1. General Functions
A non-periodic function can be represented as a sum of sin’s and cos’s of (possibly) all frequencies:
F() is the spectrum of the function f(x)

2. Fourier Transform
F() is computed from f(x) by the Fourier Transform:

3. Example: Box Function

4. Box Function and Its Transform

5. Cosine and Its Transform -1 1
If f(x) is even, so is F()

6. Sine and Its Transform  -1 1 -
If f(x) is odd, so is F()

7. Delta Function and Its Transform
Fourier transform and inverse Fourier transform are
qualitatively the same, so knowing one direction gives you the other

8. Shah Function and Its Transform
Moving the spikes closer together in the spatial domain moves them farther apart in the frequency domain!

9. Gaussian and Its Transform

10. Qualitative Properties
The spectrum of a functions tells us the relative amounts of high and low frequencies
Sharp edges give high frequencies
Smooth variations give low frequencies
A function is bandlimited if its spectrum has no frequencies above a maximum limit
sin, cos are bandlimited
Box, Gaussian, etc are not

11. Functions to Images Images are 2D, discrete functions
2D Fourier transform uses product of sin’s and cos’s (things carry over naturally)
Fourier transform of a discrete, quantized function will only contain discrete frequencies in quantized amounts
Numerical algorithm: Fast Fourier Transform (FFT) computes discrete Fourier transforms

12. 2D Discrete Fourier Transform

13. Filters
A filter is something that attenuates or enhances particular frequencies
Easiest to visualize in the frequency domain, where filtering is defined as multiplication:
Here, F is the spectrum of the function, G is the spectrum of the filter, and H is the filtered function. Multiplication is point-wise

14. Qualitative Filters F G H
Low-pass  =
High-pass  =
Band-pass  =

15. Low-Pass Filtered Image

16. High-Pass Filtered Image

17. Filtering in the Spatial Domain
Filtering the spatial domain is achieved by convolution
Qualitatively: Slide the filter to each position, x, then sum up the function multiplied by the filter at that position

18. Convolution Example

19. Convolution Theorem
Convolution in the spatial domain is the same as multiplication in the frequency domain
Take a function, f, and compute its Fourier transform, F
Take a filter, g, and compute its Fourier transform, G
Compute H=FG
Take the inverse Fourier transform of H, to get h
Then h=fg
Multiplication in the spatial domain is the same as convolution in the frequency domain

20. Sampling in Spatial Domain
Sampling in the spatial domain is like multiplying by a spike function 

21. Sampling in Frequency Domain
Sampling in the frequency domain is like convolving with a spike function 

22. Reconstruction in Frequency Domain
To reconstruct, we must restore the original spectrum
That can be done by multiplying by a square pulse 

23. Reconstruction in Spatial Domain
Multiplying by a square pulse in the frequency domain is the same as convolving with a sinc function in the spatial domain 

24. Aliasing Due to Under-sampling
If the sampling rate is too low, high frequencies get reconstructed as lower frequencies
High frequencies from one copy get added to low frequencies from another  

25. Aliasing Implications
There is a minimum frequency with which functions must be sampled – the Nyquist frequency
Twice the maximum frequency present in the signal
Signals that are not bandlimited cannot be accurately sampled and reconstructed
Not all sampling schemes allow reconstruction
eg: Sampling with a box

26. More Aliasing Poor reconstruction also results in aliasing
Consider a signal reconstructed with a box filter in the spatial domain (which means using a sinc in the frequency domain):  