Structuring Element (SE)
A flat structuring element is a set of points at given coordinates:
B = { (0,-2),
(-1,-1),(0,-1),(1,-1),
(-2, 0),(-1, 0),(0, 0),(1, 0),(2, 0),
(-1, 1),(0, 1),(1, 1),
(0, 2)}
Display of a flat structuring element
The set of points can be represented by a binary image. The 1-pixels are the points in the set and the 0-pixels are the rest. The convention here is to have the central pixel at the (0,0) coordinates. In this way, it is possible to represent negative coordinates without the need of any additional parameter.
b=mmsecross(2);
mmseshow(b)
ans =
0 0 1 0 0
0 1 1 1 0
1 1 1 1 1
0 1 1 1 0
0 0 1 0 0
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mmseshow can also return an image suitable for displaying (Note that the pixel at the origin is marked):
A=mmsecross(2); B = mmseshow(A,'EXPAND'); mmshow(B); ;
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Minkowski Addition
Below is an example to compute the Minkowski addition of two 3x3 cross structuring element. This is implemented by the dilation of a larger image of the first structuring element by the second structuring element.
B = mmsecross;
Bi = mmseshow(B)
Bi =
0 1 0
1 1 1
0 1 0
Ai=uint8(zeros(5));
Ai(2:4,2:4) = Bi
Ai =
0 0 0 0 0
0 0 1 0 0
0 1 1 1 0
0 0 1 0 0
0 0 0 0 0
Ci=mmdil(Ai,B)
Ci =
0 0 1 0 0
0 1 1 1 0
1 1 1 1 1
0 1 1 1 0
0 0 1 0 0
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In the example below, we have the Minkowski addition of D and E:
D=uint8(zeros(7,9)); D(4,5)=1; D(3,6)=1; D(3,7)=1; D(2,8)=1; mmshow(D); ; str=mmsecross; E = mmseshow(str,'EXPAND'); mmshow(E); ; F=mmdil(D,str); mmshow(F); ;
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Digital Disks
Digital disks of radius r can be created by successive Minkowski additions of a primitive SE. mmsesum implements this Minkowski additions.
B1=mmsebox(1); BD1=mmseshow(B1,'EXPAND'); mmshow(BD1); ; % B2=mmsesum(B1,2); BD2=mmseshow(B2,'EXPAND'); mmshow(BD2); ; % B3=mmsesum(B1,3); BD3=mmseshow(B3,'EXPAND'); mmshow(BD3); ; % B4=mmsesum(B1,4); BD4=mmseshow(B4,'EXPAND'); mmshow(BD4); ;
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bi = logical(uint8([ 0 0 1; 1 1 1; 0 1 1])); B1=mmimg2se(bi); BD1=mmseshow(B1,'EXPAND'); mmshow(BD1); ; % B2=mmsesum(B1,2); BD2=mmseshow(B2,'EXPAND'); mmshow(BD2); ; % B3=mmsesum(B1,3); BD3=mmseshow(B3,'EXPAND'); mmshow(BD3); ; % B4=mmsesum(B1,4); BD4=mmseshow(B4,'EXPAND'); mmshow(BD4); ;
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Exercises
- What is the main difference between the Minkowski addition and the binary dilation?
- Make a MATLAB function that implements the Minkowski addition between two structuring elements.
function A=mink(B,C). Use the functionsmmdil,mmseshow,mmimg2se.