advanced digital signal processingyildiz.edu.tr/~naydin/adp/lectures/pdf/adp_10.pdf · advanced...

1

Prof. Nizamettin AYDIN

[email protected]

http://www.yildiz.edu.tr/~naydin

Advanced Digital Signal Processing

1

Identification and Detection of Embolic Signals Using DWT and Fuzzy Logic

2

Introduction

• Stroke is an illness causing partial or total paralysis, or death. • The most common type of stroke (80% of all strokes) occurs

when a blood vessel in or around the brain becomes plugged. • The plug can originate in an artery of the brain or somewhere

else in the body, often the heart, where it breaks off and travels up the arterial tree to the brain, until it lodges in a blood vessel.

• These "travelling clots" are called emboli.• Strokes caused by emboli from the heart are often seen in

people with an irregular heartbeat (a condition called atrial fibrillation), as well as after a heart attack or heart surgery.

3

Embolic signals (ES) are;• Within audio range• Very short duration• Non-stationary• Frequency focused• High intensity transient

like• Unidirectional

Embolic Signals

4

Gated

transmiter

Master

osc.

Band-pass

filter

Sample

& holdDemodulator

Receiver

amplifier

RF

filter

Band-pass

filter

Sample

& holdDemodulator

Logic

unit

si

sq

V

Transducer

sin

cos

Typical Doppler System for Detecting Emboli

• Transcranial Doppler (TCD) ultrasound• Operating frequency=1-2 MHz

Further processing

5

Detection System

sq(k) sr(k)

Quadrature to directionalconversion

DWT(n scale)

DWT(n scale)

ForwardCoeff.filtering

IDWT(n scale)

[.]2

(n scale)

ReverseCoeff.filtering

IDWT(n scale)

[.]2

(n scale)

Detectionand

Classifi-cation

sf(k)si(k)

6

2

DWT of ES

time (ms) time (ms) time (ms) time (ms)

7

Fuzzy Membership Function and Detection Rules

113,)(

112

1 zzzthth

thnx−=−

−=

Membership value (MV):

If x(n) < th1 ; MV ⇒ 0 or 1

If x(n) ≥ th1 & ≤ th2 ;

MV ⇒

If x(n) > th2 & < th3 ; MV ⇒ 0 or 1

If x(n) ≥ th3 & ≤ th4 ;

MV ⇒

If x(n) > th4 ; MV ⇒ 0 or 1

th1 th2 th3 th4 x(n)

0

1

Trapezoidal membership function used for the derivation of membership values

214,3)(

234

zzzthth

thnx−=−

−=

8

Some Parameters Used in Detection

Ath : threshold value

P2TR: peak value to threshold ratio

TP2TR: total peak value to threshold ratio

RR : rise rate

FR : fall rate

F2RM : peak forward power to reverse power ratio

TF2R : total forward power to reverse power ratio

)()(

)(log102

)()(

)(log102

)/(2

)/(2

)()(

log102

)(log102

loglog 22

dBkA

kARTF

dBtA

tARMF

msdBtt

TRPFR

msdBtt

TRPRR

dBA

kATRTP

dBA

ATRP

NNA

off

on

off

on

off

on

t

tk r

t

tk f

pkr

pkf

pkoff

onpk

th

t

tk f

th

pk

rnfnth

∑∑

∑

=

=

=

=

=

−=

−=

=

=

+= σσ

Ath

Apk

ton tpk toff time

power A sketch of an instantaneous power.

