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CHAPTER 7
SHAPE DESCRIPTION
DIGITAL IMAGE PROCESSINGDIGITAL IMAGE PROCESSINGDIGITAL IMAGE PROCESSING
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Contents
♦♦ INTRODUCTIONINTRODUCTION
♦♦ CHAIN CODESCHAIN CODES
♦♦ POLYGONAL APPROXIMATIONSPOLYGONAL APPROXIMATIONS
♦♦ FOURIER DESCRIPTORSFOURIER DESCRIPTORS
♦♦ QUADTREESQUADTREES
♦♦ PYRAMIDSPYRAMIDS
♦♦ SHAPE FEATURESSHAPE FEATURES
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Contents
♦♦ MOMENT DESCRIPTORSMOMENT DESCRIPTORS
♦♦ THINNING ALGORITHMSTHINNING ALGORITHMS
♦♦ MATHEMATICAL MORPHOLOGYMATHEMATICAL MORPHOLOGY
♦♦ GREYSCALE MORPHOLOGYGREYSCALE MORPHOLOGY
♦♦ SKELETONSSKELETONS
♦♦ SHAPE DECOMPOSITIONSHAPE DECOMPOSITION
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Introduction
Two-dimensional shapes can be described in two differentways:
Á) Use of the object boundary and its Use of the object boundary and its featuresfeatures (e.g. (e.g. boundary length)boundary length). This method is directly connected to edge and line detection. The resulting description schemes are called external represantationsexternal represantations..
Â) Description of the region occupied by the object Description of the region occupied by the object on on thethe image plane.image plane. This method is linked to the region segmentation techniques. The resulting representation schemes are called internal representations.internal representations.
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Introduction
Shape representation schemes must have certain desirableproperties:
UniquenessUniqueness. This is of crucial importance in object recognition, because each object must have a unique representation
CompletenessCompleteness. This refers to unambiguous representations
Invariance under geometrical transformationsInvariance under geometrical transformations.Invariance under translation, rotation, scaling and reflection is very important for object recognition applications.
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Introduction
SensitivitySensitivity. This is the ability of a representation schemeto reflect easily the differences between similar objects
Abstraction from detailAbstraction from detail. This refers to the ability of therepresentation to represent the basic features of a shapeand to abstract from detail. This property is directly related to the noise robustness of the representation.
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• The chain code depends on the start point of boundaryfollowing.
• An advantage of chain code is that it is translation invariant.
• Scale invariance can be obtained by changing the size of the sampling grid, producing seldom, however, exactly the same chain code.
• Rotation invariance is obtained by using the difference difference chain chain codecode..
Chain codes
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Chain codes
Chain code of the digital boundary
Figure 1: Directions of boundary segments of a chain code for (a) a 4-connected chain; (b) an 8-connected chain.
(a) (b)
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The difference code chaindifference code chain is given by:
{1iif)x,x(diff
1iif)x,x(diffi
1ii
Ni
d≠
=
−=
• Chain codes provide a good compression of boundary description.
• Chain codes can also be used to calculate certain boundary features.
Chain codes
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The boundary perimeter T is given by:
∑=
=N
iinT
1
{02mod1
12mod2
=
== i
i
xif
xifinwhere:(in case of an8-connectedchain code)
The object width w and height h are given by:
∑∑==
==N
ii
N
ii hhww
11
,
{{3,2,00
11
3,2,10
01
=
=
=
=== i
i
i
i
xif
xifi
xif
xifi hw
(in case of an4-connectedchain code)
Chain codes
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Chain codes can be used in the calculation of object area
The boundaries of binary objects can be easilly followed byemploying an algorithm similar to Papert’s turtle:Papert’s turtle:
Figure 2: Turtle procedure in binary object boundary following
Chain codes
• Fir pixel value turn and advance one pixel
1
0
left
right
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Polygonal approximations
•Digital boundaries carry information which may be superfluous for certain applications. Boundary approximations can be sufficient in such cases. Linear Linear piecewisepiecewise (polygonal) approximations(polygonal) approximations are the most frequently used.
••The optimal linear piecewise approximationThe optimal linear piecewise approximation can be obtained by choosing the polygon vertices in such a way that the overall approximation error is minimized.•Error measures:
•Mean square 2ii
1
22 |dx| −= ∑
−
=
Ν
ι
Ε
|dx|max ii12
max −=−≤≤ Νι
Ε•Maximal
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••Splitting techniquesSplitting techniques divide a curve segment recursively into smaller segments, until each curve segment can be approximated by a linear segment within an acceptable error range.
