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Vision Lab, Dept. of EE, NCTU Jui-Nan Chang 2009.4.6. The Fuzzy Transformation and Its Applications in Image Processing. Outline. Introduction Basic Concepts Properties of Fuzzy Transformation Filter Generalization Using the FZT and Applications Conclusion References. - PowerPoint PPT Presentation
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Vision Lab, Dept. of EE, NCTUJui-Nan Chang
2009.4.6
1
Outline
Introduction Basic Concepts Properties of Fuzzy Transformation Filter Generalization Using the FZT
and Applications Conclusion References
2
Introduction (1/2) Nonlinear signal processing methods
- heavy tailed distribution or non-stationary statistics
Spatial & Rank (SR) orderings- center weighted median (CWM)- weighted median (WM)- permutation
Spatial correlation and rank order information crisp (binary) SR relations
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Introduction (2/2) Fuzzy SR relations
- crisp SR relations sample spread (diversity)- fuzzy spatial samples- fuzzy order statistics- fuzzy spatial indexes- fuzzy rank
crisp SR space
fuzzy SR space
fuzzy transformation
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Basic Concepts (1/4)
1 2, , ,l Nx x x x (1) (2) ( ), , ,L Nx x x x spatial sample
20,17,19,50,53,48,58,55,51lx 17,19,20,48,50,51,53,55,58Lx
3,1,2,5,7,4,9,8,6r 2,3,1,6,4,9,5,8,7s
0 0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1 0
0 0 0 0 0 1 0 0 0
R
crisp SR relations
( )
,( )( )
1,
0,i j
i ji j
for x xR
for x x
Tl Lx x R L lx x R
1: Tr N R 1:s N Rwe get
order statistic
rank index spatial index
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Basic Concepts (2/4)
Combined with spread information- membership functionsGaussian membership function
Uniform membership function
Triangular membership function
Note: they are all monotonically non-decreasing function
and
( , )F a b
2
2
( )( , ) exp[ ]
2G
a ba b
1,( , )
0,U
a ba b
otherwise
1 ,( , )
0,T
a ba b
a b
otherwise
0lim ( , ) 1, lim ( , ) 0F Fa b a b
a b a b
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Basic Concepts (3/4) Combined with spread information
- fuzzy SR relations
we get
They are represented the weighted averages of
the crisp order statistics , spatial samples ,spatial indexes and rank indexes.
1 (1) 1 ( )
(1) ( )
( , ) ( , )
( , ) ( , )
F F N
F N F N N
x x x x
R
x x x x
row normalizedcolumn normalized
LR
lR
Tll Lx x R L
L lx x R
1:Tlr N R 1: Ls N R
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Basic Concepts (4/4)
20,17,19,50,53,48,58,55,51lx
17,19,20,48,50,51,53,55,58Lx
3,1,2,5,7,4,9,8,6r
2,3,1,6,4,9,5,8,7s
0 0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1 0
0 0 0 0 0 1 0 0 0
R
Example (Gaussian membership function)
18.9,18.4,18.8,50.8,52.5,50.0,55.9,53.9,51.3lx
18.4,18.8,18.9,50,50.8,51.3,52.5,53.9,55.9Lx
2.14,1.85,2.05,5.65,6.59,5.18,8.19,7.3,5.94r
2.07,2.02,1.98,6.06,6.22,6.32,6.55,6.79, 7.05s
0.64 0.95 1 0 0 0 0 0 0
1 0.82 0.64 0 0 0 0 0 0
0.82 1 0.95 0 0 0 0 0 0
0 0 0 0.82 1 0.95 0.64 0.29 0.04
0 0 0 0.29 0.64 0.82 1 0.82 0.29
0 0 0 1 0.82 0.64 0.29 0.09 0.01
0 0 0 0.01 0.04 0.09 0.29 0.64 1
0 0 0 0.09 0.29 0.45 0.82 1 0.64
0 0 0 0.64 0.95 1 0.82 0.45 0.09
R
fuzzy SR spacecrisp SR space
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Properties of Fuzzy Transformation
Element Invariant Property
- the crisp SR relations are fully preserved by the FZT
Order Invariant Property
- the fuzzy SR space contains SR information consistent with that in the crisp SR space
Mean preserving an unbiased operator
( ) ( )if , i.e., i j i jx x x x
if r , i j i jr r r
E x E x9
Filter Generalization Using the FZT and Applications Fuzzy identity filer
- remove the blocking artifact with preserving edge- use Gaussian membership function- use MSE criteria to estimate the parameter
( ) ,( )1
,( )1
N
k c kk
IF c N
c kk
x RO x
R
: the spatial index of the center sample in the filtering windowc
,1
N
k c kk
x R
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Filter Generalization Using the FZT and Applications Fuzzy identity filer
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Filter Generalization Using the FZT and Applications Fuzzy identity filer
blocking artifact QF=10 result of fuzzy IF 12
Filter Generalization Using the FZT and Applications LUM filter – impulse noise removal(lower-upper-middle)
The LUM smoother may cause over smoothing when there are no outliers, or under smoothing when corrupted samples have ranks within the range [k,N-k+1 ]
( )
( 1)
,
, 1
, 1
k c
LUM c c
N k c
x r k
O x k r N k
x r N k
13
Filter Generalization Using the FZT and Applications FLUM filter – impulse noise removal(fuzzy lower-upper-middle)
The FLUM filter incorporates sample spread information, and thus more effectively identifies true outliers and improve filer performance
( )
( 1)
,
, 1
, 1
k c
FLUM c c
N k c
x r k
O x k r N k
x r N k
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Filter Generalization Using the FZT and Applications FLUM filter – impulse noise removal
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Filter Generalization Using the FZT and Applications FLUM filter – impulse noise removal
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Filter Generalization Using the FZT and Applications FLUM filter – impulse noise removal
5% impulse noise crisp LUM filter fuzzy LUM filter
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Conclusion
FZT retains the consistent SR information of the samples
FZT effectively reflects sample spread information
The FZT is utilized to generalize conventional filters to exploit the joint spatial-rank-spread information
It has potential to be exploited in novel techniques for other signal processing applications
18
References
Yao Nie and K. E. Barner, "The fuzzy transformation and its applications in image processing," Image Processing, IEEE Transactions on, vol. 15, pp. 910-927, 2006.
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