chapter 9 morphological image processing. preview morphology: denotes a branch of biology that deals...
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Chapter 9
Morphological Image Processing
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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.
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ScopeWill focus on binary images.Applicable to other situations. (Higher-dimensional space)
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Set TheoryEmpty setSubsetUnionIntersectionDisjoint setsComplementDifferenceReflection of set B:Translation of set A by point z=(z1,z2):
}for |{ˆ BbbwwB
}for ,|{)( AazaccA z
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Logic OperationsANDOR NOT
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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
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Illustration
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Example
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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)(
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Illustration
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Example: Removing image components
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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 )(
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OpeningOpening generally smoothes the contour of an object, breaks narrow isthmuses, eliminate thin protrusions.
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ClosingClosing tends to smooth contours, fuse narrow breaks and long thin gulfs, eliminate small holes, fill gaps in the contour.
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Illustration
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Example
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Hit-or-Miss TransformShape detection tool )]([)( XWAXABA c
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Boundary ExtractionDefinition: )()( BAAA
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Region FillingBeginning with a point p inside the boundary, repeat:
with X0=pUntil Xk=Xk-1
Conditional dilation
ckk ABXX )( 1
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Example
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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
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Illustration
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Example
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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.
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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
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Illustration
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ThinningThe thinning of a set A by a structuring element B is defined as:
cBAABAABA )()(
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Illustration
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Thickening)( BAABA
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Skeleton
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Skeleton: Definition
times ...)))(...()(
)()()(
)()(0
kBBBAkBA
BkBAkBAAS
ASAS
k
K
kk
})(|max{ kBAkK
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Illustration
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Pruning
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Extension to Gray-Scale Images
Dilation Max Erosion Min
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Illustration
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Opening and Closing
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Smoothing and Gradient