinusoid rotated at 60 degrees and its FFT

32. 2D FFT

33. Image analysis with FFT

34. Image analysis with FFT

35. 1 0 -1 2 0 -2 1 0 -1 Filtering in spatial domain * =

36. Filtering in frequency domain FFT FFT = Inverse FFT Slide: Hoiem

37. FFT in Matlab • Filtering with fft • Displaying with fft im = double(imread('cameraman.tif'))/255; [imh, imw] = size(im); hs = 50; % filter half-size fil = fspecial('gaussian', hs*2+1, 10); fftsize = 1024; % should be order of 2 (for speed) and include padding im_fft = fft2(im, fftsize, fftsize); % 1) fft im with padding fil_fft = fft2(fil, fftsize, fftsize); % 2) fft fil, pad to same size as image im_fil_fft = im_fft .* fil_fft; % 3) multiply fft images im_fil = ifft2(im_fil_fft); % 4) inverse fft2 im_fil = im_fil(1+hs:size(im,1)+hs, 1+hs:size(im, 2)+hs); % 5) remove padding figure, imshow(im); figure, imshow(im_fil); figure(1), imagesc(log(abs(fftshift(im_fft)))), axis image, colormap jet

38. Filtering Why does the Gaussian give a nice smooth image, but the square filter give edgy artifacts? Gaussian Box filter

39. Gaussian

40. Box Filter

41. Sampling Why does a lower resolution image still make sense to us? What do we lose?

42. Subsampling by a factor of 2 Throw away every other row and column to create a 1/2 size image

43. Aliasing problem • 1D example (sinewave):

44. Aliasing problem • 1D example (sinewave):

45. Aliasing problem • Sub-sampling may be dangerous…. • Characteristic errors may appear: • “Wagon wheels rolling the wrong way in movies” • “Checkerboards disintegrate in ray tracing” • “Striped shirts look funny on color television”

46. Aliasing in video

47. Aliasing in graphics Source: A. Efros

48. Resample the checkerboard by taking one sample at each circle. In the case of the top left board, new representation is reasonable. Top right also yields a reasonable representation. Bottom left is all black (dubious) and bottom right has checks that are too big. Sampling scheme is crucially related to frequency

49. Constructing a pyramid by taking every second pixel leads to layers that badly misrepresent the top layer

50. Nyquist-Shannon Sampling Theorem • When sampling a signal at discrete intervals, the sampling frequency must be  2  fmax • fmax = max frequency of the input signal • This will allows to reconstruct the original perfectly from the sampled version good v v v bad