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Definitions ^ MorphologyA branch of biology which deals with the form andstructure of animals and plants ^ Mathematical Morphology^ ^ A tool for extracting image components that are useful inthe representation and description of region shapes^ ^ The language of mathematical morphology is Set Theory
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Set theory ^ The set space of binary image is Z
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i.e. each element of the set is a 2D vector whosecoordinates are the (x,y) of a black (or white,depending on the convention) pixel in the image The set space of gray level image is Z
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i.e. each element of the set is a 3D vector: (x,y) andintensity level. NOTE:Set Theory and Logical operations are covered in:Section 9.1, Chapter # 9, 2
nd^ Edition DIP by Gonzalez Section 2.6.4, Chapter # 2, 3
rd^ Edition DIP by Gonzalez
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Set theory ^ Subset: if every element of A is also an element ofanother set B, the A is said to be a subset of B ^ Union: The set of all elements belonging either to A,B or both ^ Intersection: The set of all elements belonging toboth A and B
A^ B C A^ B ^ D A^
B ^
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Set theory ^ Two sets A and B are said to be disjoint or mutuallyexclusive if they have no common element ^ Complement: The set of elements not contained inA ^ Difference of two sets A and B, denoted by A – B, isdefined asi.e. the set of elements that belong to A, but not to B
A^ B^
{ | } c A w w
A ^
{^ |^
,^
}^
c
A^ B^
w w^
A w^
B^ A
B
^ ^
^
^
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Set theory ^ Reflection of set Bi.e. the set of element w, such that w isformed by multiplying each of twocoordinates of all the elements of set B by -1 ^ Translation of set A by point z = (z
,z^ ), denoted 1 2
^ {^ (A), is defined asz^
|^
,^ for^
}
B^ w w
b^
b^ B ^
^
(^ )^
{^ |^
,^ for^
}
A^ w wz
a^ z
a
A
^
^ ^
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Logical operations
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Dilation ^ Definition 1: The dilation of two sets A and B isdefined as:i.e. when the reflection of set B about its origin is shiftedby z, the dilation of A by B is the set of all displacementssuch that
overlaps A by at least one element
^ Definition 2: Minkowski’s Addition DefinitionIt states that the dilation of A by B is obtained by the union ofall translates of A, with the translation distance equal to therow and column index of pixels of B that are logical 1
{ | ( )^
z
A^ B^
z^ B^
A
^ ^
B
(^ ) b b B
A^ B^
A
^ ^
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Dilation ^ Set B is commonly referred to as the structuring elementin dilation as well as in many morphological operations ^ Effects^ ^ Expands the size of 1-valued objects^ ^ Smoothes object boundaries^ ^ Closes holes and gaps
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Dilation ^ Dilation process is analogous to convolution process^ ^ B (structuring element) is considered as convolutionmask^ ^ B is flipped around the origin and then successivelydisplacing it so that it slides over set (image) A^ ^ But, Dilation is based on set operations whereasconvolution is based on arithmetic operations ^ Rule for Dilation^ In a binary image, if any of the pixel (in the neighborhooddefined by structuring element) is 1, then output is 1
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Dilation: Example^1
0 0
1 1
1 0
0
0 1
0 1
1
1 0
1 1
1 0
0 0
1
0 0
1 1
1
^
1 0
0
0 1
1 0
0
0 1
0 0
1
1 0
0 1
1 0
0 0
1
0 0
0 1
1
^
^
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Dilation: Example
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Dilation: Example
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Erosion ^ Effects^ ^ Shrinks the size of 1-valued objects^ ^ Smoothes object boundaries^ ^ Removes small objects ^ Rule for Erosion^ In a binary image, if any of the pixel (in the neighborhooddefined by structuring element) is 1, then output is 1
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Erosion: Example^1
1 1
1 1
1 1
1
1 0
1 1 1
1 1
0 0
0 0
1 0
0
1 0
0 0
0
