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. What is the Fourier Transform of ? Examples *

5. Image DomainFrequency Domain

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

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

8. Image Pyramid High resolution Low resolution

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

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

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

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 (CT) 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

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