Wavelet Based Subband Shrinkage Models and their Applications in Denoising of Biomedical Signals

Wavelet Based Subband Shrinkage Models and their Applications in Denoising of

Biomedical Signals

ByDr. S. Poornachandra

Dean IQACSNS College of Engineering

04/21/23S. Poornachandra

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Objective Denoising of biomedical signals with

better performance


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Types of noises The muscle artifacts

Respirator muscles Cardiac muscle Moving artifacts

Electro-magnetic radiations Power line frequency noise Instrument noise Interference of other physiological signals


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Statistical Estimations

Mean Variance Risk

M

1k

2

kk θθER ˆ


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ECG Specification

The practical ECG was downloaded from the PhysioBank

Sampling rate is 360Hz Resolution is 11 Bits/Samples Bit rate is 3960 bps Length of ECG data: 650000 Length of ECG data considered: 5000


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Other biosignals used. . . .

EEG PCG Pulse Waveform


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Parameters for Analysis

Signal to Noise Ratio (SNR) =

Percentage Root Mean-Squared Difference(PRD)

= SNR Improvements

= Input SNR – Output SNR RMS Error

= RMS (Recovered Signal – Original Signal) PSNR

=

VarianceNoise VarianceSignal

100% SignalOrigianl

SignalRecovered SignalOriginal

SignalRecovered - SignalOriginal SignalOriginal


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Time-domain

Advantages Simple Easy to implement Lower computational complexity

Disadvantages Slow convergence when the input is highly

colored


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Need for Transform-domain Advantages

Better convergence Parallism

Disadvantages Complexity increases as order of the filter increases

Exhibit slow convergence High minimum mean square error

Remedy Subbanding – reduced coefficients at each subband


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Advantages of Wavelet

Works on non-stationary data Time-frequency aspect gives information

about frequency composition of a signal at a particular time

Short signal pieces also have significance


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Wavelets Defined . . . . .

“The wavelet transform is a tool that cuts up data, functions or operators into different frequency components and then studies each component with a resolution matched to its scales”

Dr. Ingrid Daubechies, Lucent, Princeton U


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DWT – Demystified

d3: Level 3 DWTCoeff.

Length: 512B: 0 ~

g[n] h[n]

g[n] h[n]

g[n] h[n]

2


Length: 256B: 0 ~ /2 Hz

Length: 256B: /2 ~ Hz

Length: 128B: 0 ~ /4 HzLength: 128

B: /4 ~ /2 Hz


…a3….

Length: 64B: 0 ~ /8 HzLength: 64

B: /8 ~ /4 Hz

2

2 2

22

|H(j)|

/2-/2

|G(j)|

- /2-/2

a2

a1

Level 3 approximation

Coefficients


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Shrinkage ?

A shrinkage method compares empirical wavelet coefficient with a threshold and is set to zero if its magnitude is less than the threshold value.


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Condition & Characteristics of Shrinkage

The magnitude of signal component must be larger than existing noise component

It does not introduce artifacts The wavelet transform localizes the most

important spatial and frequential features of a regular signal in a limited number of wavelet coefficients.

Observations suggest that small coefficients should be replace by zero, because they are dominated by noise and carry only a small amount of information.


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Pioneers …

Donoho and Johnstone (1994) – Soft Shrinkage Coifman and Donoho (1995) – Cycle Spinning Nason (1996) – Cross Validation Shrinkage Bruce and Gao (1997) –Garrote Shrinkage


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Shrinkage functions

xx

xx

x0,

xS

,

,

||

)(

|| ,

|| )(

xx

x0,xH

λ |x| /x)x-(λ

λ|x| 0,xG

,

)(

2


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Shrinkage Algorithm

Apply DWT to the vector y and obtain the empirical wavelet coefficients cj,k at scale j, where j = 1, 2, . . , J.

Estimated coefficients are obtained based on the threshold

= [1, 2, . . . . j]T.

Apply shrinkage to the empirical wavelet coefficients at each scale j.

The estimate of the function can be obtained by taking inverse DWT.


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Threshold methods

The rigrsure uses for the soft shrinkage estimator, which is a shrinkage solution rule based on Stein’s Unbiased Risk Estimate (SURE).

