1. 2D Image
Fourier Spectrum

2. Fourier Transform -- Examples
Image
Fourier spectrum

3. Phase and Magnitude Curious fact Demonstration
All natural images have very similar magnitude transform.
So why do they look different…?
Demonstration
Take two pictures, swap the phase transforms, compute the inverse - what does the result look like?
 Phase in images matters a lot (more than magnitude)

4. Slide: Freeman & Durand

5. Slide: Freeman & Durand

6. Reconstruction with zebra phase, cheetah magnitude
Slide: Freeman & Durand

7. Reconstruction with cheetah phase, zebra magnitude
Slide: Freeman & Durand

8. Convolution

9. Spatial Filtering Operations
Example
3 x 3
h(x,y) = 1/9 S f(n,m)
(n,m) in the 3x3 neighborhood of (x,y)

10. Noise Cleaning Salt & Pepper Noise 3 X 3 Average 5 X 5 Average
Median

11. Noise Cleaning Salt & Pepper Noise 3 X 3 Average 5 X 5 Average
Median

12. Gradient magnitude
x derivative
y derivative

13. Vertical edges
Horizontal edges
Edge Detection
Image

14. Convolution Properties
Commutative:
f*g = g*f
Associative:
(f*g)*h = f*(g*h)
Homogeneous:
f*(g)=  f*g
Additive (Distributive):
f*(g+h)= f*g+f*h
Shift-Invariant
f*g(x-x0,y-yo)= (f*g) (x-x0,y-yo)

15. The Convolution Theorem
and similarly:

16. Going back to the Noise Cleaning example…
3 X 3 Average
Salt & Pepper Noise
Convolution with a rect  Multiplication with a sinc in the Fourier domain
= LPF (Low-Pass Filter)
7 X 7 Average
5 X 5 Average
Wider rect  Narrower sinc = Stronger LPF

17. Examples
What is the Fourier Transform of ? *

18. Image Domain
Frequency Domain

19. (developed on the board) Nyquist frequency, Aliasing, etc…
The Sampling Theorem
(developed on the board)
Nyquist frequency, Aliasing, etc…

20. Multi-Scale Image Representation
Gaussian pyramids
Laplacian Pyramids
Wavelet Pyramids
Good for:
- pattern matching
- motion analysis
- image compression
- other applications

21. Image Pyramid
High resolution
Low resolution

22. Fast Pattern Matching
search
search
search
search

23. The Gaussian Pyramid Low resolution down-sample blur down-sample blur
High resolution

24. - = - = - = The Laplacian Pyramid Gaussian Pyramid Laplacian Pyramid
expand - =
expand - =
expand - =

25. Laplacian ~ Difference of Gaussians- =
DOG = Difference of Gaussians
More details on Gaussian and Laplacian pyramids
can be found in the paper by Burt and Adelson
(link will appear on the website).

26. Computerized Tomography (CT)
f(x,y) u v
F(u,v)

27. Computerized Tomography
Original
(simulated)
2D image
8 projections-
Frequency
Domain
120 projections-
Frequency
Domain
Reconstruction from
8 projections
Reconstruction from
120 projections

28. End of Lesson... Exercise#1 -- will be posted on the website.
(Theoretical exercise: To be done and submitted individually)