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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 f -- frequency (number of times the wave repeats itself in a given length) p --phase (position that the wave starts)

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The Frequency Domain

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  1. 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 Image Processing

  2. 240-373 Image Processing

  3. 240-373 Image Processing

  4. 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 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”) • Fast Hartley Transform (FHT) • M and N must be power of 2 • Much faster than DHT • Equation: 240-373 Image Processing

  5. 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 • Discrete Fourier Transform (DFT) • Imaginary part • Real part • The actual complex result is Fi(u,v) + Fr(u,v) 240-373 Image Processing

  6. A C B D D B C A Fourier Power Spectrum and Inverse Fourier Transform • Fourier power spectrum • 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) • Optical transformation • A common approach to view image in frequency domain Original image Transformed image 240-373 Image Processing

  7. Power and Autocorrelation Functions • Power function: • Autocorrelation function • Inverse Fourier transform of or • Hartley transform of 240-373 Image Processing

  8. Hartley vs Fourier Transform 240-373 Image Processing

  9. Interpretation of the power function 240-373 Image Processing

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