1. Frequency Domain – Homomorphic Filters & Nyquist
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2. References Gonzales and Wood second edition
Gonzales and Wood first edition
Jain
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3. Homomorphic Filtering
Based on illumination – reflectance model
f(x,y) = i(x,y) r(x,y)
Heuristic used (the following is usually true)
Illumination across the scene varies (corresponds to lower frequencies)
Reflectance varies more abruptly, e.g. at the junction of 2 dissimilar objects (corresponds to higher frequencies)
If the signal can be properly separated we can focus on the reflectance
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4. Homomorphic Filtering – Strategy / Parameters
From [1]
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5. Homomorphic Filtering - Example
From [1]
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6. DFT Periodicity
From [1]
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7. Separability
From [1]
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8. Need for correct padding
From [1]
Compare
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- need more padding

9. Padded inputs From [1] A=period of f B=period of h Padding P>=A+B-1
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10. 2-D Padding
From [1]
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11. Convolution Theorem Definition
M-1 N-1
f(x,y)*h(x,y) = (1/MN) Σ Σ f(m,n)h(x-m,y-n)
m=0 n=0
Transform Pairs
f(x,y)*h(x,y) F(u,v)H(u,v)
f(x,y)h(x,y) F(u,v)*H(u,v)
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12. Correlation Theorem Definition
M-1 N-1
f(x,y) h(x,y) = (1/MN) Σ Σ f*(m,n)h(x+m,y+n)
m=0 n=0
Transform Pairs
f(x,y) h(x,y) F*(u,v)H(u,v)
f*(x,y)h(x,y) F(u,v) H(u,v)
Autocorrelation theorem: Set h(x,y) = f(x,y)
f(x,y) f(x,y) |F(u,v)|2
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13. Fourier Properties
From [1]
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14. Fourier Properties
From [1]
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15. Fourier Properties
From [1]
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16. Fourier Properties
From [1]
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17. Fourier vs FFT Computational Advantage
From [1]
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18. 5/28/2019

19. From [2]
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20. From [2]
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21. 5/28/2019

22. From [3]
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