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 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? ?

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) = +

12. Frequency spectra

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

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. Phase and magnitude Fourier transform of a real function is complex –difficult to plot, visualize –instead, we can think of the phase and magnitude of the transform Phase is the phase of the complex transform Magnitude is the magnitude of the complex transform Curious fact –all natural images have about the same magnitude transform –hence, phase seems to matter, but magnitude largely doesn’t Demonstration –Take two pictures, swap the phase transforms, compute the inverse - what does the result look like?

26. This is the magnitude transform of the cheetah pic

27. This is the phase transform of the cheetah pic

29. This is the magnitude transform of the zebra pic

30. This is the phase transfor m of the zebra pic

31. Reconstructio n with zebra phase, cheetah magnitude

32. Reconstructi on with cheetah phase, zebra magnitude

33. 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!

34. 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)

35. 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

36. 2D FFT Sinusoid with frequency = 1 and its FFT

37. 2D FFT Sinusoid with frequency = 3 and its FFT

38. 2D FFT Sinusoid with frequency = 5 and its FFT

39. 2D FFT Sinusoid with frequency = 10 and its FFT

40. 2D FFT Sinusoid with frequency = 15 and its FFT

41. 2D FFT Sinusoid with varying frequency and their FFT

42. Rotation Sinusoid rotated at 30 degrees and its FFT

43. 2D FFT Sinusoid rotated at 60 degrees and its FFT

44. 2D FFT

45. Image analysis with FFT

46. http://www.cs.unm.edu/~brayer/vision/fourier.html

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

48. Filtering in frequency domain FFT Inverse FFT =

49. 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

50. 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?

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

52. Gaussian Gaussian filter

53. Box Filter Box filter

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

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

56. 1D example (sinewave): Aliasing problem

57. 1D example (sinewave): Aliasing problem

58. 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

59. Aliasing in video

60. Aliasing in graphics

61. 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

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

63. 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 from the sampled version without aliasing good bad vvv Nyquist-Shannon Sampling Theorem

64. 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

65. 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);

66. 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 Magnitude of the Fourier transform of each image displayed as a log scale (constant component is at the center). Fourier transform of a resampled image is obtained by scaling the Fourier transform of the original image and then tiling the plane

67. 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

68. 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

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

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

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

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

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

76. 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,”

77. 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

78. 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

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

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

81. 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

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

83. Practice question 1. Match the spatial domain image to the Fourier magnitude image 1 54 A 3 2 C B D E 1 – D, 2 – B, 3 – A, 4 – E, 5 - C