EECE253_09_Sharpening

# EECE253_09_Sharpening - 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: Sharpening and Edge Enhancement 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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Sunday, 27 April, 2008 2 Sunday, 27 April, 2008 2 1999-2007 by Richard Alan Peters II Sharpening O results from high frequency enhancement since small features correspond to short wavelength sinusoids. O Relative amplification of high frequencies in the Fourier domain corresponds to differentiation in the spatial domain. O On a discrete image, differentiation corresponds to pixel differencing.
Sunday, 27 April, 2008 3 Sunday, 27 April, 2008 3 1999-2007 by Richard Alan Peters II () {} ( ) 2( ) ) ) 2 ) , , , 2, 22 , . iu c v r r c v r r c v r iv c v r I I r rce dcdr I r c e dcdr I rc e i v I r c e dcdr I F u v π ππ ∞∞ −+ −∞ −∞ −∞ −∞ −∞ −∞ −∞ −∞ ⎧⎫ = ⎨⎬ ⎩⎭ =⋅ =− ∫∫ 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

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Sunday, 27 April, 2008 4 Sunday, 27 April, 2008 4 1999-2007 by Richard Alan Peters II Differentiation is Highpass Filtering () { } { } ,, I uv u I uv c I v r ⎧⎫ ⎨⎬ ⎩⎭ FF 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
Sunday, 27 April, 2008 5 Sunday, 27 April, 2008 5 1999-2007 by Richard Alan Peters II Fourier Transforms of Sums of Derivatives () { } ( ) 22 , . rc iu v I v F u v I ππ ∂∂ + ⎧⎫ ⎡⎤ =− + + ⎨⎬ ⎢⎥ ⎣⎦ ⎩⎭ FF Sum of first-order partial derviatives… Sum of first-order partial derviatives… …linear amplification of high frequencies …linear amplification of high frequencies {} 2 2 44 , . uv I F u v I + + + Sum of second-order partial derviatives… Sum of second-order partial derviatives… …quadratic amplification of high frequencies …quadratic amplification of high frequencies

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Sunday, 27 April, 2008 6 Sunday, 27 April, 2008 6 1999-2007 by Richard Alan Peters II Sharpening: Differencing / Highpass Filtering Sharpening results from adding to the image a copy of itself that has been: O Pixel-differenced in the spatial domain: Each pixel in the output is a difference between itself and a weighted average of its neighbors.
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EECE253_09_Sharpening - EECE\CS 253 Image Processing...

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