Chapter 9

Morphological Image Processing

PreviewMorphology: denotes a branch of biology that deals with the form and structure of animals and planets.Mathematical morphology: tool for extracting image components that are useful in the representation and description of region shapes.Filtering, thinning, pruning.

ScopeWill focus on binary images.Applicable to other situations. (Higher-dimensional space)

Set TheoryEmpty setSubsetUnionIntersectionDisjoint setsComplementDifferenceReflection of set B:Translation of set A by point z=(z1,z2):

}for |{ˆ BbbwwB

}for ,|{)( AazaccA z

Logic OperationsANDOR NOT

DilationWith A and B as sets in Z2, the dilation of A by B is defined as:

Or, equivalently,B is commonly known as the structuring element.

})ˆ(|{ ABzBA z

}])ˆ[(|{ AABzBA z

Illustration

Example

ErosionWith A and B as sets in Z2, the erosion of A by B is defined as:

Dilation and erosion are duals:

})(|{ ABzBA z

BABA cc ˆ)(

Illustration

Example: Removing image components

Opening and ClosingOpening of set A by structuring element B:

Erosion followed by dilationClosing of set A by structuring element B:

Dilation followed by erosion

BBABA )(

BBABA )(

OpeningOpening generally smoothes the contour of an object, breaks narrow isthmuses, eliminate thin protrusions.

ClosingClosing tends to smooth contours, fuse narrow breaks and long thin gulfs, eliminate small holes, fill gaps in the contour.

Illustration

Example

Hit-or-Miss TransformShape detection tool )]([)( XWAXABA c

Boundary ExtractionDefinition: )()( BAAA

Region FillingBeginning with a point p inside the boundary, repeat:

with X0=pUntil Xk=Xk-1

Conditional dilation

ckk ABXX )( 1

Example

Extraction of Connected Component

Beginning with a point p of the connected component, repeat:

with X0=pUntil Xk=Xk-1

The connected component Y=Xk

ABXX kk )( 1

Illustration

Example

Convex HullA 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 H of an arbitrary set S is the smallest convex set containing S.H-S is called the convex deficiency of S.C(A): convex hull of a set A.

AlgorithmFour structuring elements: Bi, i=1,2,3,4Repeatwith X0

i =A until Xki=Xk-1

i to obtain Di

The convex hull of A is:

ABXX ik

ik )( 1

4

1

)(

i

iDAC

Illustration

ThinningThe thinning of a set A by a structuring element B is defined as:

cBAABAABA )()(

Illustration

Thickening)( BAABA

Skeleton

Skeleton: Definition

times ...)))(...()(

)()()(

)()(0

kBBBAkBA

BkBAkBAAS

ASAS

k

K

kk

})(|max{ kBAkK

Illustration

Pruning

Extension to Gray-Scale Images

Dilation Max Erosion Min

Illustration

Opening and Closing

Smoothing and Gradient