1. 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
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Image Processing, Leture #14

2. 240-373 Image Processing, Leture #14
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Image Processing, Leture #14

3. 240-373 Image Processing, Leture #14
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Image Processing, Leture #14

4. 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:
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Image Processing, Leture #14

5. 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)
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Image Processing, Leture #14

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)
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Image Processing, Leture #14

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