1 / 14

Filters

Filters. A filter is something that attenuates or enhances particular frequencies Easiest to visualize in the frequency domain, where filtering is defined as multiplication:

tad-deleon
Download Presentation

Filters

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

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

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

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

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

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

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

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

More Related