Morphological Image Processingresearch.iaun.ac.ir/pd/pourghassem/pdfs/UploadFile_4849.pdf · 9...

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Morphological Image Processing

Chapter 9

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Instructor: Hossein Pourghassem


Morphology deals with form and structure

Mathematical morphology is a tool for extracting image components useful in:

representation and description of region shape (e.g. boundaries)pre or post processing (filtering thinning etc )

Islamic Azad University, Najafabad Branch, Department of Electrical Engineering, Dr. H. Pourghassem, 2

pre- or post-processing (filtering, thinning, etc.)

Based on set theory

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Morphology

Sets represent objects in imagesSets in binary images (x y)Sets in binary images (x,y)Sets in gray scale images (x,y,g)Some morphological operations:

Dilation & Erosion


Opening & ClosingHit-or-Miss Transform

Basic Algorithms

Basic Concepts of Set Theory

Assume A is a set in If a=(a1,a2) an element of A, then we write a∈A

2ZIf a (a1,a2) an element of A, then we write a∈AIf not, then a∉A∅: null (empty) setTypical set specification: C={w|w=-d, for d ∉ D}If A is the subset of B then we write A⊆B


⊆Union of A and B: C=A∪BIntersection of A and B: D=A∩B

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Disjoint sets: A∩B= ∅Complement of A:

}|{ AwwAc ∉

Basic Concepts of Set Theory

Difference of A and B:A-B={w|w ∈ A, w ∉ B}=Reflection of B:

}|{ AwwA ∉=

A∩ Bc

ˆ B = {w | w = −b,b∈ B}


Translation of A by z=(z1,z2): B {w | w b,b∈ B}

(A)z = {c | c = a+ z,a ∈ A}



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Logical Operation


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Dilation & Erosion

Dilation:

∅: empty set; A,B: sets in Z2

Dilation of A by B:

})ˆ(|{ ∅≠∩=⊕ ABzBA


})(|{ ∅≠∩=⊕ ABzBA z

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Dilation & Erosion

Dilation:

Obtaining the reflection of B about its origin and thenshifting this reflection by z

The dilation of A by B then is the set of all zdisplacements such that and A overlap by at leastone nonzero element

B̂


one nonzero element…

Dilation & Erosion

Dilation:

}])ˆ[(|{ AABxBA x ⊆∩=⊕

B is the structuring element in dilation.


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Dilation & Erosion

Erosion:

i.e. the erosion of A by B is the set of all points xsuch that B, translated by x, is contained in A.

})(|{ ABxBA x ⊆=


In general:

BABA cc ˆ)( ⊕=

Dilation & Erosion


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Opening & Closing

In essence, dilation expands an image and erosion shrinks it.

Opening:generally smoothes the contour of an image, breaks isthmuses, eliminates protrusions.

Cl i


Closing:smoothes sections of contours, but it generally fuses breaks, holes, gaps, etc.

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Opening & Closing

Opening of A by structuring element B:

BBABA ⊕= ) (o

• Closing:


BBABA )( ⊕=•



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Opening & Closing

BDBCDCifABA

⊆→⊆⊆

oo

o

BAABDBCDCif

BABBACCif

•⊆•⊆•→⊆

=⊆→⊆

)( ooo


BABBABAA

•=•••⊆)(

Hit-or-Miss Transform

Morphological hit or miss transform is aMorphological hit-or-miss transform is a basic tool for shape detection.

Definitions:B (B1,B2)


B1 is the set of elements of B associated with an objectB2 is the set of elements of B associated with thecorresponding background.

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Hit-or-Miss Transform

A * B contains all the origin points at which, simultaneously:

B1 found a match (“hit”) in A and, B2 found a match in Ac.

)()(* BABABA c

)(21 XWBandXB −==


or

) () (* 21 BABABA c∩=*

)ˆ() (* 21 BABABA ⊕−=*

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Example Basic Morphological Algorithms

Purpose: to extract image components that are useful in the representation and description of shape.

