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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
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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Erosion
Definition 1: The erosion of two sets A and B isdefined as:i.e. The Erosion of A by B is the set of all points z, such thatB, translated by z, is contained in A
Definition 2: Minkowski’s Subtraction DefinitionIt states that the erosion of A by B is obtained by theintersection of all translates of A, with the translation distanceequal to the row and column index of reflection pixels of B thatare logical 1.
z
A
B
z
B
A
(^
)^
b
b^
B
A
B
A
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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 0, then output is 0
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Erosion: Example
9
Erosion: Example
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Dilation and Erosion: Application
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Structuring element: Shapes
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Opening and Closing: Example
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Opening and Closing: Example
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Hit or Miss Transform
A tool for shape detection or for the detection of adisjoint region in an image
Idea
Suppose we have a binary image that contains certainshapes (circles, squares, lines, etc,….) called image A
We use another image or matrix to search image A for aparticular pattern of bits. We will call this pattern “shape D”
We then search image A for shape D
Whenever there is a ‘hit’, we indicate where the center ofshape D was on image A.
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Hit or Miss Transform: Idea
We take an binary image
We pick a shape to look for (an MxN matrix of 1’s and 0’s
We then look for those certain patterns of pixels
Every time we find the pattern, we make a mark at thecenter of the mask
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Hit or Miss Transform ^
We want to detect disjoint region D in space A ^
Let D be enclosed by a small window W ^
The local background of D with respect to W is (W-D) ^
D is found by
(^
)^
(^
(^
))
c
A
D
A
W
D
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Hit or Miss Transform
If B denotes the set composed of D and itsbackground, the match/hit (set of matches/hits) of Bin A is
Generalized notation: B = (B
, B 1
) where^2
B
: the set formed from elements of B associated with an 1 object i.e. B
1
= D
B
: the set formed from elements of B associated with the 2 background i.e. B
2
= W-D
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(
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))
c
A
B
A
D
A
W
D
1
2
(
)
(
)
c
A
B
A
B
A
B
Thus,
contains all the origin points at which,
simultaneously, B
1
found a hit in A and B
2
found a hit in A
c
A
B
