Erosion (morphology)
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Erosion is one of two fundamental operations in Morphological image processing from which all other morphological operations are based. The operation is a subset of set theory, where each pixel in an image is considered to be a member of a set of pixels, rather than the usual interpretation of an image being a strict function of two dimensions.
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[edit] Definition
Erosion: Let A denote a binary image and B denote a structuring element. Then the erosion of A by B is given by: 
The concept of erosion can also be extended to greyscale images. See, for example, Gonzalez (2002).
[edit] Example
Suppose A is a 13 * 13 matrix and B is a 5 * 1 matrix:
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
Assuming that the origin B is at its center, for each pixel in A superimpose the origin of B, if B is completely contained by A the pixel is retained, else deleted.
The Erosion of A by B is given by
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 1 1 1 1 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
This means that only when B is completely contained inside A that the pixels values are retained, else it gets deleted or in other words it gets eroded.
[edit] See also
[edit] References
R. C. Gonzalez and R. E. Woods, Digital image processing, 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 2002.
