1. Morphological Image Processing การทำงานกับรูปภาพด้วยวิธีมอร์โฟโลจิคัล
Chapter 9
Morphological Image Processing
การทำงานกับรูปภาพด้วยวิธีมอร์โฟโลจิคัล

2. Meaning of “Morphology”
Commonly a branch of biology that deals with the form and structure of animals and plants.
“mathematical morphology” as a tool for extracting image components that representation and description of region shape, such as boundaries, skeletons, and the convex hull.
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3. Mathematical morphology
The language of mathematical morphology is set theory.
Sets in mathematical morphology represent objects in an image.
Example: the set of all back pixels in a binary image is a complete morphological description of the image. Each element of set is a tuple(2D vector) whose coordinates are the (x,y) coordinates of a black pixel in the image.
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4. Digital Image Processing by K.Ratchadaporn
Example
Binary Image
Set A is set of black pixels 7
A = {(3,1),(4,1),(2,2),(5,2),
(2,3),(5,3),(1,4),(2,4),(3,4),
(4,4),(5,4),(6,4),(1,5),(6,5),
(1,6),(6,6)} 7
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5. Basic Concepts of Set Theory
Definition of Elements
What Subset is
Union Operation
Intersection Operation
Mutually exclusive Property
Complement Operation
Difference Operation
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6. Definition of Elements
Let A be a set in Z2. If a = (a1,a2) is an element of A, then we write
Similarly, if a isn’t an element of A we write
An arbitrary set in Zn has elements n-tuples as (z1,z2,. . .,zn)
The set with no elements is called the null or empty set
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7. Digital Image Processing by K.Ratchadaporn
What subset is
If every element of a set A is also an element of another set B then A is said to be a subset of B, denoted as
Example:
X={(1,1),(1,2),(1,3),(2,1),(2,2),
(2,3),(3,1),(3,2),(3,3)}
and
Y={(1,2),(2,1),(2,2),(2,3),(3,2)}
So,
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8. Digital Image Processing by K.Ratchadaporn
Union Operation
The union of two sets A and B denoted by
Set C is the set of all elements belonging to either A, B, or both
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9. Intersection Operation
The intersection of two sets A and B denoted by
Set D is the set of all elements belonging to both A and B
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10. Mutually Exclusive Property
Two sets A and B is disjoint or mutually exclusive if they have no common elements A B
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11. Complement & Difference
The complement of a set A is the set of elements not contained in A:
Difference of two sets A and B, denoted A-B, is defined as
This is the set of elements that belong to A, but not to B.
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12. Digital Image Processing by K.Ratchadaporn
Summary
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13. Digital Image Processing by K.Ratchadaporn
Addition definition
Two additional definition that are used extensively in morphology
The reflection of set B is defined as
The translation of set A by point z=(z1,z2) is defined as
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14. Digital Image Processing by K.Ratchadaporn
Reflection
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15. Digital Image Processing by K.Ratchadaporn
Translation z1 z2
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16. Logic Operations & Binary Images
The principal logic operations used in image processing are AND, OR, and NOT(Complement)
Logic operations are preformed on a pixel by pixel basis between corresponding pixels of two or more images(except NOT)
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17. Logic Operations & Binary Images
AND OR
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18. Logic Operations & Binary Images
NAND
XOR
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19. Fundamental morphological processing
Two Operation are fundamental to morphological processing:
Dilation
Erosion
Many of the morphological algorithms are based on these two primitive operations
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20. Digital Image Processing by K.Ratchadaporn
Dilation
Let A and B as set in Z2,
The dilation of A by B is defined as
Then it is the set of all displacements, z
Such that B and A overlap by at least one element
Note : Set B is commonly referred to as the “structuring element”
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21. Digital Image Processing by K.Ratchadaporn
Example : Dilation d d x y
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22. Digital Image Processing by K.Ratchadaporn
Example : Dilation
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23. Application : Dilation1
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24. Digital Image Processing by K.Ratchadaporn
Erosion
Let A and B as set in Z2,
The erosion of A by B is defined as
Then it is the set of all points z
Such that B translated by z, is contained in A
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25. Digital Image Processing by K.Ratchadaporn
Example : Erosion d d x y
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26. Digital Image Processing by K.Ratchadaporn
Example : Erosion
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27. Digital Image Processing by K.Ratchadaporn
Application : Erosion
(a) Image of squares of size 1,3,5,7,9 and 15 pixels on the side
(b) Erosion of (a) with a square structuring element of 1’s, 13 pixels on the side
(c) Dilation of (b) with a same structuring element
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28. Digital Image Processing by K.Ratchadaporn
Erosion Complement
Dilation and Erosion are duals of each other with respect to set complementation and reflection
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29. Digital Image Processing by K.Ratchadaporn
Proving
Starting with the definition of erosion
If set (B)z is contained in set A, then
thus
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30. About Dilation & Erosion
Dilation expands an image.
Erosion shrinks an image.
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31. Digital Image Processing by K.Ratchadaporn
Opening and Closing
Opening generally smoothes the contour of an object, breaks narrow isthmuses, and eliminates thin protrusions.
Closing also tends to smooth sections of contours but, as opposed to opening, it generally fuses narrow breaks and long thin gulfs, eliminates small holes, and fills gaps in the contour
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32. Digital Image Processing by K.Ratchadaporn
Opening
The opening of set A by structuring element B is defined as
Thus, the opening A by B is the erosion of A by B, followed by a dilation of the result by B.
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33. Digital Image Processing by K.Ratchadaporn
Opening A B
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34. Digital Image Processing by K.Ratchadaporn
Closing
The closing of set A by structuring element B is defined as
Thus, the opening A by B is the dilation of A by B, followed by a erosion of the result by B.
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35. Digital Image Processing by K.Ratchadaporn
Closing A B
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36. Digital Image Processing by K.Ratchadaporn
Opening and Closing
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37. Digital Image Processing by K.Ratchadaporn
Apply for Problem
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38. HIS-or-MISS Translation
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39. HIS-or-MISS Translation
If B denotes the set composed of X and its background, The match (or set of matches) of B in A, denoted is
If B1=X and B2=(W-X)
By using the definition of set differences given
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