(C) 2002 University of Wisconsin, CS 559

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Presentation transcript:

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

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

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

(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

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

(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