FiltersQualitative FiltersLow-Pass Filtered ImageHigh-Pass Filtered ImageFiltering in the Spatial DomainConvolution ExampleConvolution TheoremSampling in Spatial DomainSampling in Frequency DomainReconstruction in Frequency DomainReconstruction in Spatial DomainAliasing Due to Under-samplingAliasing ImplicationsMore Aliasing02/07/02(C) 2002 University of Wisconsin, CS 559Filters•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 559Qualitative FiltersF G===HLow-passHigh-passBand-pass02/07/02(C) 2002 University of Wisconsin, CS 559Low-Pass Filtered Image02/07/02(C) 2002 University of Wisconsin, CS 559High-Pass Filtered Image02/07/02(C) 2002 University of Wisconsin, CS 559Filtering 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 559Convolution Example02/07/02(C) 2002 University of Wisconsin, CS 559Convolution 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 domain02/07/02(C) 2002 University of Wisconsin, CS 559Sampling in Spatial Domain•Sampling in the spatial domain is like multiplying by a spike function02/07/02(C) 2002 University of Wisconsin, CS 559Sampling in Frequency Domain•Sampling in the frequency domain is like convolving with a spike function02/07/02(C) 2002 University of Wisconsin, CS 559Reconstruction in Frequency Domain•To reconstruct, we must restore the original spectrum•That can be done by multiplying by a square pulse02/07/02(C) 2002 University of Wisconsin, CS 559Reconstruction in Spatial Domain•Multiplying by a square pulse in the frequency domain is the same as convolving with a sinc function in the spatial domain02/07/02(C) 2002 University of Wisconsin, CS 559Aliasing 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 another02/07/02(C) 2002 University of Wisconsin, CS 559Aliasing 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 box02/07/02(C) 2002 University of Wisconsin, CS 559More 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
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