Polygonal approximations
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Figure 3: Splitting methodfor polygonalapproximations
Figure 4:Splitting methodfor the linear picewiseapproximationof a closed curve
Polygonal approximations
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•A basic advantage of the splitting approach is that it candetect the inflection points on a curve and can use themin curve representation.
••Merge techniques in the polygonal approximationMerge techniques in the polygonal approximationoperate in the opposite way.
•The primary disadvantage of the merge algorithm is thatpolygon vertices do not coincide with curve inflectionpoints.
•This problem can be solved by combining split and mergetechniques.
Polygonal approximations
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Fourier descriptors
Signal representation using Fourier descriptorsFourier descriptors
∑−
=
−=
1
0
2exp)()(
N
n N
nkinzkZ
π
∑−
=
=
1
0
2exp)(
1)(
N
k N
nkiKZ
Nnz
π
Figure 5: Parametric curve representation
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Fourier representation propertiesFourier representation properties
A)A) The coefficient Æ(0) represents the centre of gravitycentre of gravityof the curve.
B)B) Fourier coefficients Z(k) represent slowly and rapidlyvarying shape trends for small and large indices krespectively.
C)C) A translation in curve coordinates by z0 :zt(n)=z(n)+z0 , z0=x0+iy0
affects only the term Z(0) of the representation:Zt(0)=Z(0)+z0
Fourier descriptors
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D)D) A rotation of the curve coordinates by angle è :zr(n)=z(n)eiè
results in a phase shift of the transform coefficients by anequal amount:
Zr(k)=Z(k)eiè
E)E) A scaling operation by a factor á, results in a scaling ofFourier coefficients by an equal amount:
zs(n)=áz(n)Zs(k)=áZ(k)
Fourier representation propertiesFourier representation properties
Fourier descriptors
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F) F) A change in the starting point of curve traversal:zt(n)=z(n-n0)
produces modulationmodulation of the Fourier descriptors:Zt(k)=Z(k)e-i2ðn0k/N
Fourier descriptors have interesting invariance propertiesthat can be used in object recognition applications.
∑−
=
−=1
0
221 |))(||)((|
N
k
kZkZE( Error measure formatching two curvesz1(n) , z2(n) )
Fourier representation propertiesFourier representation properties
Fourier descriptors
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Quadtrees
•Quadtrees are based on the following recursive approach:if a binary image region of size 2n × 2n consists of both 0sand 1s, it is declared inhomogeneous and is split into foursquare subregions R0, R1, R2, R3, having size 2n-1×2n-1 each.•This procedure continues until all subregions arehomogeneous. •The resulting representation is a quadtreequadtree.
∑=
≈=n
0k
nk 43
44N
Maximalnumberof nodes
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Figure 6: (a) Binary image (b) Quadtree representation
Quadtrees
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Pyramids
•• MultiresolutionMultiresolution representations employ several copies ofthe same image at different resolutions.• Multiresolution techniques applied to greyscale or binaryimages lead to the so-called image pyramids.image pyramids.• An image pyramid is a series fk(i,j), k=0,…,n of imagearrays, each having size 2k × 2k .
fk(i,j)=g(fk+1(2i,2j), fk+1(2i,2j+1), fk+1(2i+1,2j), fk+1(2i+1,2j+1))
g(·) is a mapping function
∑∑= =
+ ++=1
0l
1
0m1kk )mj2,li2(f
4
1)j,i(f
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Figure 7:(a) Image pyramid
(b) Mapping from onepyramid level to thenext level.
Pyramids
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•Pyramids techniques enjoy a certain popularity for image analysis and compression applications, because they offer abstraction from image details.
•Binary image pyramids can be used in multiresolution edge detection and region segmentation.
•The total space required for the storage of a pyramid (and of a quadtree) is 4/3 × (2n × 2n) where 2n × 2n is the size of the original image. Of course, the pyramid can be simply stored on n+1 arrays of size 2k × 2k, k=0,..,n.
Pyramids
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Pyramids
Figure 8:(a) Originalbinaryimage
(b) Imagepyramid
(c) Output ofthe pyramidedge detector
(d) Edgepyramid
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Shape features
Geometrical shapes possess certain features (e.g. perimeter)that carry sufficient information for some object recognitionapplications. Such features can be used as object descriptorsresulting in a significant data compression, because they canrepresent the geometrical shape by a relatively small featurevector.
