chapter9 morphological image processing

7/31/2019 Chapter9 Morphological Image Processing

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Chapter 9: Morphological

Image Processing

Digital Image Processing


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Mathematic Morphology used to extract image components that are

useful in the representation and description of

region shape, such as boundaries extraction

skeletons

convex hull

morphological filtering thinning

pruning


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Mathematic Morphologymathematical framework used for:

pre-processing noise filtering, shape simplification, ...

enhancing object structure skeletonization, convex hull...

Segmentation watershed,

quantitative description area, perimeter, ...


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Z2

and Z3

set in mathematic morphology representobjects in an image binary image (0 = white, 1 = black) : the

element of the set is the coordinates (x,y)of pixel belong to the object Z2

gray-scaled image : the element of the setis the coordinates (x,y) of pixel belong to theobject and the gray levels Z3


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Basic Set Theory


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Reflection and Translation},|{ Bfor bbwwB

},|{)( Afor azaccA z


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Logic Operations


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Example


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Structuring element (SE)

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small set to probe the image under study for each SE, define origo shape and size must be adapted to geometricproperties for the objects


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Basic idea in parallel for each pixel in binary image:

check if SE is satisfied

output pixel is set to 0 or 1 depending onused operation

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How to describe SE many different ways!

information needed: position of origo for SE

positions of elements belonging to SE

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Basic morphological operations Erosion

Dilation

combine to Opening object

Closening background

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keep general shape but

smooth with respect to


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Erosion Does the structuring element fit theset?

erosion of a set A by structuring elementB: all z in A such that B is in A whenorigin of B=z

shrink the object

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}{ Az|(B)BA z


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Erosion

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Erosion

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Erosion

}{ Az|(B)BAz


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Dilation Does the structuring element hit theset?

dilation of a set A by structuringelement B: all z in A such that B hits Awhen origin of B=z

grow the object17

}{ A)Bz|(BAz


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Dilation

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Dilation

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Dilation

}{ A)Bz|(BA z

B = structuring element


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Dilation : Bridging gaps


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useful erosion

removal of structures of certain shape and

size, given by SE Dilation

filling of holes of certain shape and size,

given by SE

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Combining erosion and

dilation WANTED:

remove structures / fill holes

without affecting remaining parts

SOLUTION:

combine erosion and dilation

(using same SE)

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Erosion : eliminating irrelevant

detail

structuring element B = 13x13 pixels of gray level 1


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Openingerosion followed by dilation, denoted

eliminates protrusions

breaks necks smoothes contour

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BBABA )(


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Opening

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Opening

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Opening

BBABA )(

})(|){( ABBBA zz


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Closingdilation followed by erosion, denoted

smooth contour

fuse narrow breaks and long thin gulfs eliminate small holes

fill gaps in the contour

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BBABA )(


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Closing

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Closing

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Closing

BBABA )(


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PropertiesOpening(i) AB is a subset (subimage) of A

(ii) If C is a subset of D, then C

B is a subset of D

B(iii) (A B) B = A BClosing(i) A is a subset (subimage) of AB

(ii) If C is a subset of D, then C B is a subset of D B(iii) (A B) B = A B

Note: repeated openings/closings has no effect!


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Duality Opening and closing are dual with respect

to complementation and reflection

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)()( BABAcc


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Useful: open & close

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Application: filtering

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Hit-or-Miss Transformation

(HMT) find location of one shape among a set of shapes

template matching

composite SE: object part (B1) and backgroundpart (B2)

does B1 fits the object while, simultaneously,B2 misses the object, i.e., fits the background?

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Hit-or-Miss Transformation

)]([)( XWAXABAc


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Boundary Extraction

)()( BAAA


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Example


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Region Filling

,...3,2,1)( 1

kABXX

c

kk


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Example


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Extraction of connected

components


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Example

iii


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Convex hull

A set Ais issaid to be

convex ifthe straightline segmentjoining anytwo pointsin A liesentirely

within A.

i

iDAC

4

1)(

,...3,2,1and4,3,2,1)( kiABXXii

k

i

k

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Thinning

cBAA

BAABA

)(

)(


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Thickening

)( BAABA

K

ASAS )()(


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Skeletons kkASAS

0

)()(

BkBAkBAASk

)()()(

})(|max{ kBAkK

))((0

kBASA k

K

k


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}{BAX H 3 3 i l f 1


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Pruning

}{1 BAX

AHXX )( 23

314 XXX

H = 3x3 structuring element of 1s)( 1

8

12

k

k

BXX


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5 basic structuring elements

chapter9 morphological image processing

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