1. BYST Xform-1 DIP - WS2002: Fourier Transform Digital Image Processing Bundit Thipakorn, Ph.D. Computer Engineering Department Fourier Transform and Image Filtering in the Frequency Domain

2. BYST Xform-2 DIP - WS2002: Fourier Transform Image Transforms Used of Image Transforms: To reorder or rearrange the information in the image in such the way that the information is easier to perform by the image processing operations.

3. BYST Xform-3 DIP - WS2002: Fourier Transform Image Transforms The common steps of using the image transforms in the image processing process. Forward Transform Image f(x,y) Image Processing Process Inverse Transform Processed Image f’(x,y)

4. BYST Xform-4 DIP - WS2002: Fourier Transform Image Transforms A Generic Image Transform Equation: T(u,v) = wheref(x,y) is the input N x N image T(u,v) are the transform coefficients andB(x,y; u,v) is a set of basis images.

5. BYST Xform-5 DIP - WS2002: Fourier Transform Image Transforms A Generic Inverse Transform Equation: f(x,y) = wheref(x,y) is the input N x N image T(u,v) are the transform coefficients andB -1 (x,y; u,v) is a set of inverse basis images.

6. BYST Xform-6 DIP - WS2002: Fourier Transform Similarity Index: T(u,v) A high value of T(u,v) f(x,y) and B(x,y; u,v) are alike. A low value of T(u,v) f(x,y) and B(x,y; u,v) are difference.

7. BYST Xform-7 DIP - WS2002: Fourier Transform Fourier Transform Definition: F(u,v) = Forward Fourier Transform Inverse Fourier Transform f(x,y) =

8. BYST Xform-8 DIP - WS2002: Fourier Transform Fourier Transform wheref(x,y) is the input N x N image F(u,v) are the Fourier transform coefficients is a set of basis images. is a set of inverse basis images. By Euler's identity: = - j

9. BYST Xform-9 DIP - WS2002: Fourier Transform Fourier Transform decompose an image into its sine and cosine components. the transformed image is represented as a set of spatial frequencies |F(u,v)|. i.e.

10. BYST Xform-10 DIP - WS2002: Fourier Transform The frequency of the sinusoidal is given by the distance of the point (u,v) from the origin: Fourier Transform and the orientation is given by:

11. BYST Xform-11 DIP - WS2002: Fourier Transform Frequency = how rapidly the signal is changing in space. Fourier Transform Frequency = 0 Frequency = 1 Frequency = 4

12. BYST Xform-12 DIP - WS2002: Fourier Transform Spatial Domain Frequency Domain

13. BYST Xform-13 DIP - WS2002: Fourier Transform Spatial Domain Frequency Domain

14. BYST Xform-14 DIP - WS2002: Fourier Transform Spatial Domain Frequency Domain

15. BYST Xform-15 DIP - WS2002: Fourier Transform Spatial Domain Frequency Domain A Nature Image A Man-Made Image

16. BYST Xform-16 DIP - WS2002: Fourier Transform Spatial Domain Frequency Domain The Man-Made Texture Images

17. BYST Xform-17 DIP - WS2002: Fourier Transform Fourier Transform The Important Observations: q |F(0,0)| of most images is the highest. q The Fourier transform exhibits the coherent structures reflected some structures in the original image as lines or spokes passing through the origin.

18. BYST Xform-18 DIP - WS2002: Fourier Transform Fourier Transform The Important Observations: q The Fourier transform of the natural scenes tend to contain no coherent structures. q The coherence of the texture in the original image and the structures in the Fourier image is obviously clear.

19. BYST Xform-19 DIP - WS2002: Fourier Transform Power Spectrum: P(u,v) P(u,v) = |F(u,v)| 2 Total Power: P T P T = The image power is concentrated in the low frequency components. Fourier Transform

20. BYST Xform-20 DIP - WS2002: Fourier Transform Fourier Transform Radius (pixels) % image power 816 32 64 1289597 98 99.4 99.8

21. BYST Xform-21 DIP - WS2002: Fourier Transform Some Properties: ► Separability F(u,v) = F(x,v) = where Fourier Transform Properties

22. BYST Xform-22 DIP - WS2002: Fourier Transform ► Translation F(u-u 0, v-v 0 ) f(x-x 0, y-y 0 ) Fourier Transform Properties

23. BYST Xform-23 DIP - WS2002: Fourier Transform ► Periodicity and Conjugate Symmetry F(u,v) = F(u+N, v) = F(u, v+N) = F(u+N, v+N) If f(x,y) is real, |F(u,v)| = F*(-u, -v) or |F(u,v)| = |F(-u, -v)| Fourier Transform Properties

24. BYST Xform-24 DIP - WS2002: Fourier Transform ► Distributivity and Scaling F[f 1 (x,y) + f 2 (x,y)] = F[f 1 (x,y)] + F[f 2 (x,y)] af(x,y)aF(u,v) f(ax,by) Fourier Transform Properties