The sqtwolog threshold uses a fixed form threshold yielding minimax performance multiplied by a small factor proportional to log(length(x)).

The heursure threshold is the hybridization of both rigrsure and sqtwolog threshold.

The minimax threshold uses a fixed threshold chosen to yield minimax performance for MSE against an ideal procedure.


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Median Absolute Deviation

Prof. Donoho proposed

Where, is the estimate of noise variance

Median Absolute Deviation MAD(v)=[|v1-vmed|, … , |v1-vmed|]med

)ln(2ˆ N

6745.0

ˆ , jicmedian


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Alpha-trim Filter

The alpha-trim filter is a special type of L-filter,

A particular choice of aj coefficient yields a alpha-trim filter

where T is the largest integer which is less than or equal to αM, 0 ≤ α ≤ 0.5. When α = zero, the α-trim filter becomes the running mean filter; When α = 0.5, the α-trim filter becomes the median filter.

jxja ii

M

j1

jxii

TM

Tj TN

1 )(2

1


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Threshold at each subband

The Threshold values at each subband for 20% noise level is given in tables

SNR (dB) PRD (%) ECG Signal


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Wavelet level analysis


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Wavelet level analysis


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Hybrid Model ….

Analysis Filter

a3

d2

d3

d1

Hard shrinkage

Xshrinkage

Xshrinkage

Xshrinkage


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Basic shrinkage (ECG)


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Basic shrinkage (PCG)


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BSWTAF-I (Scale-Domain Analysis)

ATI Model

Analysis Filter

Adaptive Filter

Shrinkage Function

Synthesis Filter


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BSWTAF-II (Scale-Domain Analysis)

TAI Model

Analysis Filter

Shrinkage Function

Adaptive Filter

Synthesis Filter


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ASWTAF Model (Time-Domain Analysis)

TIA Model

Analysis Filter

Shrinkage Function

Adaptive Filter

Synthesis Filter


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ECG Simulation….


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ECG Simulation….


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EEG Simulation….


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EEG Simulation….


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PCG Simulation….


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PCG Simulation….


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Shrinkage Distribution


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Hyper Shrinkage Function

Where

λ |x| x) tanh(ρ

λ |x| 0txxxhyp ) tanh()(

xmaxΔρ


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Modified-hyper shrinkage function

k is the scaling function

x

xxkxXmh

,0

,6

1)(

2


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Subband Adaptive shrinkage function

j

j

j

SA

λ |x| 0

λ |x| -

1

--1

x jx2

j

x2

j

)(


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ECG Denoising - Noise level is 20%


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ECG Denoising - Noise level is 20%


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EEG Denoising - Noise level is 20%


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EEG Denoising - Noise level is 20%


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PCG Denoising - Noise level is 20%


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PCG Denoising - Noise level is 20%


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Objective…

Reduce the minimum mean square error (MMSE)

between original ECG f and denoised ECG . y = [y1, y2, . . . , yN] N

Thenyi = f(xi) + ni, i = 1, 2, . . ,N

The risk function,

f̂

2ˆ1),ˆ( ff

NffR


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Estimation of Mean, Variance and Risk

)()( XEM

)()( XVarV

2)()(

XER2

)()(

MV


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Mean estimation for Hyper Shrinkage

Let X ~ N(θ,1), and be the probability distribution and the density function for standard Gaussian random variable respectively, then the Mean estimation is given by

kM hyp 13)(

k13

k11 2 k11 2


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Variance estimation for Hyper Shrinkage

The Variance estimation is given by

1)(hypV

2)(mhM


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Risk estimation for Hyper Shrinkage

The Risk estimation is given by

1)(hypR


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Mean estimation for Subband Adaptive Shrinkage

)(SAM 21 32)]()(2[ 2

)]()([2 2

)Φ( )Φ(- 22

The Mean estimation is given by


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Variance estimation for Subband Adaptive Shrinkage

The Variance estimation is given by

)(SAV )Φ(- )Φ(

)]()([31 2

)]1)(1([2 22 )Φ( )Φ(

)Φ(- )Φ(- )])[8 3 Φ(-Φ(

)Φ( )Φ(- 24 2)(SAM


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Powerline Frequency Interference Cancellation


