1. Frequency Domain Analysis
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Frequency Domain Analysis

2. Frequency Domain Analysis
Basic Fourier Results
Any periodic signal can be uniquely represented as a linear combination of sines and cosines with frequencies that are harmonics of the fundamental frequency of the signal.
Any function (constrained by some nice math properties) can be represented as a linear combination of sines and cosines.
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Frequency Domain Analysis

3. Fourier Series Example
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Frequency Domain Analysis

4. Frequency Domain Analysis
Math Review
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Frequency Domain Analysis

5. 1-D Fourier – Continuous f(x)
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Frequency Domain Analysis

6. Frequency Domain Analysis
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Frequency Domain Analysis

7. Fourier – Polar Coordinate Representation
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Frequency Domain Analysis

8. Frequency Domain Analysis
Example
Higher Δx leads to lower Δu
Lower Δx leads to higher Δu
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Frequency Domain Analysis

9. Example – 1D Fourier Transform
From[1]
Peak depends on area of pulse.
Higher Δx leads to lower Δu.
Higher pulse width leads to lower high freq content.
Δu = 1/M Δx
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Frequency Domain Analysis

10. Frequency Domain Analysis
2-D Fourier Transform
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Frequency Domain Analysis

11. Example 2D – Fourier Transform
From[1]
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Frequency Domain Analysis

12. Properties of Fourier Transform
F(u,v) is a global measure, it depends on all values of f(x,y).
u,v measure frequency in x, y directions. It is often possible to relate the rate of change in intensity with the frequency.
F(0,0) = average of f(x,y) over the whole image
{f(x,y)|x=0,1,..M-1; y=0,1,..N-1}
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Frequency Domain Analysis

13. Frequency Domain Analysis
Example 2D DFT
From[1]
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Frequency Domain Analysis

14. Properties of Fourier Transform
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Frequency Domain Analysis

15. Filtering in Frequency Domain
More intuitive – reduce lower frequency, reduce noise, reduce higher frequency, etc
Design a filter H(u, v) to incorporate this behavior
Typical approach
Input image is f(x,y) – real values
Compute F(u,v)
G(u,v) = F(u,v) H(u,v)
Compute g(x,y) – if f(x,y) is real and h(x,y) is real then g(x,y) should be real. Computation errors may yield non-zero imaginary components
Often supplemented with operations to center the image, image cropping, integer to floating point conversion and vice versa ……
Often use multiple stage filters
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Frequency Domain Analysis

16. Filtering in Freq Domain
From[1]
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Frequency Domain Analysis

17. Frequency Domain Analysis
Filter Example
Filter such that the result image has an average value of zero. So one approach:
Select a filter so that G(0,0)=0
Compute Fourier Inverse, and that will yield an image with an average of 0.
Such a filter will highlight the non-uniform parts of the image
Display image after adding a constant – required to display the negative values.
From[1]
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Frequency Domain Analysis

18. Low Pass and High Pass Filter Examples
From[1]
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Frequency Domain Analysis

19. Frequency Domain Analysis
Gaussian filters
From[1]
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Frequency Domain Analysis

20. Frequency Domain Analysis
Typical filters
Low Pass
Ideal low pass: Not physically realizable, but can be modeled on computers
Butterworth: Approximations to ideal
Gaussian
High Pass
Ideal high pass
Butterworth
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Frequency Domain Analysis

21. Frequency Domain Analysis
Ideal Low Pass
Passes a range of low frequencies without attenuation, and blocks all the higher frequencies
H(u,v) = 1 if D(u,v) <= D0
0 if D(u,v) > D0
D(u,v) = [(u-M/2)2 + (v-N/2)2]1/2 Euclidean distance
Cutoff frequency
From[1]
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Frequency Domain Analysis

22. Using Power Spectrum for Filter Design
Power at a frequency P(u,v) = |F(u,v)|2
Total image power PT = ∑u∑vP(u,v)
u = 0,1,…, M-1; v = 0,1,….,N-1
A circle of radius r will cover a% of the power
a = 100 [∑u∑vP(u,v)/ PT]
From[1]
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Frequency Domain Analysis

23. Image Power, ILPF, Fidelity
From[1]
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Frequency Domain Analysis

24. Frequency Domain Analysis
ILPF Ringing
From[1]
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Frequency Domain Analysis

25. Frequency Domain Analysis
High Pass Filters
Hhp(u,v) = 1 – Hlp(u,v)
From[1]
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Frequency Domain Analysis

26. Laplacian in Frequency Domain
From[1]
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Frequency Domain Analysis

27. Frequency Domain Analysis
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28. Frequency Domain Analysis
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29. Frequency Domain Analysis
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Frequency Domain Analysis