# ch03c - Ch3 Spatial Filtering 1 Overview Correlation and...

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Ch3 – Spatial Filtering 1 • Overview • Correlation and Convolution • Neighborhood Averaging • Binomial Filtering • Gaussian Blurring • Outlier Removal • Median Filtering • k-Nearest Neighbors • Conclusion

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Overview • Spatial filters make use of a fixed sized neighborhood in an input image to calculate output intensities • A large number of techniques have been invented to smooth or sharpen images • We will consider pros/cons of some popular methods
Overview • Linear filters use a weighted sum of pixels in the input image to calculate the output pixel • In most cases, the sum of weights is one, so the output brightness = input brightness • Nonlinear filters can not be calculated using just a weighted sum • Other operations (e.g. sqrt, log, sorting, selection) are involved in calculation

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Correlation and Convolution • We can formalize the phrase “weighted sum of pixels” using correlation and convolution
Correlation and Convolution f(x) 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 w(x) 1 1 1 g(x) 4 8 6 10 15 16 17 13 14 13 16 22 24 25 19 12

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Correlation and Convolution f(x) 6 6 6 5 4 3 2 1 1 1 1 1 1 6 6 6 w(x) -1 1 g(x) 0 0 -1 -1 -1 -1 -1 0 0 0 0 0 5 0 0 -6
Correlation and Convolution • Both formulas extend easily to two dimensions

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Correlation and Convolution 0.25 0.25 0.25 0.25 3 5 7 9 5 3 9 7 3 5 7 9 5 3 9 7 0.75 2 3 4 2.25 2 4 6 8 4 2 4 6 8 4 2 4 6 8 4 1.25 2 3 4 1.75 w(x,y) f(x.y) g(x,y)=f(x,y)*w(x,y)
Correlation and Convolution • Some important properties of convolution: f * g = g * f f * (g * h) = (f * g) * h f * (g + h) = f * g + f * h a(f * g) = (af) * g = f * (ag), where a=scalar f * δ = f, where δ =delta function d/dx(f * g) = (df/dx) * g = f * (dg/dx)

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• The easiest spatial filter to implement is neighborhood averaging • Each (x,y) pixel in the image is replaced by the average of the pixels in an NxN neighborhood centered at (x,y) • This will smooth an image and remove noise and small details
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ch03c - Ch3 Spatial Filtering 1 Overview Correlation and...

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