View
221
Download
4
Tags:
Embed Size (px)
Citation preview
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