25. BYST Xform-25 DIP - WS2002: Fourier Transform Fourier Transform ► Convolution Properties f(x,y) * g(x,y)F(u,v)G(u,v) ► Correlation F*(u,v)G(u,v) f(x,y) g(x,y)F(u,v) * G(u,v) f*(x,y)g(x,y)

26. BYST Xform-26 DIP - WS2002: Fourier Transform Discrete Cosine Transform (DCT) Definition: C(u,v) = Forward DCT Transform Inverse DCT Transform f(x,y) =

27. BYST Xform-27 DIP - WS2002: Fourier Transform Definition where DCT DCT coefficients C(u,v) Real numbers Fourier transform coefficients F(u,v) Complex numbers. Fourier V.S. DCT

28. BYST Xform-28 DIP - WS2002: Fourier Transform Walsh and Hadamard Transform Square Wave Fourier and DCT Transforms Basis Function Sinusoidal Wave Walsh and Hadamard Transforms

29. BYST Xform-29 DIP - WS2002: Fourier Transform Walsh Transform Definition: W(u,v) = Forward Walsh Transform Inverse Walsh Transform f(x,y) =

30. BYST Xform-30 DIP - WS2002: Fourier Transform Walsh Transform b k (z) is the kth bit in the binary representation of z. where The set of basic functions g(x,y;u,v) of Walsh transform is defined as following: g(x,y;u,v) = i.e. the value of g(x,y;u,v) will be either +1 or -1. Cont’d.

31. BYST Xform-31 DIP - WS2002: Fourier Transform Walsh Transform Thus Walsh’s basis functions = Square waves where the width of pulse may vary. Note: g(x,y;u,v) is called “sequency component”. Cont’d.

32. BYST Xform-32 DIP - WS2002: Fourier Transform Hadamard Transform Definition: H(u,v) = Forward Hadamard Transform Inverse Hadamard Transform f(x,y) =

33. BYST Xform-33 DIP - WS2002: Fourier Transform Hadamard Transform b k (z) is the kth bit in the binary representation of z. where The set of basic functions g(x,y;u,v) of Hadamard transform is defined as following: g(x,y;u,v) = i.e. the value of g(x,y;u,v) will be either +1 or -1. Cont’d.

34. BYST Xform-34 DIP - WS2002: Fourier Transform Hadamard Transform The Hadamard matrix of lowest order (N=2) is: The Hadamard transform is a symmetric, separable unitary transformation. It exists for N = 2 n (n = integer). and for successively larger 2N, these can be generated recursively by the expression: Cont’d.

35. BYST Xform-35 DIP - WS2002: Fourier Transform Hadamard Transform Ex: The Hadamard matrices of order four and eight are: Cont’d.

36. BYST Xform-36 DIP - WS2002: Fourier Transform Hadamard Transform Cont’d. Where + and - indicate +1 and -1, respectively. Number of Sign Changes 07 3 4 1 6 2 5

37. BYST Xform-37 DIP - WS2002: Fourier Transform Hadamard Transform Cont’d. Where + and - indicate +1 and -1, respectively. Number of Sign Changes 01 2 3 4 5 6 7 The ordered Hadamard transform

38. BYST Xform-38 DIP - WS2002: Fourier Transform Image Filtering in Frequency Domain DFT Image f(x,y) X Inverse DFT Inverse DFTF(u,v) F(u,v)H(u,v) Filter H(u,v) Processed Image f’(x,y)

39. BYST Xform-39 DIP - WS2002: Fourier Transform Image Filtering Cont’d. where F(u,v) is the Fourier transform of the input image to The basic “model” for filtering in the frequency domain is expressed as: be filtered, F’(u,v) is the Fourier transform of the enhanced image, H(u,v) is the Frequency response of the filter.

40. BYST Xform-40 DIP - WS2002: Fourier Transform Image Filtering Cont’d. Select H(u,v) that yields F’(u,v) Objective: Basics of filtering in the frequency domain: 1. Multiply the input image by (-1) x+y to center the transform. 2. Determine H(u,v) and Compute F(u,v).

41. BYST Xform-41 DIP - WS2002: Fourier Transform Image Filtering Cont’d. 3. Multiply F(u,v) with H(u,v). 4. Compute the inverse Fourier transform of the result in (3). 5. Obtain the real part of the result in (4). 6. Multiply the result in (5) by (-1) x+y.

42. BYST Xform-42 DIP - WS2002: Fourier Transform Image Filtering Cont’d. Lowpass Filter H(u,v) = 1 if u 2 + v 2 < r 2 H(u,v) = 0 if u 2 + v 2 ≥ r 2

43. BYST Xform-43 DIP - WS2002: Fourier Transform Image Filtering Cont’d. Highpass Filter H(u,v) = 0 if u 2 + v 2 < r 2 H(u,v) = 1 if u 2 + v 2 ≥ r 2

44. BYST Xform-44 DIP - WS2002: Fourier Transform Image Filtering Cont’d. Band Pass Filter H(u,v) = 1 if a 2 ≤ u 2 + v 2 ≥ b 2 H(u,v) = 0 otherwise