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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