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EECE253_11_SamplingAliasing

EECE253_11_SamplingAliasing - 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: Sampling and Aliasing 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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Tuesday, December 6, 2011 2 Tuesday, December 6, 2011 2 1999-2007 by Richard Alan Peters II Image Resampling Can Lead to the Jaggies ! zo o m e d  ×  2 zo o m e d  ×  2 The jaggies! Warning:
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Tuesday, December 6, 2011 3 Tuesday, December 6, 2011 3 1999-2007 by Richard Alan Peters II >> J = I(1:2:R,1:2:C,:); >> J = I(1:2:R,1:2:C,:); Downsampling (Decimation) E.g.: every 2 nd pixel in every 2 nd row This is a bad way to do it.
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Tuesday, December 6, 2011 4 Tuesday, December 6, 2011 4 1999-2007 by Richard Alan Peters II >> J = I(1:2:R,1:2:C,:); >> J = I(1:2:R,1:2:C,:); Downsampling (Decimation) E.g.: every 2 nd pixel in every 2 nd row Bad, bad, very bad.
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Tuesday, December 6, 2011 5 Tuesday, December 6, 2011 5 1999-2007 by Richard Alan Peters II Power Spectrum from Discrete Fourier Transform Recall: DFT
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Tuesday, December 6, 2011 6 Tuesday, December 6, 2011 6 1999-2007 by Richard Alan Peters II Power Spectrum from Discrete Fourier Transform The DFT of an image is the same size as the image. DFT decimated image power spectrum Recall:
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Tuesday, December 6, 2011 7 Tuesday, December 6, 2011 7 1999-2007 by Richard Alan Peters II The Scaling Property of the FT { } ( 29 ( 29 { } ( 29 2 ( ) If , I , , then I , , . I I i uc vr v u r c e dcdr r c ab av bu a b π - + -∞ -∞ = ÷ ∫ ∫ F F This  im plie s  that if an im ag e  is  re duc e d in s ize , its   fe ature s  in the  s patial do m ain be c o m e  s m alle r and its   fe ature s  in the  fre q ue nc y do m ain be c o m e  larg e r. This  im plie s  that if an im ag e  is  re duc e d in s ize , its   fe ature s  in the  s patial do m ain be c o m e  s m alle r and its   fe ature s  in the  fre q ue nc y do m ain be c o m e  larg e r. Recall:
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Tuesday, December 6, 2011 8 Tuesday, December 6, 2011 8 1999-2007 by Richard Alan Peters II The Uncertainty Relation FT FT space frequency FT FT space frequency A s m all o bje c t in s pac e   has  a larg e  fre q ue nc y  e xte nt and vic e -ve rs a. A s m all o bje c t in s pac e   has  a larg e  fre q ue nc y  e xte nt and vic e -ve rs a. Recall:
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Tuesday, December 6, 2011 9 Tuesday, December 6, 2011 9 1999-2007 by Richard Alan Peters II Effect of Decimation on the DFT of an Image 1. Decimation of an R × C image, I, by a factor of n results in an R / n × C / n image, J. 2. The DFT of image J is the same size as J. 3. The uncertainty relation implies that the FT of J should be n R / n × n C / n = R × C.
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