Unlocking Image Frequencies: Discover the Magic of Fourier Transform

1. The Frequency Domain Somewhere in Cinque Terre, May 2005 15-463: Computational Photography Alexei Efros, CMU, Fall 2012 Many slides borrowed from Steve Seitz

2. Salvador Dali “Gala Contemplating the Mediterranean Sea, which at 30 meters becomes the portrait of Abraham Lincoln”, 1976 Salvador Dali, “Gala Contemplating the Mediterranean Sea, which at 30 meters becomes the portrait of Abraham Lincoln”, 1976 Salvador Dali, “Gala Contemplating the Mediterranean Sea, which at 30 meters becomes the portrait of Abraham Lincoln”, 1976

5. A nice set of basis Teases away fast vs. slow changes in the image. This change of basis has a special name…

6. Jean Baptiste Joseph Fourier (1768-1830) 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 ...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. Laplace Legendre Lagrange

7. A sum of sines • Our building block: • Add enough of them to get any signal f(x) you want! • How many degrees of freedom? • What does each control? • Which one encodes the coarse vs. fine structure of the signal?

8. Inverse Fourier Transform Fourier Transform F(w) f(x) F(w) f(x) Fourier Transform • We want to understand the frequency w of our signal. So, let’s reparametrize the signal by w instead of x: • For every w from 0 to inf, F(w) holds the amplitude A and phase f of the corresponding sine • How can F hold both? Complex number trick! We can always go back:

9. Time and Frequency • example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)

10. Time and Frequency • example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t) = +

11. Frequency Spectra • example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t) = +

12. Frequency Spectra • Usually, frequency is more interesting than the phase

13. Frequency Spectra = + =

14. Frequency Spectra = + =

15. Frequency Spectra = + =

16. Frequency Spectra = + =

17. Frequency Spectra = + =

18. Frequency Spectra =

19. Frequency Spectra

20. FT: Just a change of basis M * f(x) = F(w) = * . . .

21. IFT: Just a change of basis M-1 * F(w)= f(x) = * . . .

22. Finally: Scary Math

23. Finally: Scary Math • …not really scary: • is hiding our old friend: • So it’s just our signal f(x) times sine at frequency w phase can be encoded by sin/cos pair

24. Extension to 2D in Matlab, check out: imagesc(log(abs(fftshift(fft2(im)))));

25. Fourier analysis in images Intensity Image Fourier Image http://sharp.bu.edu/~slehar/fourier/fourier.html#filtering

26. Signals can be composed + = http://sharp.bu.edu/~slehar/fourier/fourier.html#filtering More: http://www.cs.unm.edu/~brayer/vision/fourier.html

27. Man-made Scene

28. Can change spectrum, then reconstruct

29. Low and High Pass filtering

30. The greatest thing since sliced (banana) bread! The Fourier transform of the convolution of two functions is the product of their Fourier transforms The inverse Fourier transform of the product of two Fourier transforms is the convolution of the two inverse Fourier transforms Convolution in spatial domain is equivalent to multiplication in frequency domain! The Convolution Theorem

31. 2D convolution theorem example |F(sx,sy)| f(x,y) * h(x,y) |H(sx,sy)| g(x,y) |G(sx,sy)|

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

33. Gaussian

34. Box Filter

35. Fourier Transform pairs

36. Low-pass, Band-pass, High-pass filters low-pass: High-pass / band-pass:

37. Edges in images

38. What does blurring take away? original

39. What does blurring take away? smoothed (5x5 Gaussian)

40. High-Pass filter smoothed – original

41. Band-pass filtering • Laplacian Pyramid (subband images) • Created from Gaussian pyramid by subtraction Gaussian Pyramid (low-pass images)

42. Laplacian Pyramid • How can we reconstruct (collapse) this pyramid into the original image? Need this! Original image

43. Why Laplacian? Gaussian Laplacian of Gaussian delta function

44. Project 2: Hybrid Images unit impulse Laplacian of Gaussian Gaussian Gaussian Filter! Laplacian Filter! A. Oliva, A. Torralba, P.G. Schyns, “Hybrid Images,” SIGGRAPH 2006 Project Instructions: http://www.cs.illinois.edu/class/fa10/cs498dwh/projects/hybrid/ComputationalPhotography_ProjectHybrid.html

45. Clues from Human Perception Early processing in humans filters for various orientations and scales of frequency Perceptual cues in the mid frequencies dominate perception When we see an image from far away, we are effectively subsampling it Early Visual Processing: Multi-scale edge and blob filters

46. Frequency Domain and Perception Campbell-Robson contrast sensitivity curve

47. Da Vinci and Peripheral Vision

48. Leonardo playing with peripheral vision

49. - = + a = Unsharp Masking

50. Freq. Perception Depends on Color R G B