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EECE253_07_Convolution

# EECE253_07_Convolution - 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 2011 Lecture Notes: Spatial Convolution 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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2011-09-29 2 1999-2011 by Richard Alan Peters II Spatial Filtering ( ) [ ]( ) ( ) { } { } { } ( ) , T , , ,..., ,... , ,..., ,... . r c r c f r s r r s c d c c d r c r c = = Î - + Î - + J I I That is, the value of the transformed image, J , at pixel location ( r,c ) is a function of the values of the original image, I , in a 2 s +1 × 2 d +1 rectangular neighborhood centered on pixel location ( r,c ). Let I and J be images such that J = T[ I ]. T[·] represents a transformation, such that,
2011-09-29 3 1999-2011 by Richard Alan Peters II Moving Windows The value, J ( r,c ) = T[ I ]( r,c ), is a function of a rectangular neighborhood centered on pixel location ( r,c ) in I . There is a different neighborhood for each pixel location, but if the dimensions of the neighbor- hood are the same for each location, then trans- form T is sometimes called a moving window transform .

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2011-09-29 4 1999-2011 by Richard Alan Peters II Moving-Window Transformations Neutral Buoyancy Facility at NASA Johnson Space Center We’ll take a section of this image to demonstrate the MWT photo: R.A.Peters II, 1999
2011-09-29 5 1999-2011 by Richard Alan Peters II Moving-Window Transformations operate on this region

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2011-09-29 6 1999-2011 by Richard Alan Peters II Moving-Window Transformations apply a pixel grid Pixelize the section to better see the effects.
2011-09-29 7 1999-2011 by Richard Alan Peters II Moving-Window Transformations sample (average in the squares). Pixelize the section to better see the effects.

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2011-09-29 8 1999-2011 by Richard Alan Peters II Moving-Window Transformations lets get some perspective on this
2011-09-29 9 1999-2011 by Richard Alan Peters II Moving-Window Transformations a neighborhood defined by a weight matrix

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2011-09-29 10 1999-2011 by Richard Alan Peters II Moving-Window Transformations neighborhoods at other pixel locations
2011-09-29 11 1999-2011 by Richard Alan Peters II Linear Moving-Window Transformations ( i.e. convolution) The output of the transform at each pixel is the (weighted) average of the pixels in the neighborhood.

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2011-09-29 12 1999-2011 by Richard Alan Peters II Moving-Window Transformations result of a 9 x 9 uniform averaging
2011-09-29 13 1999-2011 by Richard Alan Peters II Convolution: Mathematical Representation If a MW transformation is linear then it is a convolution : [ ] ( , ) ( , ) ( , ) ( , ) , r c r c r c d d r c r c r c ¥ ¥ -¥ -¥ = * = - - ò ò J I h I h [ ] ( , ) ( , ) ( , ) ( , ) s d s d r c r c B r c r c r c r c =- =- = * = - - å å J I h h for a real image ( I : × ), or for a digital image ( I : × ):

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2011-09-29 14 1999-2011 by Richard Alan Peters II Convolution Mask ( Weight Matrix ) The object, h ( ρ , χ ), in the equation is a weighting function, or in the discrete case, a rectangular matrix of numbers.
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EECE253_07_Convolution - EECE\CS 253 Image Processing...

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