240-373 Image Processing, Leture #14 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 30/11/61 240-373 Image Processing, Leture #14

240-373 Image Processing, Leture #14 30/11/61 240-373 Image Processing, Leture #14

240-373 Image Processing, Leture #14 30/11/61 240-373 Image Processing, Leture #14

240-373 Image Processing, Leture #14 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: 30/11/61 240-373 Image Processing, Leture #14

240-373 Image Processing, Leture #14 The Fourier Transform The Furier 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) 30/11/61 240-373 Image Processing, Leture #14

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) 30/11/61 240-373 Image Processing, Leture #14

Power and Autocorrelation functions Power function: Autocorrelation function Inverse Fourier transform of or Hartley transform of Interpretation of the power functions 30/11/61 240-373 Image Processing, Leture #14