1. Vector Spaces Space of vectors, closed under addition and scalar multiplication

2. Image Averaging as Vector addition

3. Scaler product, dot product, norm

4. Norm of Images

5. Orthogonal Images, Distance,Basis

6. Roberts Basis: 2x2 Orthogonal

8. Cauchy Schwartz Inequality  U+V  ≤  U  +  V 

9. Schwartz Inequality

10. Quotient: Angle Between two images

11. Fourier Analysis

12. Fourier Transform Pair Given image I(x,y), its fourier transform is

13. Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

14. Complex Arithmetic

15. Fourier Traansform of an Image is a complex matrix Let F =[F(u,v)] F = Φ MM I(x,y) Φ NN I(x,y)= Φ* MM F Φ* MM Where Φ JJ (k,l)= [Φ JJ (k,l) ] and Φ JJ (k,l) = (1/J) exp(2Πjkl/J) for k,l= 0,…,J-1

16. Fourier Transform

17. Properties Convolution Given the FT pair of an image f(x,y) F(u,v) and mask pair h(x,y) H(u,v) f(x,y)* h(x,y) F(u,v). H(u,v) and f(x,y) h(x,y) F(u,v)* H(u,v)

18. Properties of Fourier Transform

22. Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

23. Design of H(u,v) İdeal Low Pass filter H(u,v) = 1 if |u,v |< r 0 o.w. Ideal High pass filter H(u,v) = 1 if |u,v |> r 0 o.w Ideal Band pass filter H(u,v) = 1 if r1<|u,v |< r2 0 o.w

24. İmage Enhancement Spatial Smoothing Low Pass Filtering

25. Ideal Low pass filter

26. Ideal Low Pass Filter

27. Output of the Ideal Low Pass Filter

28. Gaussian Low Pass Filyer

29. Gaussian Low Pass Filter

32. High Pass Filter: Ideal and Gaussian

33. Ideal High Pass

34. Fourier Transform-High Pas Filtering

35. Frequency Spectrum of Damaged Circuit

36. Gaussian Low Pass and High Pass

37. Output of Gaussian High Pass

38. Gaussian Filters: Space and Frequency Domain

39. Spatial Laplacian Masks and its Fourier Transform

40. Laplacian Filter

41. Laplacian Filtering