9

More Parameters

These parameters are based on the narrow-band assumption

ts : averaged time centre of the signal

fs : averaged frequency centre of the signal

Ts2: time spreading

Bs2: frequency spreading

a(t) : instantaneous amplitude

f(t) : instantaneous frequency

sa(t) : complex quadrature signal given as

sa(t)=s(t)+jH{s(t)}

Where H{s(t)} is Hilbert transform of s(t) dt

tsdtf

tsta

dttsE

dftSffE

B

dttsttE

T

dffSfE

f

dttstE

t

a

a

s

ss

s

ss

s

ss

ss

)(arg

2

1)(

)()(

)(

)()(1

)()(1

)(1

)(1

2

222

222

2

2

π=

=

+∞<=

−=

−=

=

=

∫

∫

∫

∫

∫

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

10

Method

• Two independent data sets comprising 100 ES, 100 artefacts and 100 Doppler speckle were used (parameters were optimised using the 1st data set)

• Sampling frequecy was 7150 Hz• 8 scales DWT was applied to directional Doppler signals• 8th order Daubechies wavelet filter was used

• Individual wavelet scales were reconstructed

• Instantaneous powers for each scale were calculated

• Thresholds for each scale were determined

• Certain parameters were evaluated for each scales

• Membership values were derived from the membership functions

• Final decision for the type of the signal is based on average membership values of all the parameters for each signal types

11

Table 2: Membership rules for the parametersx(n) < th1 ≥ th1&≤ th2 > th2&< th3 ≥ th3&≤ th4 > th4

ar em sp ar em sp ar em sp ar em sp ar em sp

P2TR 0 0 1 0 z1 z3 0 1 0 z2 z4 0 1 0 0TP2TR 0 0 1 0 z1 z3 0 1 0 z2 z4 0 1 0 0F2RM 1 0 0 z3 0 z1 0 0 1 0 z2 z4 0 1 0TF2R 1 0 0 z3 0 z1 0 0 1 0 z2 z4 0 1 0RR 1 0 0 z3 z1 0 0 1 0 0 z4 z2 0 0 1FR 1 0 0 z3 z1 0 0 1 0 0 z4 z2 0 0 1ts 0 0 1 0 z3 z1 0 1 0 z2 z4 0 1 0 0fs 0 0 1 0 z1 z3 0 1 0 z2 z4 0 1 0 0T 2

s 0 0 1 0 z3 z1 0 1 0 z2 z4 0 1 0 0B2

s 1 0 0 z3 z1 0 0 1 0 0 z4 z2 0 0 1V IE 0 0 1 z1 0 z3 1 0 0 z4 z2 0 0 1 0V IF 1 0 0 z3 z1 0 0 1 0 0 z4 z2 0 0 1

ar: artifact, em: emboli, sp: speckle; z1 = (th2 − th1)−1(x(n) − th1),

z2 = (th4 − th3)−1(x(n) − th3), z3 = 1 − z1, z4 = 1 − z1

Membership Values for Parameters

12

3

ES Artifact DSMean SD Mean SD Mean SD

SMP 2.45 0.83 5.96 0.77 3.27 0.72TP2TR 25.13 3.17 34.57 6.62 18.32 3.13P2TR 12.1 3.01 14.16 6.67 6.29 1.82F2RM 25.44 7.01 9.4 11.15 23.18 7.59F2R8 1.29 7.13 1.79 11.25 -0.58 6.18RR 3.88 2.53 0.95 0.59 3.52 2.05FR 4.65 3.15 0.97 0.63 4.00 2.12

TF2R 15.30 6.24 4.05 9.82 14.13 4.03ts 51.59 31.7 146.38 76.8 6.96 4.9fs 0.114 0.049 0.014 0.007 0.108 0.189T 2

s 73.87 43.3 185.08 106.6 10.86 8.6B2

s 0.08 0.038 0.033 0.031 0.492 0.438V IE 101.34 96.1 43.47 63.3 12.3 5.6V IF 0.016 0.008 0.012 0.021 0.021 0.012

Mean and Standard Deviations of Parameters

13

Threshold Values of Parameters

th1 th2 th3 th4

P2TR (dB) 6 12 14 20TP2TR (dB) 17 23 26 38F2RM (dB) 10 20 22 26TF2R (dB) 4 8 10 20RR (ms) 0.6 1.4 2 5FR (ms) 0.6 1.4 2 6ts (ms) 10 20 60 120

fs/Fs (unit) 0.01 0.035 0.08 0.1T 2

s (ms2) 6 18 40 100B2

s/Fs (unit) 0.03 0.06 0.1 0.4V IE (unit) 12 60 100 140

V IF/Fs (unit) 0.008 0.016 0.021 0.04Fs= Sampling frequency

14

Detection Results

Data set 1 Data set 2100 ES 100 ES

98% as embolic signal 95% as ES1% as artifact 3% as artifact1% disputed 2% disputed100 artifacts 100 artifacts