Shape features can be grouped in two large classes:boundary featuresboundary featuresregionregion featuresfeatures
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Object perimeterperimeter::
∫ += dttytxT )()( 22
∑∑−
=+
−
=
−==1
11
1
1
||N
iii
N
ii xxdT x1,…,xN : boundary
coordinate list
Curvature Curvature magnitudemagnitude::2
2
2
2
22|)(|
+
=∆
dt
yd
dt
xdtk
Shape features
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Curvature Curvature magnitudemagnitude::
ds
sodsk
)()( /= 22 dydxds +=where
Shape features
])1n(y)n(y2)1n(y[)]1n(x)n(x2)1n(x[1
|)n(k| 222
++−−+++−−⋅=∆
Another curvature definitionAnother curvature definition::
Approximaton of the local curvatureApproximaton of the local curvature::
= oddfor x,
2
2
even,for x2
1
)x(L
i
i
i
)x(L)x(L
xx)n(k
1nn
1nn
−
−
−−
≅ where
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Bending Bending energyenergy::
∫=T
dttkT
E0
2|)(|1
∑−
=
=1
0
2|)(|1 n
i
ikT
E where 1<n<N
∑
=
42 2|)(|
T
kkZE
π Calculated from boundaryFourier descriptors
Shape features
2
T
2E
=
πCircle bending energy
∑=
−=−=n
1i
2
2
object
circleN
|)i(k|T
41
E
E1E
πNormalization ofbending energy
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ObjectObject areaarea
∫∫=R
dxdyA
∫∂
−=
Rdt
dt
dytx
dt
dxtyA )()(
using a differentialgeometry formula
Compactness or circularityCompactness or circularity
A4
T 2
πγ = 2N T
A41
πγ −= normalized version
Shape features
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Object width and heightwidth and height
)(min)(max
)(min)(max
tytyh
txtxw
tt
tt
−=
−=
Object diameterdiameter
),(max,
lkRXX
XXdDlk ∈
=
where XkXl is the direction of the line segment
Shape features
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Topological descriptors can give useful global informationabout an object. Two important topological features are theholes Hholes H and the connected components Cconnected components C of an object.
EulerEuler numbernumberE = C E = C -- HH
Letters A, B, C, have Euler numbers 0, -1, 1, respectively.
Shape features
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Moment descriptors
The moments moments of an image f(x,y)f(x,y) are given by:
∫ ∫∞
∞−
∞
∞−== ,...2,1,0q,p,dxdy)y,x(fyxm qp
pq
Centre of gravityCentre of gravity of an object
00
01
00
10
m
my,
m
mx ==
Central momentsCentral moments
∫ ∫∞
∞−
∞
∞−=−−= ,...2,1,0,,),()()( qpdxdyyxfyyxx qp
pqµ
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Moment relations for discrete imagesMoment relations for discrete images
∑∑
∑∑−−=
=
i j
qppq
i j
qppq
jifyjxi
jifjim
),()()(
),(..
µ
Moment relations for binary imagesMoment relations for binary images
∑∑
∑∑−−=
=
i j
qppq
i j
qppq
yjxi
jim
)()(
..
µ
Moment descriptors
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Coordinates of the centre of gravityCoordinates of the centre of gravity
∑∑∈∈
==RjiRji
jN
yiN
x),(),(
11
Object orientation èorientation è: can be derived by minimizing thefunction:
∑∑∈
−−−=Rji
yjxiS),(
2]sin)(cos)[()( θθθ
where Í is the area of an image in pixels
−
=0220
112arctan
21
µµµ
θ
Moment descriptors
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Object eccentricityeccentricity
−+−+
=θµθµθµθµθµθµ
ε2coscossin2sinsincos
112
202
02
112
202
02
+−=
A11
22002 4)( µµµ
ε
Object spread or sizespread or size
)( 2002 µµ +=S
Moment descriptors
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Moment descriptors
Figure 9: Definition of object orientation
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Thinning algorithms
ThinningThinning can be defined heuristically as a set of succesiveerosions of the outermost layers of a shape, until aconnected unit-width set of lines (skeleton) is obtained.
Thinning algorithms satisfy the following two constraints:
1. They maintain connectivity at each iteration. They do1. They maintain connectivity at each iteration. They donot remove border pixels that may cause discontinuitiesnot remove border pixels that may cause discontinuities
2. They do not shorten the end of thinned shape limbs.2. They do not shorten the end of thinned shape limbs.
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Figure 10:(a) Border pixel whose removal may cause discontinuities;(b) border pixel whose removal will shorten an object limb;(c) local pixel notation used in connectivity check.
Thinning algorithms
(a) (b) (c)
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Figure 11: Central window pixels belonging to: (a) an East boundary; (b) a South boundary; (c) a North-West corner point.
