Filters A filter is something that attenuates or enhances particular frequencies Easiest to visualize in the frequency domain where filtering is defined as multiplication H F G 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 Qualitative Filters F 02 07 02 G H Low pass High pass Band pass 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 h x f g f u g x u du 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 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 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 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
View Full Document
Unlocking...