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EECE253_16_MedianFilters

# EECE253_16_MedianFilters - EECE\CS 253 Image Processing...

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EECE\CS 253 Image Processing Richard Alan Peters II Department of Electrical Engineering and Computer Science Fall Semester 2007 Lecture Notes Lecture Notes on Mathematical Morphology: The Median Filter This work is licensed under the Creative Commons Attribution-Noncommercial 2.5 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/2.5/ or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA.

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December 6, 2011 2 December 6, 2011 2 1999-2007 by Richard Alan Peters II The Median Filter Returns the median value of the pixels in a neighborhood Is non-linear Is a morphological filter Is similar to a uniform blurring filter which returns the mean value of the pixels in a neighborhood of a pixel Unlike a mean value filter the median tends to preserve step edges original median filtered
December 6, 2011 3 December 6, 2011 3 1999-2007 by Richard Alan Peters II This can be computed as follows: 1. Let I be a monochrome (1-band) image. 2. Let Z define a neighborhood of arbitrary shape. 3. At each pixel location, p = ( r , c ), in I 4. … select the n pixels in the Z -neighborhood of p , 5. … sort the n pixels in the neighborhood of p, by 6. The output value at p is L ( m ), where m = n / 2 +1. value, into a list L ( j ) for j = 1,…, n . Median Filter: General Definition { } ( 29 ( 29 ( 29 { } supp med , median Z I Z I + = q p p q

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December 6, 2011 4 December 6, 2011 4 1999-2007 by Richard Alan Peters II Median Filter: General Definition sorted intensity values from neighborhood of p . 131 133 133 136 140 143 147 152 154 157 160 162 163 164 165 171 p median assigned to pixel loc p in output image.
December 6, 2011 5 December 6, 2011 5 1999-2007 by Richard Alan Peters II A Noisy Step Edge ( 29 = + - 33 25 . 1 32 25 . 0 25 . 0 5 . 32 n n n H for for ( 29 ( 29 ( 29 ( 29 25 . 0 , 25 . 0 unif 5 . 32 - = + - where n u n u n H

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December 6, 2011 6 December 6, 2011 6 1999-2007 by Richard Alan Peters II Blurred Noisy 1D Step Edge ( 29 ( 29 - = + = 4 4 9 1 k k n h n h
December 6, 2011 7 December 6, 2011 7 1999-2007 by Richard Alan Peters II Blurred Noisy 1D Step Edge J(32-4:32+4)= 0.1920 0.3416 0.0464 0.0177 0.3062 1.3043 1.0079 1.0082 1.0950 J(33-4:33+4)= 0.3416 0.0464 0.0177 0.3062 1.3043 1.0079 1.0082 1.0950 1.2935 0.5910 0.7134 mean mean

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December 6, 2011 8 December 6, 2011 8 1999-2007 by Richard Alan Peters II Median Filtered Noisy 1D Step Edge ( 29 ( 29 { } 4 4 med - = + = k k n h n h
December 6, 2011 9 December 6, 2011 9 1999-2007 by Richard Alan Peters II Median Filtered Noisy 1D Step Edge J(32-4:32+4)= 0.1920 0.3416 0.0464 0.0177 0.3062 1.3043 1.0079 1.0082 1.0950 0.0177 0.0464 0.1920 0.3062 0.3416 1.0079 1.0082 1.0950 1.3043 J(33-4:33+4)= 0.3416 0.0464 0.0177 0.3062 1.3043 1.0079 1.0082 1.0950 1.2935 0.0177 0.0464 0.3062 0.3416 1.0079 1.0082 1.0950 1.2935 1.3043 sorted sorted median median

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December 6, 2011 10 December 6, 2011 10 1999-2007 by Richard Alan Peters II Median vs. Blurred step noisy blurred median The median filter preserves  the step edge better than the  blurring filter.
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