1. 240-373: Chapter 14: The Frequency Domain 1 Montri Karnjanadecha montri@coe.psu.ac.th http://fivedots.coe.psu. ac.th/~montri 240-373 Image Processing

2. 240-373: Chapter 14: The Frequency Domain 2 Chapter 14 The Frequency Domain

3. 240-373: Chapter 14: The Frequency Domain 3 The Frequency Domain Any wave shape can be approximated by a sum of periodic (such as sine and cosine) functions. a--amplitude of waveform f-- frequency (number of times the wave repeats itself in a given length) p--phase (position that the wave starts) Usually phase is ignored in image processing

4. 240-373: Chapter 14: The Frequency Domain 4

5. 5

6. 6 The Hartley Transform Discrete Hartley Transform (DHT) –The M x N image is converted into a second image (also M x N) –M and N should be power of 2 (e.g..., 128, 256, 512, etc.) –The basic transform depends on calculating the following for each pixel in the new M x N array

7. 240-373: Chapter 14: The Frequency Domain 7 The Hartley Transform where f(x,y) is the intensity of the pixel at position (x,y) H(u,v) is the value of element in frequency domain –The results are periodic –The cosine+sine (CAS) term is call “the kernel of the transformation” (or ”basis function”)

8. 240-373: Chapter 14: The Frequency Domain 8 The Hartley Transform Fast Hartley Transform (FHT) –M and N must be power of 2 –Much faster than DHT –Equation:

9. 240-373: Chapter 14: The Frequency Domain 9 The Fourier Transform The Fourier transform –Each element has real and imaginary values –Formula: –f(x,y) is point (x,y) in the original image and F(u,v) is the point (u,v) in the frequency image

10. 240-373: Chapter 14: The Frequency Domain 10 The Fourier Transform Discrete Fourier Transform (DFT) –Imaginary part –Real part –The actual complex result is F i (u,v) + F r (u,v)

11. 240-373: Chapter 14: The Frequency Domain 11 Fourier Power Spectrum and Inverse Fourier Transform Fourier power spectrum Inverse Fourier Transform

12. 240-373: Chapter 14: The Frequency Domain 12 Fourier Power Spectrum and Inverse Fourier Transform Fast Fourier Transform (FFT) –Much faster than DFT –M and N must be power of 2 –Computation is reduced from M 2 N 2 to MN log 2 M. log 2 N (~1/1000 times)

13. 240-373: Chapter 14: The Frequency Domain 13 Fourier Power Spectrum and Inverse Fourier Transform Optical transformation –A common approach to view image in frequency domain Original image Transformed image AB DC CD BA

14. 240-373: Chapter 14: The Frequency Domain 14 Power and Autocorrelation Functions Power function: Autocorrelation function –Inverse Fourier transform of or –Hartley transform of

15. 240-373: Chapter 14: The Frequency Domain 15 Hartley vs Fourier Transform

16. 240-373: Chapter 14: The Frequency Domain 16 Interpretation of the power function

17. 240-373: Chapter 14: The Frequency Domain 17 Applications of Frequency Domain Processing Convolution in the frequency domain

18. 240-373: Chapter 14: The Frequency Domain 18 Applications of Frequency Domain Processing –useful when the image is larger than 1024x1024 and the template size is greater than 16x16 –Template and image must be the same size

19. 240-373: Chapter 14: The Frequency Domain 19 –Use FHT or FFT instead of DHT or DFT –Number of points should be kept small –The transform is periodic zeros must be padded to the image and the template minimum image size must be (N+n-1) x (M+m-1) –Convolution in frequency domain is “real convolution” Normal convolution Real convolution

20. 240-373: Chapter 14: The Frequency Domain 20

21. 240-373: Chapter 14: The Frequency Domain 21 –Convolution in frequency domain is “real convolution” Normal convolution Real convolution

22. 240-373: Chapter 14: The Frequency Domain 22 Convolution using the Fourier transform Technique 1: Convolution using the Fourier transform USE: To perform a convolution OPERATION: –zero-padding both the image (MxN) and the template (m x n) to the size (N+n-1) x (M+m-1) –Applying FFT to the modified image and template –Multiplying element by element of the transformed image against the transformed template

23. 240-373: Chapter 14: The Frequency Domain 23 Convolution using the Fourier transform OPERATION: (cont’d) –Multiplication is done as follows: F(image) F(template) F(result) (r 1,i 1 ) (r 2, i 2 ) (r 1 r 2 - i 1 i 2, r 1 i 2 +r 2 i 1 ) i.e. 4 real multiplications and 2 additions –Performing Inverse Fourier transform

24. 240-373: Chapter 14: The Frequency Domain 24 Hartley convolution Technique 2: Hartley convolution USE: To perform a convolution OPERATION: –zero-padding both the image (MxN) and the template (m x n) to the size (N+n-1) x (M+m-1) image template

25. 240-373: Chapter 14: The Frequency Domain 25 Hartley convolution –Applying Hartley transform to the modified image and template image template

26. 240-373: Chapter 14: The Frequency Domain 26 Hartley convolution –Multiplying them by evaluating:

27. 240-373: Chapter 14: The Frequency Domain 27 Hartley convolution: Cont’d Giving: –Performing Inverse Hartley transform, gives:

28. 240-373: Chapter 14: The Frequency Domain 28 Hartley convolution: Cont’d

29. 240-373: Chapter 14: The Frequency Domain 29 Deconvolution Convolution R = I * T Deconvolution I = R * -1 T Deconvolution of R by T = convolution of R by some ‘inverse’ of the template T (T’)

30. 240-373: Chapter 14: The Frequency Domain 30 Deconvolution Consider periodic convolution as a matrix operation. For example

31. 240-373: Chapter 14: The Frequency Domain 31 Deconvolution is equivalent to A B C AB = C ABB -1 = CB -1 A = CB -1