Approaches 2D adaptive filters
1 approaches
1.1 2d least mean square fir adaptive filters
1.2 2d least mean square iir adaptive filters
1.3 recursive least square adaptive filters
1.4 lexicographic ordering
1.5 mcclellan transformations
1.6 block diagonal 2d adaptive filters
approaches
2d least mean square fir adaptive filters
least mean square (lms) adaptive filters use common error measure method, mean square error. 2d lms adaptive filters derived 1d lms adaptvie filters main method minimizes output mean square value adjusting coefficients of filter. filter has primary 2d input signal d , reference input signal x. primary input signal d consists of ideal signal , noise component. filter n n causal fir filter impulse response
w
{\displaystyle w}
. can filter output given by
y
(
n
1
,
n
2
)
=
∑
m
1
=
0
n
−
1
∑
m
2
=
0
n
−
1
w
j
(
m
1
,
m
2
)
x
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n
1
−
m
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,
n
2
−
m
2
)
{\displaystyle y(n_{1},n_{2})=\sum _{m_{1}=0}^{n-1}\sum _{m_{2}=0}^{n-1}w_{j}(m_{1},m_{2})x(n_{1}-m_{1},n_{2}-m_{2})}
where j iteration number adaptive filters.
the error signal
e
j
{\displaystyle e_{j}}
@ j-th iteration defined as
e
j
=
d
(
n
1
,
n
2
)
−
∑
m
1
=
0
n
−
1
∑
m
2
=
0
n
−
1
w
j
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m
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m
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)
x
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n
1
−
m
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n
2
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m
2
)
{\displaystyle e_{j}=d(n_{1},n_{2})-\sum _{m_{1}=0}^{n-1}\sum _{m_{2}=0}^{n-1}w_{j}(m_{1},m_{2})x(n_{1}-m_{1},n_{2}-m_{2})}
the weight matrix @ next iteration equal present weight matrix plus change proportional negative gradient of mean square error. two-dimensional lms adaptive filter, filter coefficients updated follows:
w
j
+
1
(
n
1
,
n
2
)
=
w
j
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n
1
,
n
2
)
+
2
μ
e
j
x
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n
1
,
n
2
)
{\displaystyle w_{j+1}(n_{1},n_{2})=w_{j}(n_{1},n_{2})+2\mu e_{j}x(n_{1},n_{2})}
where
μ
{\displaystyle \mu }
scaler multiplier controlling can control rate of convergence , filter stability.
advantages: tdlms adaptive filter can implemented without form of matrix operations or averaging or differentiation. algorithm convergence not depend on initial conditions , converge arbitrarily initial value, hence, provides performance in nonstationary images.
disadvantages: exact values of expectations of tdlms adaptive filter not converges fixed value, if need maintain tracking ability. therefore, design choice of μ depends on particular application , involves tradeoff between convergence speed, tracking ability, , steady-state mse.
2d least mean square iir adaptive filters
for two-dimensional lms iir adaptive filter, basic idea same 2d lms fir adaptive filters, except using iir filter, can reduce filter order requirements. two-dimensional iir filter`s difference equation can written as
y
(
n
1
,
n
2
)
=
∑
m
1
=
0
m
1
∑
m
2
=
0
m
2
a
(
m
1
,
m
2
)
x
(
n
1
−
m
1
,
n
2
−
m
2
)
−
∑
m
1
=
0
l
1
∑
m
2
=
0
l
2
(
m
1
,
m
2
)
≠
(
0
,
0
)
b
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m
1
,
m
2
)
y
(
n
1
−
m
1
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n
2
−
m
2
)
{\displaystyle y(n_{1},n_{2})=\sum _{m_{1}=0}^{m_{1}}\sum _{m_{2}=0}^{m_{2}}a(m_{1},m_{2})x(n_{1}-m_{1},n_{2}-m_{2})-{\sum _{m_{1}=0}^{l_{1}}\sum _{m_{2}=0}^{l_{2}}}_{(m_{1},m_{2})\neq (0,0)}b(m_{1},m_{2})y(n_{1}-m_{1},n_{2}-m_{2})}
where
y
(
n
1
,
n
2
)
{\displaystyle y(n_{1},n_{2})}
,
x
(
n
1
,
n
2
)
{\displaystyle x(n_{1},n_{2})}
are, respectively, output , input of adaptive filter.
b
(
m
1
,
m
2
)
{\displaystyle b(m_{1},m_{2})}
,
a
(
m
1
,
m
2
)
{\displaystyle a(m_{1},m_{2})}
masks of filter`s input , output. error signal given by
e
(
n
1
,
n
2
)
=
d
(
n
1
,
n
2
)
−
y
(
n
1
,
n
2
)
{\displaystyle e(n_{1},n_{2})=d(n_{1},n_{2})-y(n_{1},n_{2})}
where
d
(
n
1
,
n
2
)
{\displaystyle d(n_{1},n_{2})}
is primary output signal.
the mean square error
e
{
e
2
(
n
1
,
n
2
)
}
{\displaystyle e\{e^{2}(n_{1},n_{2})\}}
minimized updating filter weights in manner converge optimum filter weight.
advantages: iir filters can satisfy prescribed frequency response because can reduce filter`s order requirements.
disadvantages: performance surfaces of adaptive lms iir adaptive filters nonquadratic , may have local minima. meanwhile, adaptive iir filters may become unstable during adaptation, kind of stability monitoring needed.
recursive least square adaptive filters
2d recursive least square adaptive filters can developed applying 1d recursive least squares filters along both horizontal , vertical directions. rls adaptive algorithm finds filter coefficients recursively minimize weighted least squares cost function. rls algorithm different least mean squares algorithm aim reduce mean square error, input signal considered deterministic. reason, rls algorithm has fast convergence characteristic.
advantages: rls algorithm has fast convergence property. accuracy of image denoising based on rls algorithm better 2d lms adaptive filters.
disadvantages: rls algorithm needs large amount of computations, in two-dimensional , multidimensional case.
lexicographic ordering
one convenient approach implement 2d adaptive filters transform 2d problem 1d problem lexicographic ordering. simplifies implementation , makes possible benefit extensive literature available 1d adaptive filters , utilize of existing 1d algorithms.
mcclellan transformations
mcclellan transformations can used transform 1d filter design 2d filter design using transformation function. theory allows design of 2d adaptive filters out of existing 1d prototype filters. compared direct approach, system has advantages of lower computational complexity , faster convergence rate. however, in order work properly, needs priori information system correctly select transformation function parameters, making system pre-constrained.
block diagonal 2d adaptive filters
block diagonal 2d adaptive filters alternative approach scans signal through blocks , applies weight adjustments each block, instead of each sample in traditional adaptive filters. advantage of kind of system takes account signal correlations along both dimensions. on other hand, assumes higher local stationarity of signal.
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