1. Convolution

2. 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)

3. The Convolution Theorem
and similarly:

4. Examples
What is the Fourier Transform of ? *

5. Image Domain
Frequency Domain

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

7. Multi-Resolution Image Representation
Gaussian pyramids
Laplacian Pyramids
Wavelet Pyramids

8. Image Pyramid
Low resolution
High resolution

9. Fast Pattern Matching Also good for: - motion analysis
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Also good for:
- motion analysis
- image compression
- other applications

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

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

12. 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).

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

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