96% detected as artifact 98% as artifact4% disputed 2% as ES

100 DS 100 DS93% as DS 95% as DS6% as ES 1% as artifact

1% disputed 4% disputed

When it was tested on a third data set, 198 ES out of 202 were detected as ES

15

Denoising of Embolic Doppler Signals

Wavelet denoising involves:•Taking DWT •Estimating noise level and an appropriate threshold•Shrinking the coefficients•Taking inverse DWT

16

DWT and Denoising

∑−

=

−=1

0 0

00)(1

),(

0

N

km

m

ms

a

anbkks

anmW ψ

• Scaling and positions are dyadic

• Fast algorithms exist• Scaling function and

wavelet must satisfy the following conditions

S(k)

A1LPF

D1HPF 2

2

S(k)

LPF

HPF2

2N

N

N N

N

N

N/2

N/2

0)(,1)( == ∫∫ dttdtt ψφ

Wavelet denoising involves;•Taking DWT •Estimating noise level and an appropriate threshold•Shrinking the coefficients•Taking inverse DWT

17

Method

• 50 low intensity ES from patients with symptomatic carotid stenosis were recorded by using a TCD system.

• The data length was 2048 point with a sampling frequency of 7150 Hz.

• For wavelet denoising, Daubechies’ 8th order wavelet with 8 scales was used. Wavelet denoising rules were specifically adapted to suppress Doppler speckle and artifacts, by utilizing ES characteristics.

• ES were analyzed using both a 128 point complex FFT with Hanning window and a 64 scale complex Morlet WT.

• TF and TS representations of ES before and after denoising were compared by calculating EBR, HWM, and ESO.

18

4

EBR, HWM, ESO

• Embolic signal to background power ratio (EBR) is given as

• Apeak is the power at frequency with maximum power intensity. Bavg is the average power of the background intensity and calculated by time and frequency/scale averaging of the TF/TS results.

• HWM is an estimation of temporal resolution, defined as half width maximum of the embolic signal power increase in the time domain.

• ESO is absolute time of embolic signal onset as an estimation of the accuracy of temporal localization.

)(log10 dBB

AEBR

avg

peak=

19

DWT of ES


20

Results

Mean (and standard deviations) of the EBR, HWM, and ESO for the 50 embolic signals

EBR(dB) HWM(ms) ESO(ms)

WFT 15.31(2.65) 11.98(11.77) 57.82(22.91)

WFTdenoise 21.90 (3.44) 10.77(11.07) 67.76(7.67)

WT 15.45(2.37) 22.00(47.46) 53.07(28.10)

WTdenoise 22.40(4.23) 13.21(27.41) 68.39(7.70)

ESO measured from time domain signals 67.85(7.64)

21

Improvement on EBR

0

5

10

15

20

25

30

35

40

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Signal no

EB

R (

DB

) EBRf

EBRfd

EBRw

EBRwd

22

Improvement on HWM

0

50

100

150

200

250

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Signal no

HW

M (

ms) HWMf

HWMfd

HWMw

HWMwd

23

Improvement on ESO

0102030405060708090

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Signal no

ES

O (

ms)

ESOf

ESOfd

ESOt

ESOw

ESOwd

24

5

Red: WFT, green: WFT_d, blue: WT, yellow: WT_d

time (ms)

Denoising Examples

25


time (ms)

Denoising Examples

26


time (ms)

