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240-373 Image Processing

240-373 Image Processing. Montri Karnjanadecha montri@coe.psu.ac.th http://fivedots.coe.psu.ac.th/~montri. Chapter 14. The Frequency Domain. The Frequency Domain. Any wave shape can be approximated by a sum of periodic (such as sine and cosine) functions. a --amplitude of waveform

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240-373 Image Processing

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

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

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

  5. 240-373: Chapter 14: The Frequency Domain

  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 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”) 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: 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 240-373: Chapter 14: The Frequency Domain

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

  11. Fourier Power Spectrum and Inverse Fourier Transform • Fourier power spectrum • Inverse Fourier Transform 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 M2N2 to MN log2 M.log2 N (~1/1000 times) 240-373: Chapter 14: The Frequency Domain

  13. A C B D D B C A Fourier Power Spectrum and Inverse Fourier Transform • Optical transformation • A common approach to view image in frequency domain Original image Transformed image 240-373: Chapter 14: The Frequency Domain

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

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

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

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

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

  21. Convolution in frequency domain is “real convolution” Normal convolution Real convolution 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 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) (r1,i1) (r2, i2) (r1r2 - i1i2, r1i2+r2i1) i.e. 4 real multiplications and 2 additions • Performing Inverse Fourier transform 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 240-373: Chapter 14: The Frequency Domain

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

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

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

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

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

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

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

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