02/07/02 (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

)()()( GFH

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

Qualitative Filters

F G

=

=

=

H

Low-pass

High-pass

Band-pass

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

Low-Pass Filtered Image

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

High-Pass Filtered Image

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

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

duuxgufgfxh )()()(

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

Convolution Example

02/07/02 (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=FG– Take the inverse Fourier transform of H, to get h– Then h=fg

• Multiplication in the spatial domain is the same as convolution in the frequency domain

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

Sampling in Spatial Domain

• Sampling in the spatial domain is like multiplying by a spike function

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

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

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

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

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

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

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):