07-1 - Wavelets fundamentals

07-1 - Wavelets fundamentals - 4/28/2008 Wavelets...

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4/28/2008 1 Wavelets fundamentals Spring 2008 ELEN 4304/5365 DIP 1 by Gleb V. Tcheslavski: gleb@ee.lamar.edu http://ee.lamar.edu/gleb/dip/index.htm Preliminaries When looking at images, we generally see connected regions of similar texture and intensity levels combined to form objects. Small or low- contrast objects are better viewed at high resolution. If small and large objects are present it can be Spring 2008 ELEN 4304/5365 DIP 2 are present, it can be advantageous to study them at different resolutions. From math viewpoint, images are 2D arrays of intensity values with locally varying statistics that result from different features.
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4/28/2008 2 Image pyramids An image pyramid is a collection of decreasing resolution images arranged in the shape of a pyramid. The base of a pyramid is a high resolution image being processed; the apex contains a low-resolution approximation. While moving up, both size and resolution decrease Spring 2008 ELEN 4304/5365 DIP 3 resolution decrease. Base level J is of size where 22 JJ NN =⋅ 2 log J N = Image pyramids The apex level 0 is of size 1x1. Most pyramids are truncated to P + 1 levels, where 1 P J. The total number of pixels in a P + 1 level pyramid is 11 1 4 ⎛⎞ 2 1. . . 44 4 3 P ++ ++ ⎜⎟ ⎝⎠ On the diagram for constructing two image pyramids, the “level j- 1 approximation output” provides the images needed to build an approximation pyramid , while the “level j prediction residual output” is used to build a complementary prediction residual pyramid . Unlike Spring 2008 ELEN 4304/5365 DIP 4 approximation pyramids, prediction residual pyramids contain only one reduced-resolution approximation of the input image (top of the pyramid, level J-P ). All other levels contain prediction residuals where the level j prediction residual ( J-P+ 1 j J ) is defined as the difference between level j approximation and its estimate.
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07-1 - Wavelets fundamentals - 4/28/2008 Wavelets...

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