Thinning algorithms
(a) (b) (c)
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Figure 12: Central window pixels belonging to: (a) a North boundary; (b) a West boundary; (c) a South-East corner.
Thinning algorithms
(a) (b) (c)
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Thinning algorithms
First thinning algorithm•Check in a local neighborhood 3×3
•If the number of the pixels of the object (except the central) Í(p0) is: 2 < Í (p0) < 8)
•we check if the removal of the central pixel would break object connectivity.
Check
•The pixel sequence is formed p1p2p3...p8p1.
•If the number of 0→1 transitions is 1, then the central pixel that has value 1 is removed.
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Thinning algorithms
where Ô(p0) denotes the number of the 0→1 transitions.
) =.p.p)&&(p=.p.p)&&(p)=)&&(T(p)N(p:(P
) =.p.p)&&(p=.p.p)&&(p)=)&&(T(p)N(p:(P
0 0 162
0 0162
7517310 0 2
753531 0 0 1
≤≤≤≤
Second thinning algorithm•Step 1: a logical rule P1 is applied in a 3×3neighbourhood and flags the border pixels that can be deleted.
•Step 2: a logical rule P2 is applied in a 3×3neighbourhood and flags the border pixels that will be deleted.
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Figure 13:Sobel edgedetectoroutput
Binaryimage
Output of the one-passthinningalgorithm
Output ofthe two-passthinningalgorithm
Thinning algorithms
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Mathematical morphology
Mathematical morphology uses a set theoretic approach toimage analysis.
The morphological transformations must possess thefollowing properties:
1. Translation invariance1. Translation invarianceØ(Ø(XXzz)=[)=[Ø(Ø(X)]X)]zz
2. Scale invariance2. Scale invarianceØØëë((X)=ëX)=ëØ(ëØ(ë--11X)X)
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3. Local knowledge. 3. Local knowledge. Transformation Ø(×) must requireonly information within a local neighbourhood for itsoperation
4. Semicontinuity. 4. Semicontinuity. The morphological transformation Ø(×)must possess certain continuity properties.
Basic morphological transformations
}{ IU 0: ≠∈==⊕∈
− XBEzXBX zBb
bs
}{ XBEzXBX zBb
bs ⊂∈==
∈− :I
dilationdilation
erosionerosion
Mathematical morphology
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Erosion and dilation are special cases of MinkowskiMinkowski setsetadditionaddition and Minkowski Minkowski set subtractionset subtraction
Mathematical morphology
UBb
bXBX∈
=⊕ IBb
bs XBX
∈
=
(a) (b) (c)
Figure 14: (a) thresholded image (b) eroded and (c) dilated image by the structuring elements SQUARE.
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Erosion, dilation, Minkowski set addition and subtractionhave the following interesting properties:
CommutativityCommutativity::
ABBA ⊕=⊕
Mathematical morphology
AssociativityAssociativity::
CBACBA ⊕⊕=⊕⊕ )()(
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Translation Translation invarianceinvariance::
zz
zz
zz
BABA
BABA
BABA
)(
)(
)(
==
⊕=⊕
Increasing Increasing propertyproperty::
DBDABA
DBDABA
⊆⇒⊆⊕⊆⊕⇒⊆
Mathematical morphology
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DistributivityDistributivity::
CBACBA
CBCACBA
CABACBA
CABACBA
CBCACBA
)()(
)()()(
)()()(
)()()(
)()()(
=⊕
=
=
⊕⊕=⊕
⊕⊕=⊕
I IIUUUUU
Mathematical morphology
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Opening ×Opening ×ÂÂ:: }{U XBBBBXX zzs
B ⊂=⊕= :)(
Closing XClosing XBB:: }{I cz
cz
sB XBBBBXX ⊂=⊕= :)(
Mathematical morphology
(a) (b)
Figure 15: (a) opened image (b) closed image.
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Opening and closing propertiesOpening and closing properties
DualityDuality::
BccB
BccB
XX
XX
)()(
)()(
=
=
Mathematical morphology
Extensivity and Extensivity and antiextensivityantiextensivity::
XX
XXB
B
⊃
⊂
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Increasing Increasing propertyproperty::
BB
BB
XXXX
XXXX
)()(
)()(
2121
2121
⊂⇒⊂
⊂⇒⊂
IdempotenceIdempotence::
BBB
BBB
XX
XX
=
=
)(
)(
Mathematical morphology
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Definition of binary dilationDefinition of binary dilation
}{ I 0: ≠∈=⊕ XBEzBX zs
Definition of binary erosionDefinition of binary erosion
}{ XBEzBX zs ⊂∈= :
An alternative way for the calculation of binary erosionAn alternative way for the calculation of binary erosionand dilationand dilation
I
U
Bbb
s
Bbb
s
XBX
XBX
∈−
∈−
=
=⊕
Mathematical morphology
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Greyscale morpholgy
The tools for greyscale morphological operations are simplefunctions g(x) having domain G. They are calledstructuring functionsstructuring functions
Figure 16: A example of a structuring function.
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Greyscale morpholgy
Greyscale dilation and erosion of a function f(x) by g(x)Greyscale dilation and erosion of a function f(x) by g(x)
}{
}{ )()(min)]([
)()(max)]([
,
,
xzgzfxgf
xzgzfxgf
DxzDz
s
DxzDz
s
−−=
−+=⊕
∈−∈
∈−∈
Greyscale opening and closingGreyscale opening and closing
)()]()([)]()[()(
)()]()([)]()[()(
xgxgxfxggfxf
xgxgxfxggfxfsg
sg
−⊕=⊕=
⊕−=⊕=
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Implementation of greyscale dilation and erosion inImplementation of greyscale dilation and erosion inpipelinepipeline
)...))(...((
)...))(...((
21
21
k
k
gggfgf
gggfgf
=⊕⊕⊕⊕=⊕
Dilation and erosion of a function by a setDilation and erosion of a function by a set
}{}{ )(...,),(),...,(min)]([)]([
)(...,),(),...,(max)]([)]([
vififvifxgfxGf
vififvifxgfxGfss
ss
+−==
+−=⊕=⊕
Opening and closing of a function by a setOpening and closing of a function by a set
)]()[()(
)]()[()(
xGGfxf
xGGfxfsG
sG
⊕=
⊕=
Greyscale morpholgy
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CloseClose--opening filter (CO)opening filter (CO)
)]()[( xfy GG=
openopen--closing filter (OC)closing filter (OC)
)]()[( xfy GG=
The algebraic difference y=f (x)-fnB(x) is a nonlinearhigh-pass filter, called toptop--hat transformation.hat transformation.
Greyscale morpholgy
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Greyscale morpholgy
Figure 17: Opening as a rolling ball transformation
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Greyscale morpholgy
(a) (b)
Figure 18: (a) Thresholded image, (b) Result of top-hat filtering
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Skeletons
Object skeletonskeleton is an important topological descriptor ofa two-dimensional binary object
(a) (b)
Figure 19: (a) Grassfire propagation model of medial axis;(b) maximal disk definition of skeleton.
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Skeletons
Figure 20: Illustration of morphological skeletonization
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Skeletons
Figure 21: (a) Fast skeletonization algorithm;(b) fast object reconstruction from skeleton subsets.
(a)
(b)
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Shape decomposition
• A complex object X can be decomposed into a union of ‘simple’ subsets X1,…, Xn, thus providing an intuitive object description scheme called shape decomposition.shape decomposition.
• Shape decomposition must use simple geometrical primitives in order to conform with our intuitive notion of simple shapes.
• The complexity of the decomposition must be small compared with the original description of X.
• A small noise sensitivity is desirable.
7.65THESSALONIKI 1998
I. Pitas Digital Image Processing FundamentalsShape description
Morphological shape decompositionMorphological shape decompositionRecursive relation:
0)(:
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conditionStopping
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Figure 22: (a) Original binary image;(b) first 16 components of its morphological shape decomposition.
Shape decomposition
7.66THESSALONIKI 1998
I. Pitas Digital Image Processing FundamentalsShape description
Blum ribbonsBlum ribbonsSimple objects Xi of the form:
BnLX iii ⊕=Disadvantages of morphological shape decompositionDisadvantages of morphological shape decomposition
It is susceptible to boundary noise.
The representation produced is not close to human shapeperception if the object consists of unions, intersectionsand differences of various geometrival primitives. Thiscan be alleviated by combining morphological techniqueswith constructive solid geometry, (CSG).constructive solid geometry, (CSG).
Shape decomposition
7.67THESSALONIKI 1998
I. Pitas Digital Image Processing FundamentalsShape description
Shape decomposition
Main advantage of CSG over skeleton representation orMain advantage of CSG over skeleton representation ormorphological shape decompositionmorphological shape decomposition
• CSG uses a multitude of geometrical primitives(e.g. squares) instead of one.
• This fact not only enhances the descriptive power of CSGbut also conforms to our intuitive notion of simplegeometrical shapes.