WEINER FILTERING AND BASIS FUNCTIONS.ppt

Wiener Filtering &
Basis Functions
Nov 4th 2004
Jukka Parviainen
parvi@hut.fi
T-61.181 Biomedical Signal Processing
Sections 4.4 - 4.5.2

Nov 4th 2004 T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
2
Outline
• a posteriori Wiener filter (Sec 4.4)
– removing noise by linear filtering in
optimal (mean-square error) way
– improving ensemble averaging
• single-trial analysis using basis
functions (Sec 4.5)
– only one or few evoked potentials
– e.g. Fourier analysis

3
Wiener - example in 2D
• model x = f(s)+v, where f(.) is a
linear blurring effect (in the
example)
• target: find an estimate s’ = g(x)
• an inverse filter to blurring
• value of SNR can be controlled
• Matlab example: ipexdeconvwnr

4
Part I - Wiener in EEG
• improving ensemble averages by
incorporating correlation
information, similar to weights
earlier in Sec. 4.3
• model: x_i(n) = s(n) + v_i(n)
• ensemble average of M records
• target: good s’(n) from x_i(n)

5
Wiener filter in EEG
• a priori Wiener filter:
)
(
)
/
1
(
)
(
)
(
)
( 



j
v
j
s
j
s
j
e
S
M
e
S
e
S
e
H


• power spectra of signal (s) and
noise (v) are F-transforms of
correlation functions r(k)

6
Interpretation of Wiener
• if ”no noise”, then H=1
• if ”no signal”, then H=0
• for stationary processes
always 0 < H < 1
• see Fig 4.22
)
(
)
/
1
(
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(
)
(
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( 



j
v
j
s
j
s
j
e
S
M
e
S
e
S
e
H



7
Wiener in theory
• design H(z), so that mean-square
error E[(s(n)-s’(n))^2] minimized
• Wiener-Hopf equations of
noncausal IIR filter lead to H(ej )
• filter gain 0 < H < 1 implies
underestimation (bias)
• bias/variance dilemma

8
A posteriori Wiener filters
• time-invariant a posteriori filtering
• estimates for signal and noise
spectra from data afterwards
• two estimates in the book:
• improvements: clipping & spectral
smoothing, see Fig 4.23
)
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(
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1
1 

j
sa
j
xa
e
MS
e
S
M
M
H 


)
(
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(
1
2 

j
sa
j
vs
e
S
e
S
H 


9
Limitations of APWFs
• contradictionary results due to
modalities: BAEP+VEP ok, SEP not
• bad results with low SNRs, see Fig
4.24
• APWF supposes stationary signals
• if/when not, time-varying Wiener
filters developed

10
APWF - What was learnt?
• authors: ”serious limitations”,
”important to be aware of possible
pitfalls”, especially when ”the
assumpition of stationarity is
incorporated into a signal model”

11
Part II - Basis functions
• often no repititions of EPs available
or possible
• therefore no averaging etc.
• prior information incorporated in
the model
• mutually orthonormal basis func.:






l
k
l
k
l
T
k
,
0
,
1



12
Orthonormal basis func.
• data is modelled using a set of
weight vectors and orthonormal
basic functions
• example: Fourier-series/transform
...
)
2
cos(
)
cos(
)
( 2
1
0 


 t
a
t
a
a
t
x 

i
N
k
k
k
i
i w w
x 

 
1
, 

13
Lowpass modelling
• basis functions divided to two sets,
”truncating” the model
• s are to be saved, size N x K
• v are to be ignored (regarded as
high-freq. noise), size N x (N-K)
i
T
s
s
i
s
i x
w
s 




^

14
Demo: Fourier-series
http://www.jhu.edu/~signals/
• rapid changes - high frequency
• value K?
• transients cannot be modelled
nicely using cosines/sines

15
Summary I: Wiener
• originally by Wiener in 40’s
• with evoked potentials in 60’s and
70’s by Walker and Doyle
• lots of research in 70’s and 80’s
(time-varying filtering by de
Weerd)
• probably a baseline technique?

16
Summary II: Basis f.
• signal can be modelled using as a
sum of products of weight vectors
and basis functions
• high-frequency components
considered as noise
• to be continued in the following
presentation

WEINER FILTERING AND BASIS FUNCTIONS.ppt

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