Chapter 9 Morphological Image Processing...
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Transcript of Chapter 9 Morphological Image Processing...
Chapter 9
Morphological Image Processingการทำ�างานก�บรปภาพด้�วยว�ธี�มอร�โฟโลจิ�คั�ล
Digital Image Processing by K.Ratchadaporn
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.
Digital Image Processing by K.Ratchadaporn
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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.
Digital Image Processing by K.Ratchadaporn
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Example
Binary Image Set A is set of
black pixelsA = {(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)}
0
0
7
7
Digital Image Processing by K.Ratchadaporn
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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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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
Aa
Aa
bydenote
Digital Image Processing by K.Ratchadaporn
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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
BA Example:
X={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)} andY={(1,2),(2,1),(2,2),(2,3),(3,2)}
So, XY
Digital Image Processing by K.Ratchadaporn
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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
BAC
Digital Image Processing by K.Ratchadaporn
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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
BAC
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Mutually Exclusive Property
BA
Two sets A and B is disjoint or mutually exclusive if they have no common elements
A B
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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.
AwwAc |
cBABwAwwBA ,|
Digital Image Processing by K.Ratchadaporn
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Summary
BA BA
cA BA
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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
BbforbwwB ,|ˆ
AaforzaccA z ,|)(
Digital Image Processing by K.Ratchadaporn
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Reflection
B
B̂
Digital Image Processing by K.Ratchadaporn
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Translation
zA)(
z1
z2
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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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Logic Operations & Binary Images
AND
OR
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Logic Operations & Binary Images
NAND
XOR
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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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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, zSuch that B and A overlap by at least one element
Note : Set B is commonly referred to as the “structuring element”
})ˆ(|{ ABzBA z
}])ˆ[(|{ AABzBA z
Digital Image Processing by K.Ratchadaporn
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Example : Dilation
2
y
2
y
2x
2x
d
d
x
y
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Example : Dilation
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Application : Dilation0 1 01 1 10 1 0
Digital Image Processing by K.Ratchadaporn
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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
})(|{ ABzBA z
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Example : Erosiond
d
x
y
2
y
2
y
2x
2x
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Example : Erosion
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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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Erosion Complement
Dilation and Erosion are duals of each other with respect to set complementation and reflection
BABA cc ˆ)(
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Proving
Starting with the definition of erosion
If set (B)z is contained in set A, then
thus
cz
c ABzBA })(|{)(
cz AB)(
BA
ABzABc
cz
c
ˆ
)(|)(
Digital Image Processing by K.Ratchadaporn
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About Dilation & Erosion
Dilation expands an image.
Erosion shrinks an image.
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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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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.
BBABA )(
Digital Image Processing by K.Ratchadaporn
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OpeningBA
B
A
BA
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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.
BBABA )(
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ClosingBA
B
A
BA
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Opening and Closing
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Apply for Problem
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HIS-or-MISS TranslationZYXA W
XY
XW
Z
cA XA
)( XWAc
)]([)( XWAXA c
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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
)]([)(* XWAXABA c
][)(* 21 BABABA c
]ˆ[)(* 21 BABABA