Erosion (morphology)

From Wikipedia, the free encyclopedia

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.

Contents

[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: A \ominus B = \{ z|(B)_z \subset A \}

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.


This applied mathematics-related article is a stub. You can help Wikipedia by expanding it.
Languages