Recap of Friday

2. Recap of Friday linear Filtering convolution differential filters filter types boundary conditions.

3. Review: questions • Write down a 3x3 filter that returns a positive value if the average value of the 4-adjacent neighbors is less than the center and a negative value otherwise • Write down a filter that will compute the gradient in the x-direction: • gradx(y,x) = im(y,x+1)-im(y,x) for each x, y Slide: Hoiem

4. Review: questions Filtering Operator a) _ = D * B b) A = _ * _ c) F = D * _ d) _ = D * D • 3. Fill in the blanks: A B E G C F D H I Slide: Hoiem

5. The Frequency Domain (Szeliski 3.4) Somewhere in Cinque Terre, May 2005 CS129: Computational Photography James Hays, Brown, Spring 2011 Slides from Steve Seitzand Alexei Efros

6. 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

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

10. Jean Baptiste Joseph Fourier (1768-1830) • had crazy idea (1807): • Any periodic 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 true! • called Fourier Series

11. 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?

12. 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? Using complex numbers. We can always go back:

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

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

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

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

17. Frequency Spectra = + =

18. Frequency Spectra = + =

19. Frequency Spectra = + =

20. Frequency Spectra = + =

21. Frequency Spectra = + =

22. Frequency Spectra =

23. Frequency Spectra

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

25. Man-made Scene

26. Can change spectrum, then reconstruct

27. Low and High Pass filtering

28. 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

29. 2D convolution theorem example |F(sx,sy)| f(x,y) * h(x,y) |H(sx,sy)| g(x,y) |G(sx,sy)|

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

31. FFT in Matlab • Filtering with fft • Displaying with fft im = double(imread(‘…'))/255; im = rgb2gray(im); % “im” should be a gray-scale floating point image [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(1), imagesc(log(abs(fftshift(im_fft)))), axis image, colormap jet Slide: Hoiem

32. Fourier Transform pairs

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

34. Edges in images

35. What does blurring take away? original

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

37. High-Pass filter smoothed – original

38. The gradient direction is given by: • how does this relate to the direction of the edge? • The edge strength is given by the gradient magnitude Image gradient • The gradient of an image: • The gradient points in the direction of most rapid change in intensity

39. Effects of noise • Consider a single row or column of the image • Plotting intensity as a function of position gives a signal How to compute a derivative? Where is the edge?

40. Look for peaks in Solution: smooth first Where is the edge?

41. Derivative theorem of convolution • This saves us one operation:

42. Laplacian of Gaussian is the Laplacian operator: 2D edge detection filters Gaussian derivative of Gaussian

43. Campbell-Robson contrast sensitivity curve

44. Depends on Color R G B

45. Lossy Image Compression (JPEG) Block-based Discrete Cosine Transform (DCT)

46. Using DCT in JPEG • The first coefficient B(0,0) is the DC component, the average intensity • The top-left coeffs represent low frequencies, the bottom right – high frequencies

47. Image compression using DCT • DCT enables image compression by concentrating most image information in the low frequencies • Lose unimportant image info (high frequencies) by cutting B(u,v) at bottom right • The decoder computes the inverse DCT – IDCT • Quantization Table • 3 5 7 9 11 13 15 17 • 5 7 9 11 13 15 17 19 • 7 9 11 13 15 17 19 21 • 9 11 13 15 17 19 21 23 • 11 13 15 17 19 21 23 25 • 13 15 17 19 21 23 25 27 • 15 17 19 21 23 25 27 29 • 17 19 21 23 25 27 29 31

48. JPEG compression comparison 89k 12k

49. 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. N2 for auto-correlation) • Images are mostly smooth • Basis for compression • Remember to low-pass before sampling

50. Summary Frequency domain can be useful for Analysis Computational efficiency Compression