Erosion (morphology)

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Erosion: Let A denote a binary image and B denote a structuring element. Then the erosion of A by B is given by: $A \ominus B = \{ z|(B)_z \subset A \}$

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

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.