EECE253_18_GrayMorphology

EECE253_18_GrayMorphology - EECE\CS 253 Image Processing...

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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 on Mathematical Morphology: Grayscale Images 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. December 6, 2011 2 December 6, 2011 2 1999-2007 by Richard Alan Peters Grayscale Morphology Grayscale morphology is a multidimensional generalization of the binary operations. Binary morphology is defined in terms of set-inclusion of pixel sets. So is the grayscale case, but the pixel sets are of higher dimension. In particular, standard R C, 1- band intensity images and the associated structuring elements are defined as 3-D solids wherein the 3 rd axis is intensity and set-inclusion is volumetric. set inclusion (explained on p. 11 ) set inclusion (explained on p. 11 ) (a) binary, (b) & (c) grayscale (a) binary, (b) & (c) grayscale (a) (b) (c) December 6, 2011 3 December 6, 2011 3 1999-2007 by Richard Alan Peters Recall: Types of Images A real image maps an n-dimensional Euclidean vector space into the real numbers. Pixel coordinates and pixel values are real. A discrete image maps an n-dimensional grid of points into the real numbers. Coordinates are n-tuples of integers, pixel values are real. A digital image maps an n-dimensional grid into a finite set of integers. Pixel coordinates and pixel values are integers. An image is a mapping, I, from a set, S P , of pixel coordinates to a set, G , of values such that for every coordinate vector, p , there is a value I( p ) drawn from G . An image is a mapping, I, from a set, S P , of pixel coordinates to a set, G , of values such that for every coordinate vector, p , there is a value I( p ) drawn from G . December 6, 2011 4 December 6, 2011 4 1999-2007 by Richard Alan Peters Extended Real Numbers Define the extended real numbers, * , as the real numbers plus two symbols, - and such that for all numbers x . That is if x is any real number, then is always greater than x and - is always less than x . Moreover, Let represent the real numbers. , , , = - - = - = + x x for all numbers x . , < < - x December 6, 2011 5 December 6, 2011 5 1999-2007 by Richard Alan Peters p , in an n-dimensional vector space n . Associated with each p is a value from * . The set of pixel locations together with their associated values form the image a set in n+1 : In mathematical morphology a real image, I, is defined as a function that occupies a volume in a Euclidean vector space....
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EECE253_18_GrayMorphology - EECE\CS 253 Image Processing...

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