Digital Image Processing_L8.ppt

7/27/2019 Digital Image Processing_L8.ppt

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Morphology

Morphology deals with form and structure

Mathematical morphology is a tool forextracting image components useful in:

representation and description of region shape

(e.g. boundaries)

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

Based on set theory


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Morphology

Sets represent objects in images

Sets in binary images (x,y)

Sets in gray scale images (x,y,g)

Some morphological operations:

Dilation & ErosionOpening & Closing

Hit-or-Miss Transform

Basic Algorithms


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Basic Concepts of Set Theory

A is a set in , a=(a1,a2) an element of A, aA If not, then aA

: null (empty) set

Typical set specification: C={w|w=-d, for d D}

A subset of B: AB Union of A and B: C=AB

Intersection of A and B: D=AB

Disjoint sets: AB=

Complement of A: Difference of A and B: A-B={w|w A, w B}=

Reflection of B:

Translation of A by z=(z1,z2):

Z

2

Ac{w|w

A

B{w|wb,b

(A)z{c|caz,a


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Morphological Image Processing


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Dilation & Erosion

Basic definitions:

A,B: sets in Z2with components a=(a1

,a2

)

and b=(b1,b2)

Translationof A by x=(x1,x2), denoted by (A)xis

defined as:

(A)x= {c| c=a+x, for aA}


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Dilation & Erosion

More definitions:

Reflectionof B: = {x|x=-b, for bB}

Complementof A: Ac= {x|xA}

Differenceof A & B: A-B = {x|xA, x B} = ABc

B


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Dilation & Erosion

Dilation:

: empty set; A,B: sets in Z2

Dilation of A by B:

B{x|(B)xA


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Dilation & Erosion

Dilation:

Obtaining the reflection of B about its origin and

then shifting this reflection by x

The dilation of A by B then is the set of all x

displacements such that and A overlap by at

least one nonzero element

B


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Dilation & Erosion

Dilation:

])[(|{ ABxB x

B is the structuring elementin dilation.


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Dilation & Erosion

Erosion:

i.e. the erosion of A by B is the set of all points x

such that B, translated by x, is contained in A.

In general:ABA

cc)(

)(|{ BxBAx


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Opening & Closing

In essence, dilation expands an image anderosion shrinks it.

Opening: generally smoothes the contour of an image,

breaks isthmuses, eliminates protrusions.

Closing: smoothes sections of contours, but it generally

fuses breaks, holes, gaps, etc.


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Opening & Closing

Openingof A by structuring element B:

BABA )(

Closing:

BABA )(


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Morphological hit-or-miss transform is a

basic tool for shape detection.

Definitions:

B (B1,B2)

B1is the set of elements of B associated with an object

B2is the set of elements of B associated with thecorresponding background.


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AB contains all the origin points at which,

simultaneously:

B1found a match (hit) in A and

B2found a match in Ac.

()( 21 ABAB c

or

()( 21 ABAB


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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:

()( AAA


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Region Filling Examples

k(X

k1B)


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Morphological Image ProcessingRegion Filling


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Basic Morphological Algorithms

Extraction of Connected Components:

BXkk )( 1 k=1,2,3,

Where X0=p

when Xk=Xk-1the algorithm has converged.


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Digital Image Processing_L8.ppt

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