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50 Hz Noise Cancellation in ECG


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50 Hz Noise Cancellation ECG


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50 Hz Noise Cancellation ECG


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50 Hz Noise Cancellation in EEG


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50 Hz Noise Cancellation in PCG


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Journal publication arise from this work

Poornachandra S. and N. Kumaravel, “A Wavelet coefficient smoothened RLS-Adaptive denoising model for ECG”, Journal of Biomedical Sciences Instrumentation, vol. 39, ISA vol. 437, pp. 154-157, April 2003.

Poornachandra S. and N. Kumaravel, “Hyper-trim shrinkage for denoising of ECG signal”, ELSEVIER Journal of Digital Signal Processing, Vol. 15, Issue-3, pp. 315-327, May 2005.

Poornachandra S. and N. Kumaravel, “Wavelet based Adaptive Denoising Models for Biological Signals”, Journal of Institute of Engineers, Vol. 86, Nov. 2005.

Poornachandra S. and N. Kumaravel, “Subband-Adaptive Shrinkage for Denoising of ECG Signals”, EURASIP Journal on Applied Signal Processing (Available online).


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Journal publication arise from this work

Poornachandra S. and N. Kumaravel, “Wavelet Thresholding by -Trim Mean Filter”, Journal of Institute of Engineers, Vol. 87, January 2007

Poornachandra S. and N. Kumaravel, “Statistical Estimation for Hyper Shrinkage”, ELSEVIER Journal of Digital Signal Processing, (Available online).

Poornachandra S. and N. Kumaravel, “A Novel method for the Elimination of Power Line Frequency in ECG Signal using Hyper Shrinkage Function”, ELSEVIER Journal of Digital Signal Processing, (In Press).

Poornachandra S. and N. Kumaravel, “Hyper Shrinkage for Denoising of ECG Signal with Adaptive Filter”, ELSEVIER Journal of Digital Signal Processing, (Under Review).


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Conference publication

Poornachandra S. and Dr N. Kumaravel, “A new wavelet co-efficient smoothened Adaptive filtering for bio-signal”, Proc. of ICBME, 4th –7th December 2002, Singapore.

Poornachandra S. and Dr N. Kumaravel “Wavelet coefficient smoothened LMS-adaptive denoising model for electro-cardio graph”, Proc. of NCC, 31st-2nd Jan-Feb 2003, Chennai, India.

Poornachandra S., Dr N. Kumaravel et al., “A modified Wavelet model with RLS-adaptive for denoising Gaussian Noise from ECG signal”, Proc. of NSIP, 8th –11th June 2003, Grudo-Triesta, Italy.


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Conference publication

Poornachandra S., Dr N. Kumaravel et al., “Denoising of ECG by -Trim thresholding of Wavelet coefficients”, Proc. of NCC, 31st-2nd Jan-Feb 2004, Bangalore, India.

Poornachandra S., Dr. N. Kumaravel et al, “WaveShrink using Modified Hyper-Shrinkage Function” Proc. of IEEE-EMBC, 1st – 4th September 2005, Shanghai, China.(best session paper award)

Poornachandra S., Dr. N. Kumaravel et al, “A Novel method for the elimination of powerline frequency in ECG signals using modified-hyper shrinkage”, Proc. of IFMBE, ICBME2005, 7th – 10th December 2005, Singapore.


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Conclusion…..

In real time bio-signal acquisition, the noise level is always less than the signal level. Hence shrinkage is the better choice for denoising

This thesis suggested that shrinkage can also be used to eliminate powerline frequency from bio-signals.

This thesis proposed following shrinkage function that are better than the existing ones and its mathematical models

Hyper shrinkage Modified shrinkage Subband-Adaptive shrinkage


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Suggested for future work

1. Image processing applications2. VLSI implementation3. Better shrinkage models can be proposed4. Communication signal processing5. Iterative shrinkage6. Fussy shrinkage


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Bias and Variance

Documents

Wavelet Based Subband Shrinkage Models and their Applications in Denoising of Biomedical Signals