mathematical morphology - iit bombayavikb/gnr401/dip/dip_401_lecture_5.pdf · mathematical...

MATHEMATICAL MORPHOLOGY

1NR401 Dr. A. Bhattacharya

Introduction

Mathematical morphology (MM) is the study of image processing methods based on the shape or form of objects

MM provides the ability to probe images using likely shapes of objects expected in

the image manipulate images using these probes

An approach for processing digital image based on its shape

A mathematical tool for investigating geometric structurein image

The language of morphology is set theory

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NR401 Dr. A. Bhattacharya

Goal of morphological operations

Simplify image data, preserve essential shape characteristics and eliminate noise

Permits the underlying shape to be identified and optimally reconstructed from their distorted, noisy forms

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Shape Processing and Analysis

Identification of objects, object features and assembly defects correlate directly with shape

Shape is a prime carrier of information in machine vision

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Binary Mathematical Morphology


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USED FOR :

Image pre-processing noise filtering, shape smoothing, ...

Extraction of object structure skeleton, convex hull...

Quantitative object description Object area, perimeter, ...

Presence or absence of objects Hit/Miss transform

Mathematical Tools for Binary Morphology


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Based on set theoretic operations

Union

Intersection

Complement

Negation

Translation etc.

Set theoretic Principles


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If a=(x,y) is an element in A: aA

If a=(x,y) is not an element in A: aA

Subset:every element in A is also in B (subset): AB

Let A be a set in Z2

Set theoretic Principles


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A

B

NOT A

A AND B

A OR BA XOR B

Structuring element (SE)


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Small set to probe the image under study for each SE, define origin

Shape and size must be adapted to geometric properties for the objects

Morphological Operations


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The primary morphological operations are dilation and erosion

More complicated morphological operators can be designed by means of combining erosions and dilations

Dilation


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Dilation is the operation that combines two sets using vector addition of set elements.

Let A and B are subsets in 2-D space. A: image undergoing analysis, B: Structuring element, denotes dilation

},{ 2 BbAasomeforbacZcBA

Dilation


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••• •

•

• •• •• • •

• •

• •B

A BA

Dilation


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Let A be a Subset of and . The translation of A by x is defined as

The dilation of A by B can be computed as the union of translation of A by the elements of B

2Z 2Zx

},{)( 2 AasomeforxacZcA x

Aa

aBb

b BABA

)()(

Dilation


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••• •

•

••• •

•• •• •• • •

• •

)0,0(A )1,0(A

• • B

BA

Dilation


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xB)(

BA

A

Aa

aBb

b BABA

)()(

Dilation


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SE=

Used with permission of Dr. Lucia Ballerini

[email protected]

Dilation


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SE=

Properties of Dilation


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Commutative

Associative

Extensivity

Dilation is increasing

BAABif ,0

DBDAimpliesBA

ABBA

CBACBA )()(

Properties of Dilation


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Translation Invariance

Linearity

Containment

Decomposition of structuring element

xx BABA )()(

)()()( CBCACBA

)()()( CBCACBA

)()()( CABACBA

Erosion


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Erosion is the morphological dual to dilation. It combines two sets using the vector subtraction of set elements.

Let denotes the erosion of A by BBA

){

}..,{2

2

BbeveryforAbxZx

baxtsAaanexistBbeveryforZxBA

Erosion


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• • • • ••••

• •

B

A BA

• • • •

Erosion


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Erosion can also be defined in terms of translation

In terms of intersection

))({ 2 ABZxBA x

Bb

bABA

)(

Erosion


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• • • • ••••

• •

• • • •

• • • • ••••

)1,0(1A )0,0(A

BA

Erosion


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xB)(

A

BA

))({ 2 ABZxBA x

Erosion


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Used with permission of Dr. Lucia Ballerini

[email protected]

Properties of Erosion


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Erosion is not commutative!

Extensivity

Dilation is increasing

Chain rule

ABBA

ABABif ,0

)...)(...()...( 11 kk BBABBA

CABAimpliesCBBCBAimpliesCA ,

Properties of Erosion


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Translation Invariance

Linearity

Containment

Decomposition of structuring element

)()()( CBCACBA

)()()( CBCACBA

)()()( CABACBA

xxxx BABABABA )(,)(

Duality Relationship


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Dilation and Erosion transformation bear a marked similarity, in that what one does to image foreground and the other does for the image background.

, the reflection of B, is defined as

Erosion and Dilation Duality Theorem

2ZB B

},{ bxBbsomeforxB

BABA cc )(

Duality Relationship


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Erosion and dilation are dual with respect to complementation and reflection

Erosion and dilation are not duals with respect to each other!

(A ⊝ B)c = Ac ⊕ B

Duality


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Opening and Closing are dual with respect to complementation and reflection

BABA CC ˆImportant: If origin is included in the structuring element B,

A⊝B ⊆ A ○ B ⊆ A ⊆ A ● B ⊆ A ⊕ B

Opening and Closing


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Opening and closing are iteratively applied dilation and erosion

• Opening•eliminates protrusions• breaks thin joints• smoothes contour

• Closing• smooth contour• fuse narrow breaks and long thin gulfs• eliminate small holes• fill gaps in the contour

BBABA )(

BBABA )(

Closing


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Closing by 3x3 Box SE

AΘB A BCopyright Dr. Lucia Ballerini [email protected]

Closing


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Closing by 3x3 Plus SE

A A⊕B A BCopyright Dr. Lucia Ballerini [email protected]

Physical Meaning of Closing


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See B as a ”rolling ball”

Boundary of AB = points in B that reach farthestinto A when B is rolled outside A

Closing


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0,| ABBwwBA zz

Opening and Closing


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xB

BA

ABBABA )(

}{ ABx

xx

BBA

Opening and Closing


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They are idempotent. Their reapplication has not further effects to the previously transformed result

BBABA )(

BBABA )(

Opening and Closing


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Translation invariance

Anti-extensivity of opening

Extensivity of closing

Duality

BABA x )( BABA x )(

ABA

BAA

BABA cc )(

Opening


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Opening by Box SE 3x3 A⊖B A∘B

Copyright Dr. Lucia Ballerini [email protected]

Opening


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Opening by Plus SE 3x3A⊖B A∘BCopyright Dr. Lucia Ballerini [email protected]

Physical Meaning of Opening


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Treat the structuring element as a rolling ball

Boundary of A∘B = points in B that reach farthestinto A when B is rolled inside A

Opening


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ABBBA zz |

mathematical morphology - iit bombayavikb/gnr401/dip/dip_401_lecture_5.pdf · mathematical...

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