Discrete Fourier Transform
Butterworth Lowpass Filter
D0 is cutoff freq and D(u,v) is distribution of (u,v) from centered origin. n is order
Wrap Around Error
Solved by zero padding |
If f(x) and h(x) are A and B samples respectively, pad f(x) and h(x) with zeros so both have length P>=A+B-1 |
If not zero, creates discontinuity called "frequency leakage", equivalent to convolving with sinc() function |
Reduced by multiplying with function that tapers smoothly to zero (windowing or apodizing) |
2D Continuous Fourier Transform
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Fourier Series Definition
Laplacian in Freq. Domain
Steps for Filtering
1 + 2. Given f(x,y) is MxN, zero pad to 2Mx2N (PxQ) |
3. Multiply by (-1)x+y to center |
4. Take DFT of f(x,y) to get F(u,v) |
5. Generate symmetric filter H(u,v) of size PxQ |
6. Get processed image gp(x,y)={real[ F
-1{G(u,v)}} * (-1) x+y |
Fourier Spectrum and Phase Angle
Conjugate Symetry
F*(u,v) = F(-u, -v) (Conjugate Symmetry)
F*(-u,-v) = -F(u,v) (Conjugate Asymmetry) |
Spatial Shift Theorem
Spatial transform only affects FT phase
Convolution Theorem
Space convolution = frequency multiplication
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Center DC
To shift F(0,0) (DC Component) to center, multiply by (-1)x+y |
Power Spectrum
Total power of image is just sum of P(u,v) over P-1,Q-1
a = 100[doublesum P(u,v)/Pt]
Unsharp, Highboost, High-Emphasis
gmask(x,y) = f(x,y) - flp(x,y)
g(x,y) = f(x,y) + k*gmask(x,y)
k=1, unsharp
k>1, highboost
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Created By
samclane.github.io
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