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Understanding Waveform Analysis: Transformations and Interpretations

Explore the concepts of Fourier and Hartley transforms, power functions, and autocorrelation in image processing, including the interpretation of power spectrum and the comparison between Hartley and Fourier transforms.

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Understanding Waveform Analysis: Transformations and Interpretations

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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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