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
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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:
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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)
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6. 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)
Optical transformation
A common approach to view image in frequency domain
Original image Transformed image A B D C C D B A
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7. Power and Autocorrelation Functions
Power function:
Autocorrelation function
Inverse Fourier transform of or
Hartley transform of
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8. Hartley vs Fourier Transform
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9. Interpretation of the power function
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