EECE253_11_SamplingAliasing

EECE253_11_SamplingAliasing - EECE\CS 253 Image Processing...

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Sunday, 27 April, 2008 2 Sunday, 27 April, 2008 2 1999-2007 by Richard Alan Peters II Image Resampling Can Lead to the Jaggies ! zoomed × 2 zoomed × 2 The jaggies! The jaggies! Warning: Warning:
Background image of page 2
Sunday, 27 April, 2008 3 Sunday, 27 April, 2008 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.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Sunday, 27 April, 2008 4 Sunday, 27 April, 2008 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.
Background image of page 4
Sunday, 27 April, 2008 5 Sunday, 27 April, 2008 5 1999-2007 by Richard Alan Peters II Power Spectrum from Discrete Fourier Transform Recall: DFT
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Sunday, 27 April, 2008 6 Sunday, 27 April, 2008 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:
Background image of page 6
Sunday, 27 April, 2008 7 Sunday, 27 April, 2008 7 1999-2007 by Richard Alan Peters II The Scaling Property of the FT {} () ( ) 2( ) If , I , , then I , , . I I iu c v r vu rc e dcdr rc ab av bu ab π ∞∞ −+ −∞ −∞ = ⎛⎞ ⎜⎟ ⎝⎠ ∫∫ F F This implies that if an image is reduced in size, its features in the spatial domain become smaller and its features in the frequency domain become larger. This implies that if an image is reduced in size, its features in the spatial domain become smaller and its features in the frequency domain become larger. Recall:
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Sunday, 27 April, 2008 8 Sunday, 27 April, 2008 8 1999-2007 by Richard Alan Peters II The Uncertainty Relation FT FT space frequency FT FT space frequency A small object in space has a large frequency extent and vice-versa. A small object in space has a large frequency extent and vice-versa. 2 16 1 π Δ Δ Δ Δ Δ Δ Δ Δ v u y x v u y x then, frequency in extent its is if and space in object the of extent the is If Recall:
Background image of page 8
Sunday, 27 April, 2008 9 Sunday, 27 April, 2008 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. Contradiction? Q: How can these 3 facts be true simultaneously? A: The FT of J folds over or aliases itself on the DFT of J because the DFT is defined on a torus.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon