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Soft Morphological Filter. Paper: Simple and Efficient Soft Morphological Filter in Periodic Noise Reduction Authors: Zhen Ji, Huilian Liao, Xinjun Zhang, Q.H. Wu 15 Nov. 2006. Outline. Introduction Mathematical morphology (MM) Soft morphology Soft morphological filters (SMF) - PowerPoint PPT Presentation
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Soft Morphological Filter
Paper: Simple and Efficient Soft Morphological Filter in Periodic Noise Reduction
Authors: Zhen Ji, Huilian Liao, Xinjun Zhang, Q.H. Wu
15 Nov. 2006
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Outline
Introduction Mathematical morphology (MM) Soft morphology Soft morphological filters (SMF) Soft morphological filter* (SMF*) Results Conclusions Acknowledgement
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Introduction to filters
Introduction to filters 1. Spatial filters (including MM filter)
2. Frequency filters (including spectral median filter)
Fig. 1 Fig. 2
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Mathematical morphology (MM)
Structuring element (SE) Two basic operators of
morphology: Erosion Dilation
Binary morphology & Grey-scale morphology
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MM – Erosion of Grey-scale
morphology
The erosion of f by a SE g at a point x is:
z = {(1,2),(1,3),(0,2),(0,3)}
)21)(( ,gf min , , , )3 2 2 1 7 4 5 3 min , , ,11 3 2 1
)()(min))((
]D[]D[:],D[]D[
zgzfxgf
fg xgfzx
xx
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MM — Dilation of Grey-scale
morphology
The dilation of f by a SE g at a point x is: )()(max))((
)](D[]D[ zgzfxgfzgf z
x
x
z = {(1,2),(1,1),(2,2),(2,1)}
( )( , )f g 1 2 max , , , )3 2 6 1 8 4 5 3 max , , ,5 7 12 8 12
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Soft morphology
Differences between soft morphology and standard morphology: The SE g is divided into two parts: the
hard α and the soft β= g/α The min/max operators are
substituted by other order statistics Two basic operators:
Soft erosion & soft dilation
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Soft morphology — soft erosion
Soft erosion of f by SE g (hardα and softβ) at point x is:
]D[]D[ ]D[]D[
)()())()((min)])([(
2
1
2211)(
x
x
xxk
βf zαfz
zβzfzazfkxk,a,βf
D[α]={(0,0)}; D[β]={(0,1),(0,2),(1,0),(1,1)}
)11])(2[ ( ,β,α,f 24,48,89,68572min(2)
241,2,2,2,2,min 2,4,1,22,2 min (2)(2)
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Soft morphology — soft dilation
Soft dilation of f by SE g (hard α and softβ) at point x is:
]D[\]D[]D[ ]D[
)(-)())()((max)])([(
2
1
2211)(
αgβ x-zα x-z
zβzfzαzfkxβ,α,kf xxk
D[α]={(0,0)}; D[β]={(0,1),(0,2),(1,0),(1,1)}
)21])(2[( ,,β,αf 38,49,74,95672 max(2) 133,13,1411,11,13,1max 11,13,11,1413,13max (2)(2)
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Soft morphological filter
Soft morphological filter (SMF)
) / 2
]D[]D[ ]D[]D[
)()())()((min)])([(
2
1
2211)(
x
x
xxk
βf zαfz
zβzfzazfkxk,a,βf
]D[\]D[]D[ ]D[
)(-)())()((max)])([(
2
1
2211)(
αgβ x-zα x-z
zβzfzαzfkxβ,α,kf xxk
Standard morphological filter (StdMF)
( ) (smf x
) / 2; ( =1)k
]D[]D[ ]D[]D[
)()())()((min)])([(
2
1
2211)(
x
x
xxk
βf zαfz
zβzfzazfkxk,a,βf
]D[\]D[]D[ ]D[
)(-)())()((max)])([(
2
1
2211)(
αgβ x-zα x-z
zβzfzαzfkxβ,α,kf xxk
( ) (stdmf x
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Morphological filters — comparison
1 2 3 4 5 6 7 8 9 10-5
0
5Original signal
1 2 3 4 5 6 7 8 9 10-5
0
5Periodic noise
1 2 3 4 5 6 7 8 9 10-5
0
5Noisy signal
1 2 3 4 5 6 7 8 9 10-5
0
5Median filter
1 2 3 4 5 6 7 8 9 10-5
0
5Soft morphological filter
1 2 3 4 5 6 7 8 9 10-5
0
5Standard morphological filter
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Soft morphological filter* (SMF*)
Design procedure of SMF*
) / 2; ( =2)k( ) (smf x
]D[]D[ ]D[]D[
)()())()((min)])([(
2
1
2211)(
x
x
xxk
βf zαfz
zβzfzazfkxk,a,βf
]D[\]D[]D[ ]D[
)(-)())()((max)])([(
2
1
2211)(
αgβ x-zα x-z
zβzfzαzfkxβ,α,kf xxk
Design of SMF*’s parameters: SE g and hard α are symmetric to its origin
1 1 1 1 1 0 0 0 0 0
1 1 1 1 1 0 1 1 1 0
1 1 1 1 1 0 1 1 1 0
1 1 1 1 1 0 1 1 1 0
1 1 1 1 1 0 0 0 0 0
g
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Results
Filters: SMF*; Spectral median filter; median;
Standard MM Experimental image:
Pepper image with periodic noise Measure of performance:
Peak-Signal-Noise-Ratio (PSNR) Shape error Computation time
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Results — PSNR
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Results — Shape error
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Results — Computation time
0
5
10
15
20
25
30
SMF Spectral Medi an StandardMM
Computati onti me (ms)
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Conclusions
About SMF* Purposes: Reducing the periodic
noise Properties: Preserving details of
image Advantages: Filtering quality &
Computation efficiency & Simplicity
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Acknowledgement
My supervisor: Prof. Ji
All of you
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Thank you !
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Dilation expands the image foreground and shrinks its background, whilst erosion shrinks the image foreground and expands its background.
The Erosion filter is a morphological filter that changes the shape of objects in an image by eroding (reducing) the boundaries of bright objects, and enlarging the boundaries of dark ones. It is often used to reduce, or eliminate, small bright objects.
The Dilation filter is a morphological filter that changes the shape of objects in an image by dilating (enlarging) the boundaries of bright objects, and reducing the boundaries of dark ones. The dilation filter can be used to increase the size of small bright objects.
soft erosion is anti-extensive and soft dilation is extensive, provided that the structuring element includes the origin. In particular the smaller the repetition parameter k is the more the input image shrinks or expands. (also 1st property)
