EECE253_18_GrayMorphology

EECE253_18_GrayMorphology - 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 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.

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27 April 2008 2 27 April 2008 2 1999-2007 by Richard Alan Peters II 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)
27 April 2008 4 27 April 2008 4 1999-2007 by Richard Alan Peters II Extended Real Numbers Define the extended real numbers, R * , as the real numbers plus two symbols, −∞ and such that for all numbers x R . That is if x is any real number, then is always greater than x and −∞ is always less than x . Moreover, Let R represent the real numbers. , 0 , , = −∞ = = + x x for all numbers x R . , < < x

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27 April 2008 5 27 April 2008 5 1999-2007 by Richard Alan Peters II p , in an n -dimensional vector space R n . Associated with each p is a value from R * . The set of pixel locations together with their associated values form the image – a set in R n+1 : In mathematical morphology a real image, I, is defined as a function that occupies a volume in a Euclidean vector space. I comprises a set, S p , of coordinate vectors (or pixel locations), Real Images Thus, a conventional, 1-band, R × C image is a 3D structure with S p R 2 and I( p ) R . By convention in the literature of () { } p I, I , I n S =∈ ⎡⎤ ⎣⎦ pp p p RR MM, S p R n , a real image is defined over all of R n .
27 April 2008 6 27 April 2008 6 1999-2007 by Richard Alan Peters II Support of an Image The support of a real image, I, is () () {} supp I I . n =∈ pp RR in R n where The complement of the support is, therefore, the set of pixel locations I( p ) ≠−∞ and I( p ) ≠∞ . That is, the support of a real image is the set pixel locations in R n such that I( p )= −∞ or I( p .

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27 April 2008 7 27 April 2008 7 1999-2007 by Richard Alan Peters II Grayscale Images If over its support, I takes on more than one real value, then I is called grayscale . The object commonly known as a black and white photograph is a grayscale image that has support in a rectangular subset of R 2 .
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This note was uploaded on 12/06/2011 for the course EECE 253 taught by Professor Alanpeters during the Summer '07 term at Vanderbilt.

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EECE253_18_GrayMorphology - EECE\CS 253 Image Processing...

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