Boundary Extraction:


) ()( BAAA −=β

Morphological Image Processing•Boundary Extraction Example


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•Boundary Extraction Example


Region Filling

Region filling based on set dilation, complementation andintersections.

Beginning with a point p inside the boundary the objective is

Xk = (Xk−1 ⊕ B)∩ Ac

Where X =p

Beginning with a point p inside the boundary, the objective isto fill the entire region with 1’s.


Where X0=p when Xk=Xk-1 the algorithm has converged.

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Xk = (Xk−1 ⊕ B)∩ AcRegion Filling


Morphological Image Processing• Example of Region filling


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Basic Morphological Algorithms

Extraction of Connected Components:

ABXX kk ∩⊕= − )( 1 k=1,2,3,…

Where X0=p


when Xk=Xk-1 the algorithm has converged.



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Example of Extraction of Connected Components


Convex Hull H of an arbitrary set S is the smallest convexcontaining S.

Convex Hull

Convex deficiency: H-S. these are useful for object description.The procedure consists of iteratively applying the hit-misstransform to A with B.

U

0

1 ,...3,124,3,2,1)*( −

=

===i

iik

ik

AX

kandiABXX


U4

1

1

0

)(=

−

=

==⇒=

i

i

ik

iconv

iik

ik

DAC

XXDXXif

AX

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Convex Hull


× indicate Don’t care

Thinning

CBAABAABA

)*(*

I=

−=⊗ },...,,,{}{ 321 nBBBBB =

Algorithm:

))...))((...((}{ 21 nBBBABA ⊗⊗⊗=⊗1−ii BofversionrotatedaisBwhere


1- thin with B1 to Bn

2- go to step 1 until no further change is obtained

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Thinning


Thickening

))...))((...((}{)*(

21 nBBBABABAABA

•••=•

=• U

• Structuring elements B in the above equation is theNot of B in thinning operator.

• Thickening can be performed by thinning Ac

• Connectivity check may be performed after


• Connectivity check may be performed afterthickening.

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Thickening


Skeletons


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Skeletons


Skeleton

)()(0

==

ASASK

kkU

})(|max{...)))(...()(

)()()(0

∅≠==

−==

BAkKBBBAkBABkBAkBAASk

k

o


BBBAkBA

kBASAK

kk

⊕⊕⊕=⊕

⊕==

...)))(...()(

))((0U

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Skeletons


PruningPost processing after thinning or and skeletonizing to cleanup parasitic components.M th dMethod

- Thin with proper structuring element n times

- Find end points}{1 BAX ⊗=


- Dilate end points with proper structuring element n times

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23 )(XXX

AHXXU

I

=⊕=

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Pruning


Gray Scale Dilation

})()()(|)()({),)((

DDtbtftsbf

+++++=⊕

}),(;)(),(|),(),(max{ bf DyxDytxsyxbytxsf ∈∈+++++

};)(|)()(max{))((

bf DxDxsxbxsfsbf

∈∈+++=⊕


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Gray Scale Dilation


Gray Scale Erosion

}),(;)(),(|),(),(min{),)((

bf DyxDytxsyxbytxsftsbf

∈∈++−++=

};)(|)()(min{))((

bf DxDxsxbxsfsbf

∈∈+−+=

)()(ˆ),)(ˆ(),)((

yxbyxb

tsbftsbf c

−−=

⊕=


),(),(

),(),(

yxfyxf

yxbyxbc −=

−−=

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Gray Scale Erosion


Gray scale Erosion and Dilation

Dilation produces the image with


Dilation produces the image with

brighter than and dark details have been reduced

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Gray Scale Opening and Closing

)( bbfbf o ⊕=

),(

ˆ)(

)()(

yxff

bfbf

bbfbfbbfbf

c

cc o

o

−=

=•

⊕=•⊕=


)ˆ()(

),(

bfbf

yxff

o−=•−



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Morphological Image Processingresearch.iaun.ac.ir/pd/pourghassem/pdfs/UploadFile_4849.pdf · 9...

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