1. (C) 2002 University of Wisconsin, CS 559
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
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(C) 2002 University of Wisconsin, CS 559

2. (C) 2002 University of Wisconsin, CS 559
Qualitative Filters F G H
Low-pass  =
High-pass  =
Band-pass  =
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(C) 2002 University of Wisconsin, CS 559

3. Low-Pass Filtered Image
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(C) 2002 University of Wisconsin, CS 559

4. High-Pass Filtered Image
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(C) 2002 University of Wisconsin, CS 559

5. 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
02/07/02
(C) 2002 University of Wisconsin, CS 559

6. (C) 2002 University of Wisconsin, CS 559
Convolution Example
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(C) 2002 University of Wisconsin, CS 559

7. (C) 2002 University of Wisconsin, CS 559
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
02/07/02
(C) 2002 University of Wisconsin, CS 559

8. Sampling in Spatial Domain
Sampling in the spatial domain is like multiplying by a spike function 
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(C) 2002 University of Wisconsin, CS 559

9. Sampling in Frequency Domain
Sampling in the frequency domain is like convolving with a spike function 
02/07/02
(C) 2002 University of Wisconsin, CS 559

10. Reconstruction in Frequency Domain
To reconstruct, we must restore the original spectrum
That can be done by multiplying by a square pulse 
02/07/02
(C) 2002 University of Wisconsin, CS 559

11. 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 
02/07/02
(C) 2002 University of Wisconsin, CS 559

12. 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  
02/07/02
(C) 2002 University of Wisconsin, CS 559

13. 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
02/07/02
(C) 2002 University of Wisconsin, CS 559

14. (C) 2002 University of Wisconsin, CS 559
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):  
02/07/02
(C) 2002 University of Wisconsin, CS 559