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
[0 -1/4 0;
-1/4 1 -1/4;
0 -1/4 0]
[0 -1 1]
Slide: Hoiem

4. Review: questions 3. Fill in the blanks: Filtering Operator
a) _ = D * B
b) A = _ * _
c) F = D * _
d) _ = D * D
3. Fill in the blanks: A B E G C F
a) G = D * B
b) A = B * C
c) F = D * E
d) I = D * D 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 Seitz and 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. Fourier Transform f(x) F(w) F(w) f(x)
We want to understand the frequency w of our signal. So, let’s reparametrize the signal by w instead of x:
f(x)
F(w)
Fourier
Transform
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.
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
We can always go back:
F(w)
f(x)
Inverse Fourier
Transform

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 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!
complexity argument- n log n vs n k

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

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

40. Why Laplacian?
Gaussian
delta function
Laplacian of Gaussian

41. Unsharp Masking - = =
+ a

42. Image gradient The gradient of an image:
The gradient points in the direction of most rapid change in intensity
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

43. 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?

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

45. Derivative theorem of convolution
This saves us one operation:

46. Laplacian of Gaussian Consider Where is the edge?
operator
Where is the edge?
Zero-crossings of bottom graph

47. 2D edge detection filters
Laplacian of Gaussian
Gaussian
derivative of Gaussian
How many 2nd derivative filters are there? There are four 2nd partial derivative filters.
In practice, it’s handy to define a single 2nd derivative filter—the Laplacian
is the Laplacian operator:

48. Try this in MATLAB g = fspecial('gaussian',15,2);
imagesc(g); colormap(gray);
surfl(g)
gclown = conv2(clown,g,'same');
imagesc(conv2(clown,[-1 1],'same'));
imagesc(conv2(gclown,[-1 1],'same'));
dx = conv2(g,[-1 1],'same');
imagesc(conv2(clown,dx,'same'));
lg = fspecial('log',15,2);
lclown = conv2(clown,lg,'same');
imagesc(lclown)
imagesc(clown + .2*lclown)

49. Campbell-Robson contrast sensitivity curve

50. Depends on Color R G B

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

52. 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

53. 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

54. Block size in JPEG Block size small block large block
faster
correlation exists between neighboring pixels
large block
better compression in smooth regions
It’s 8x8 in standard JPEG

55. JPEG compression comparison
89k
12k

56. 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

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