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. GaussianBox filter Gaussian 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“Hybrid Images,”

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. How is it that a 4MP image can be compressed to a few hundred KB without a noticeable change? Compression

8. Thinking in terms of frequency

9. Jean Baptiste Joseph Fourier 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 Lagrange Legendre

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

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

12. Frequency spectra

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

20. Fourier analysis in images Intensity Image Fourier Image

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

22. 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:

23. Computing Fourier transform Continuous Discrete k = -N/2..N/2 Fast Fourier Transform (FFT): NlogN

24. The convolution theorem 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!

25. Properties of Fourier transform Linearity Fourier transform of a real signal is symmetric about the origin The energy of the signal is the same as the energy of its Fourier transform See Szeliski Book (3.4)

26. 2D FFT Fourier transform (discrete case) Inverse Fourier transform: u, v : the transform or frequency variables x, y : the spatial or image variables Euler’s formula

27. 2D FFT Sinusoid with frequency = 1 and its FFT

28. 2D FFT Sinusoid with frequency = 3 and its FFT

29. 2D FFT Sinusoid with frequency = 5 and its FFT

30. 2D FFT Sinusoid with frequency = 10 and its FFT

31. 2D FFT Sinusoid with frequency = 15 and its FFT

32. 2D FFT Sinusoid with varying frequency and their FFT

33. Rotation Sinusoid rotated at 30 degrees and its FFT

34. 2D FFT Sinusoid rotated at 60 degrees and its FFT

35. 2D FFT

36. Image analysis with FFT

38. Filtering in spatial domain 01 -202 01 * =

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

40. 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

41. Questions Which has more information, the phase or the magnitude? What happens if you take the phase from one image and combine it with the magnitude from another image?

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

43. Gaussian

44. Box Filter

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

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

47. 1D example (sinewave): Aliasing problem

48. 1D example (sinewave): Aliasing problem

49. 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” Aliasing problem

50. Aliasing in video

51. Source: A. Efros Aliasing in graphics

52. 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

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

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

55. Anti-aliasing Solutions: Sample more often Get rid of all frequencies that are greater than half the new sampling frequency –Will lose information –But it’s better than aliasing –Apply a smoothing filter

56. Algorithm for downsampling by factor of 2 1.Start with image(h, w) 2.Apply low-pass filter im_blur = imfilter(image, fspecial(‘gaussian’, 7, 1)) 3.Sample every other pixel im_small = im_blur(1:2:end, 1:2:end);

57. Sampling without smoothing. Top row shows the images, sampled at every second pixel to get the next; bottom row shows the magnitude spectrum of these images. substantial aliasing

58. Sampling with smoothing (small σ). Top row shows the images. We get the next image by smoothing the image with a Gaussian with σ=1 pixel, then sampling at every second pixel to get the next; bottom row shows the magnitude spectrum of these images. Low pass filter suppresses high frequency components with less aliasing reducing aliasing

59. Sampling with smoothing (large σ). Top row shows the images. We get the next image by smoothing the image with a Gaussian with σ=2 pixels, then sampling at every second pixel to get the next; bottom row shows the magnitude spectrum of these images. Large σ  less aliasing, but with little detail Gaussian is not an ideal low-pass filter lose details

60. Subsampling without pre- filtering 1/4 (2x zoom) 1/8 (4x zoom) 1/2

61. Subsampling with Gaussian pre-filtering G 1/4G 1/8Gaussian 1/2

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

63. Why do we get different, distance- dependent interpretations of hybrid images? ?

64. Salvador Dali invented Hybrid Images? Salvador Dali “Gala Contemplating the Mediterranean Sea, which at 30 meters becomes the portrait of Abraham Lincoln”, 1976

67. Application: Hybrid images Combine low-frequency of one image with high-frequency of another one Sad faces when looked closely Happy faces when looked a few meters away A. Oliva, A. Torralba, P.G. Schyns, “Hybrid Images,” SIGGRAPH 2006“Hybrid Images,”

68. Early processing in humans filters for various orientations and scales of frequency Perceptual cues in the mid-high 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 Clues from human perception

69. Campbell-Robson contrast sensitivity curve Detect slight change in shades of gray before they become indistinguishable Contrast sensitivity is better for mid-range spatial frequencies

70. Hybrid image in FFT Hybrid ImageLow-passed ImageHigh-passed Image

71. Perception Why do we get different, distance-dependent interpretations of hybrid images? ?

72. Things to Remember Sometimes it makes sense to think of images and filtering in the frequency domain –Fourier analysis Can be faster to filter using FFT for large images (N logN vs. N 2 for auto-correlation) Images are mostly smooth –Basis for compression Remember to low-pass before sampling

73. Practice question 1. Match the spatial domain image to the Fourier magnitude image 1 54 A 3 2 C B D E