Denoising Examples

27


time (ms)

Denoising Examples

28

Denoising Examples


time (ms)

29

Denoising Examples


time (ms)

30

6

Denoising Examples


time (ms)

31

Denoising Conclusion

• ~5 dB improvement in EBR is achieved• Time localization has improved• Most of the low frequency artifacts can be

removed by simply discarding higher scale coefficients during reconstruction

• Attained a considerable improvement on analysis and detection of embolic signals

• Denoising algorithms must take into account unique specifications of a signal being analyzed

32

33

Denoising Embolic Doppler Ultrasound Signals Using Dual Tree Complex

Discrete Wavelet Transform

34

Dual tree complex wavelet transform

• The Dual Tree Complex Wavelet Transform (DTCWT) was developed to overcome the lack of shift invariance property of ordinary DWT.

• In the analysis of non-stationary Doppler signals (particularly embolic Doppler ultrasound signals which are similar to transients), any distortion in the phase of the input-signal can cause unwanted variations in different scales.

• DTCWT is approximately shift invariance and this property can cause better results in analyze part of embolic Doppler ultrasound signals.

35


36

In the figure 16 unit step functions with different phases are used as input signals for DWT and DTCWT. As it is seen, the coefficients in DTCWT are less affected.

7


• DTCWT consists of a pair of DWT trees, each representing real and imaginary parts of the transform.

• In both DWTs all the filters are real and these two real trees use two different sets of filters. These sets of filters are jointly designed so that the overall transform is approximately analytic.

• The complexity of the transform can be observed from the frequency response of wavelets at levels 1 to 4 and of the level 4 scaling function.

• In DTCWT, a real signal is applied to the both trees for decomposition and the outputs of the both reconstructed trees are added at the end of the reconstruction stage.

37


38

Decomposition

( )nh1

( )nh 0

( )nh1

( )nh 0

( )ng 0

( )ng 1

( )ng 0

( )ng 1

( )ng 0

( )ng 1

( )nh1

( )nh 0


39

Reconstruction

( )ng 1

~

( )ng 0

~

( )ng 1

~

( )ng 0

~

( )ng 1

~

( )ng 0

~

( )nh 0

~

( )nh 1

~

( )nh 0

~

( )nh 1

~

( )nh 0

~( )nh 1

~( )nh 1

~


40


41

Method

• 25 low intensity ES from patients with symptomatic carotid stenosis were recorded by using a TCD system.

• The data length was 2048 point with a sampling frequency of 7150 Hz.

• For DWT denoising, Daubechies 8th order wavelet with 8 scales was used.

• For DTCWT denoising, a DTCWT algorithm adapted from “http://taco.poly.edu/WaveletSoftware“ with 8 scales was used.

• Doppler signals were analyzed using 128 point complex FFT with Gausian window.

42

8

EBR, HWM, ESO

• TF representation of embolic signals before and after denoisingwere compared by calculating EBR, HWM, and ESO. – Embolic signal to background power ratio (EBR) is given as

• Apeak is the power at frequency with maximum power intensity. Bavg is the average power of the background intensity and calculated by time and frequency averaging of the TF results.

– HWM is an estimation of temporal resolution, defined as half width maximum of the embolic signal power increase in the time domain.

– ESO is absolute time of embolic signal onset as an estimation of the accuracy of temporal localization.

• Wavelet denoising rules were specifically adapted to reject or suppress Doppler speckle and artifacts caused by probe tapping, tissue movement, speech etc, by utilizing embolic signal characteristics.

)(log10 dBB

AEBR

avg

peak=

43

DWT of ES


44

Example of denoising using

DTCWT

45


DWT


DTCWT

46


DWT

Results

Mean of the EBR, HWM, and ESO for the 25 embolic signals

Denoising EBR(dB) HWM(ms) ESO(ms)

none 17.6 9.5 67.8

DWT 22 8.9 70

DTCWT 25.8 8.8 70.2

47

Denoising Conclusion

• ~8 dB improvement is obtained by using DTCWT compared to the improvement provided by the conventional DWT (less than 5 dB).

