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Unformatted text preview: EECE\CS 253 Image Processing Richard Alan Peters II Department of Electrical Engineering and Computer Science Fall Semester 2007 Lecture Notes Lecture Notes: Sharpening and Edge Enhancement This work is licensed under the Creative Commons AttributionNoncommercial 2.5 License. To view a copy of this license, visit http://creativecommons.org/licenses/bync/2.5/ or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA. Tuesday, December 6, 2011 2 Tuesday, December 6, 2011 2 19992007 by Richard Alan Peters Sharpening ● results from high frequency enhancement since small features correspond to short wavelength sinusoids. ● Relative amplification of high frequencies in the Fourier domain corresponds to differentiation in the spatial domain. ● On a discrete image, differentiation corresponds to pixel differencing. Tuesday, December 6, 2011 3 Tuesday, December 6, 2011 3 19992007 by Richard Alan Peters ( 29 ( 29 ( 29 ( 29 ( 29 { } ( 29 2 ( ) 2 ( ) 2 ( ) 2 2 ( ) , , , 2 , 2 2 , . i uc vr r i uc vr r i uc vr i v i uc vr I I r r c e dcdr I r c e dcdr I r c e dcdr i v I r c e dcdr i v I i v F u v π π π π π π π π ∞ ∞ ∂ + ∂∞ ∞ ∞ ∞ ∂ + ∂∞ ∞ ∞ ∞ +∞ ∞ ∞ ∞ +∞ ∞ ∂ ∂ = = × = × =  =  =  ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ F F The Derivative Property of the Fourier Transform The FT of the partial derivative w.r.t. r (in the row direction) of an image, I … The FT of the partial derivative w.r.t. r (in the row direction) of an image, I … … is equal to the product of the FT of the image and the corresponding frequency variable, v. … is equal to the product of the FT of the image and the corresponding frequency variable, v. Integration by parts Integration by parts This results in horizontal HF enhancement This results in horizontal HF enhancement Tuesday, December 6, 2011 4 Tuesday, December 6, 2011 4 19992007 by Richard Alan Peters Differentiation is Highpass Filtering ( 29 { }( 29 ( 29 { }( 29 , , , , I u v u I u v c I u v v I u v r ∂ μ ∂ ∂ μ ∂ F F F F Directional derivative in r. Directional derivative in r. Vertical HF Enhancement Vertical HF Enhancement Directional derivative in c. Directional derivative in c. Horizontal HF Enhancement Horizontal HF Enhancement Tuesday, December 6, 2011 5 Tuesday, December 6, 2011 5 19992007 by Richard Alan Peters Fourier Transforms of Sums of Derivatives ( 29 { } ( 29 ( 29 2 2 , ....
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This note was uploaded on 12/06/2011 for the course EECE 253 taught by Professor Alanpeters during the Summer '07 term at Vanderbilt.
 Summer '07
 AlanPeters
 Electrical Engineering, Image processing

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