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.


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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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


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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


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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


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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


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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 ,|


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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 ,|)(


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Reflection

B

B̂


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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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AND

OR


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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


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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


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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

)(|)(


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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 )(


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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

chapter 9 morphological image processing...

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