1. Convolution

2. Spatial Filtering Operations g(x,y) = 1/M  f(n,m) (n,m) in  S Example 3 x 3 5 x 5

3. Salt & Pepper Noise 3 X 3 Average5 X 5 Average 7 X 7 AverageMedian Noise Cleaning

4. Salt & Pepper Noise 3 X 3 Average5 X 5 Average 7 X 7 AverageMedian Noise Cleaning

5. x derivative Gradient magnitude y derivative

6. Vertical edges Horizontal edges Edge Detection Image

7. 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)

8. The Convolution Theorem and similarly:

9. What is the Fourier Transform of ? Examples *

10. Image DomainFrequency Domain

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

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

13. Image Pyramid High resolution Low resolution

14. search Fast Pattern Matching

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

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

17. - = 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).

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

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

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