Chapter 9

Morphological Image Processing

Preview Morphology

About the form and structure of animals and plants

Mathematical morphology Using set theory Extract image component Representation and description of region

shape

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Z2 and Z3

set in mathematic morphology represent objects 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 set is the coordinates (x,y) of pixel belong to the object and the gray levels Z3

Graphical examples (cont.)

AwwAc BwAwwBA ,

Logic operations on binary images

Functionally complete operations AND, OR, NOT

Special set operationsfor morphology

translation

AazaccA z for ,)(

reflection

BbbwwB for ,ˆ

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 geometricproperties for the objects

Structuring element (SE)

Outline erosion and Dilation Opening and closing Hit-or-miss transformation Some basic morphological algorithms Extensions to gray-scale images

Erosion

Does the structuring element fit the set?

erosion of a set A by structuring element B: all z in A such that B is in A when origin of B=z

shrink the object

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}{ Az|(B)BA z

Erosion

Erosion

Dilation

Does the structuring element hit the set?

dilation of a set A by structuring element B: all z in A such that B hits A when origin of B=z

grow the object13

}ˆ{ ΦA)Bz|(BA z

Dilation

Application of dilation: bridging gaps in images

Structuringelement

max. gap=2 pixels

Effects: increase size, fill gap

Application of erosion: eliminate irrelevant detail

original image

Squares of size1,3,5,7,9,15 pels

erosion

Erode with13x13 square

dilation

Dilation and erosion are duals

czc ABzBA )( ) (

ccz ABz )(

)( cz ABz

)ˆ( ABzBA z

BAc ˆ

Outline Preliminaries Dilation and erosion Opening and closing Hit-or-miss transformation Some basic morphological algorithms Extensions to gray-scale images

Opening Dilation: expands image w.r.t

structuring elements Erosion: shrink image erosion+dilation = original image ? Opening= erosion + dilation

BBABA ) (

Opening (cont.)

Opening (cont.)

Smooth the contour of an image, breaks narrow isthmuses, eliminates thin protrusions

Find contour Fill in contour

Closing Dilation+erosion = erosion + dilation ? Closing = dilation + erosion

BBABA )(

Closing (cont.)

Find contour Fill in contour

Smooth the object contour, fuse narrow breaks and longthin gulfs, eliminate small holes, and fill in gaps

Closing (cont.)

Properties of opening and closing

Opening

Closing

ABA of (subimage)subset a is (i) BDBCC ofsubset a is then D,ofsubset a is If (ii)

BABBA )( (iii)

BAA of (subimage)subset a is (i)

BDBCC ofsubset a is then D,ofsubset a is If (ii)BABBA )( (iii)

Noisyimage

openingRemoveouternoise

Removeinnernoise

closing

Outline Preliminaries Dilation and erosion Opening and closing Hit-or-miss transformation Some basic morphological algorithms Extensions to gray-scale images

Hit-or-miss transformation Find the location of certain shape

erosion

Find the set of pixels thatcontain shape X

X

Hit-or-miss transformation

Erosionwith (W-X)

Detect object via background

Hit-or-miss transformation Eliminate un-necessary parts

AND

Hit-or-miss transformation

)]([)( XWAXABA c

Basic morphological algorithms

Extract image components that are useful in the representation and description of shape

Boundary extraction Region filling Extract of connected components Convex hull Thinning Thickening Skeleton Pruning

Boundary extraction

)()( BAAA

Boundary extraction

Region filling How? Idea: place a point inside the region,

then dilate that point iteratively

,...3,2,1,)( 1 kABXX ckk

pX 0

Until 1 kk XX

Bound the growth

Region filling (cont.)

stop

Application: region filling

Extraction of connected components

Idea: start from a point in the connected component, and dilate it iteratively

,...3,2,1 ,)( 1 kABXX kk

pX 0

Until 1 kk XX

Extraction of connected components (cont.)

original

thresholding

erosion

Convex hull

A set A is said to be convex if the straight line segment joining any two points in A lies entirely within A.

The convex hull of a set A is the smallest convex set containing A.

i

iDAC

4

1)(

,...3,2,1 and 4,3,2,1 )( kiABXX iik

ik

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Convex hull

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Thinning

cBAA

BAABA

)(

)(

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Thickening

)( BAABA 1. A

2. Ac

3. Thinning Ac4. complement the result in 3

Skeletons

Set A Maximum disk

1. The largest disk Centered at a pixel2. Touch the boundaryof A at two or more places

Recall: Balls of erosion!

How to define a Skeletons?

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SkeletonsK

kk ASAS

0

)()(

BkBAkBAASk )()()(

})(|max{ kBAkK

))((0

kBASA k

K

k

K

kk ASAS

0

)()(

BkBAkBAASk )()()(

})(|max{ kBAkK

Skeletons

Skeleton

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Pruning

}{1 BAX

AHXX )( 23

314 XXX

H = 3x3 structuring element of 1’s

)( 1

8

12

k

kBXX

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52

53

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