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Basics of Signal Processing and Filtering

This article provides an overview of signal processing and filtering techniques, including box filters, Bartlett filters, Gaussian filters, high-pass filters, edge enhancement, image warping, and image size reduction and enlargement.

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Basics of Signal Processing and Filtering

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  1. Last Time • Signal Processing • Filtering basics • Homework 2 • Project 1 (c) 2002 University of Wisconsin, CS 559

  2. Today • More Filters • Sampling and Reconstruct • Image warping (c) 2002 University of Wisconsin, CS 559

  3. Box Filter • Box filters smooth by averaging neighbors • In frequency domain, keeps low frequencies and attenuates (reduces) high frequencies, so clearly a low-pass filter Spatial: Box Frequency: sinc (c) 2002 University of Wisconsin, CS 559

  4. Box Filter (c) 2002 University of Wisconsin, CS 559

  5. Filtering Algorithm • If Iinput is the input image, and Ioutput is the output image, M is the filter mask and k is the mask size: • Care must taken at the boundary • Make the output image smaller • Extend the input image in some way (c) 2002 University of Wisconsin, CS 559

  6. Bartlett Filter • Triangle shaped filter in spatial domain • In frequency domain, product of two box filters, so attenuates high frequencies more than a box Spatial: Triangle (BoxBox) Frequency: sinc2 (c) 2002 University of Wisconsin, CS 559

  7. Constructing Masks: 1D • Sample the filter function at matrix “pixels” • eg 2D Bartlett • Can go to edge of pixel or middle of next: results are slightly different 1 1 3 1 5 0 1 2 1 1 2 1 4 (c) 2002 University of Wisconsin, CS 559

  8. Constructing Masks: 2D • Multiply 2 1D masks together using outer product • M is 2D mask, m is 1D mask 0.2 0.6 0.2 0.2 0.04 0.12 0.04 0.6 0.12 0.36 0.12 0.2 0.04 0.12 0.04 (c) 2002 University of Wisconsin, CS 559

  9. Bartlett Filter (c) 2002 University of Wisconsin, CS 559

  10. Guassian Filter • Attenuates high frequencies even further • In 2d, rotationally symmetric, so fewer artifacts (c) 2002 University of Wisconsin, CS 559

  11. Gaussian Filter (c) 2002 University of Wisconsin, CS 559

  12. Constructing Gaussian Mask • Use the binomial coefficients • Central Limit Theorem (probability) says that with more samples, binomial converges to Gaussian 1 1 2 1 4 1 1 4 6 4 1 16 1 1 6 15 20 15 6 1 64 (c) 2002 University of Wisconsin, CS 559

  13. High-Pass Filters • A high-pass filter can be obtained from a low-pass filter • If we subtract the smoothed image from the original, we must be subtracting out the low frequencies • What remains must contain only the high frequencies • High-pass masks come from matrix subtraction: • eg: 3x3 Bartlett (c) 2002 University of Wisconsin, CS 559

  14. High-Pass Filter (c) 2002 University of Wisconsin, CS 559

  15. Edge Enhancement • High-pass filters give high values at edges, low values in constant regions • Adding high frequencies back into the image enhances edges • One approach: • Image = Image + [Image – smooth(Image)] Low-pass High-pass (c) 2002 University of Wisconsin, CS 559

  16. Edge-Enhance Filter (c) 2002 University of Wisconsin, CS 559

  17. Edge Enhancement (c) 2002 University of Wisconsin, CS 559

  18. Fixing Negative Values • The negative values in high-pass filters can lead to negative image values • Most image formats don’t support this • Solutions: • Truncate: Chop off values below min or above max • Offset: Add a constant to move the min value to 0 • Re-scale: Rescale the image values to fill the range (0,max) (c) 2002 University of Wisconsin, CS 559

  19. Image Warping • An image warp is a mapping from the points in one image to points in another • f tells us where in the new image to put the data from x in the old image • Simple example: Translating warp, f(x) = x+o, shifts an image (c) 2002 University of Wisconsin, CS 559

  20. Reducing Image Size • Warp function: f(x)=kx, k > 1 • Problem: More than one input pixel maps to each output pixel • Solution: Filter down to smaller size • Apply the filter, but not at every pixel, only at desired output locations • eg: To get half image size, only apply filter at every second pixel (c) 2002 University of Wisconsin, CS 559

  21. 2D Reduction Example (Bartlett) (c) 2002 University of Wisconsin, CS 559

  22. Ideal Image Size Reduction • Reconstruct original function using reconstruction filter • Resample at new resolution (lower frequency) • Clearly demonstrates that shrinking removes detail • Expensive, and not possible to do perfectly in the spatial domain… (c) 2002 University of Wisconsin, CS 559

  23. Enlarging Images • Warp function: f(x)=kx, k < 1 • Problem: Have to create pixel data • More pixels in output than in input • Solution: Filter up to larger size • Apply the filter at intermediate pixel locations • eg: To get double image size, apply filter at every pixel and every half pixel • New pixels are interpolated from old ones • Filter encodes interpolation function (c) 2002 University of Wisconsin, CS 559

  24. Enlargement (c) 2002 University of Wisconsin, CS 559

  25. Ideal Enlargement • Reconstruct original function • Resample at higher frequency • Original function was band-limited, so re-sampling does not add any extra frequency information (c) 2002 University of Wisconsin, CS 559

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