Shih_Image_Processing_and_Mathematical_Morpholo.pdf

16 Image Processing and Mathematical Morphology
Example 2.1:
A = {(0, 2), (1, 1), (1, 2), (2, 0), (2, 2), (3, 1)}
b = (0, 1)
(A) b = {(0, 3), (1, 2), (1, 3), (2, 1), (2, 3), (3, 2)}
0
1
2
3
0 1 2 3
0
0
1
0
0
1
0
1
1
1
1
0
0
0
0
0
0
1
2
3
0 1 2 3
0
0
0
0
0
0
1
0
A A (0,1)
Defi nition 2.2: Let A, B Ã E N . The binary dilation of A by B, denoted by A � b B,
is defi ned as
A � b B = {c � E N |c = a + b for some a � A and b � B}. (2.9)
The subscript of dilation “b” indicates binary. Equivalently, we may write
∪ ∪
A � B = ( A) = ( B)
.
b b a
bŒB aŒA 0
1
0
1
1
1
1
0
(2.10)
The representation, A �b B = ∪ (A) b�B b , states that the dilation of A by B can
be implemented by delaying the raster scan of A by the amounts corresponding
to the points in B and then ORing the delayed raster scans. That is
Example 2.2:
( ) OR[AND( , )].
A �bB
(,) i j = B( m, n) A( i-m, j-n) mn ,
A = {(0, 1), (1, 1), (2, 1), (3, 1)}
B = {(0, 0), (0, 2)
A �b B = {(0, 1), (1, 1), (2, 1), (3, 1), (0, 3), (1, 3), (2, 3), (3, 3)}.
0
1
2
3
0 1
0 1
0 1
0 1
0 1
2
0
0
0
0
3
0
0
0
0
0
0 1
1
0
A B A ⊕ b B
2
1
0
1
2
3
0
0
0
0
0
1
1
1
1
1
2
0
0
0
0
(2.11)
3
1
1
1
1