• Most of the low frequency artifacts can be removed by simply discarding higher scale coefficients during reconstruction

• Attained a considerable improvement on analysis and detection of embolic signals

• Denoising algorithms must take into account unique specifications of a signal being analyzed

48

9

49

Processing Complex Quadrature Signals Using Modified Complex Discrete Wavelet Transform

Quadrature Doppler Signals• A quadrature Doppler signal can be assumed as a complex signal,

in which the real and imaginary parts can be represented as the Hilbert transform of each other. Mathematically, a discrete quadrature Doppler signal can be modeled as

Y(n) = D(n) + jQ(n)

where D(n) is in-phase and Q(n) is quadrature-phase components of the signal.

• D(n) and Q(n) can also be represented in terms of the directional signals as

D(n) = ± Sf(n) ± H[Sr(n)]

Q(n) = ± H[Sf(n)] ± Sr(n)

where sf(n) and sr(n) represent forward and reverse signals respectively and H[ ] stands for the Hilbert transform.

50

Quadrature Signals

51“IWW 2009, June 5-7, 2009 Kocaeli , Turkey” “Görkem Serbes”

The information concerning flow direction is encoded in the phase relationship between D(n) and Q(n). You can see the phase difference in the above figure if you look carefully.

Extracting Directional Signals from Quadrature Signals

52

There are a number of methods for extracting directional signals from the quadrature signals, the phasing-filtering technique

(PFT), which is based on Hilbert transform, is most widely used method. Therefore in this work, the reconstructed directional

outputs of the new method are compared with the outputs of the PFT.

Analysis of Directional Signals

• After the extraction of directional signals, different signal processing methods in frequency domain and wavelet domain can be applied to these signals for diagnosis.

• Discrete Wavelet Transform(DWT) is one of the methods which provides analyzing directional signals in wavelet domain.

• With DWT, directional signals are decomposed to different frequency bands and examined.

• But DWT suffers from the lack of shift-invariance property which means that small shifts in the input signal can cause major variations in the distribution of energy between DWT coefficients at different scales .

53

Modified Dual Tree Complex Wavelet Transform

• But DTCWT has two different DWT trees inside and this causes redundancy(2m:1 for m dimensional signals).

• Quadrature Doppler ultrasound signals are dual channel signals obtained from the systems employing quadrature demodulation. Prior to processing Doppler signals by using DTCWT, directional flow signals must be obtained and then two separate transform applied, and this increases the computational complexity.

• In order to decrease computational complexity, a newcomplex discrete wavelet transform, Modified Dual Tree Complex Wavelet Transform(MDTCWT), algorithm is proposed.

54

10


• In the MDTCWT, two modifications are made to the conventional DTCWT.1. At the analysis stage, instead of applying the complex quadrature

signal to the both trees, the in-phase part is applied to the real tree through a Hilbert transformer introducing a 90 degree phase shift into the real part of the signal, and the quadrature-phase part is applied to the imaginary tree directly.

2. At the reconstruction stage, in addition to adding the outputs of reconstructed real and imaginary trees, which gives the signals caused by the blood flow in one direction, they are also subtracted resulting in the signals caused by the blood flow in the other direction.

55


56

( )nh1

( )nh 0 2

2

( )nh1

( )nh 0 2

22

2

2( )ng 0

( )ng 1 2

2( )ng 0

( )ng 1 2

2( )ng 0

( )ng 1 2

( )nh1

( )nh 0

Real Tree

Imaginary Tree

Level 1

Level 2

Level 3

Hilbert

TransformD(n)

Q(n)


57

2

2 ( )nh 1

~

( )nh 0

~

2

2 ( )nh 1

~

( )nh 0

~

2

2 ( )nh 1

~

( )nh 0

~

( )ng 1

~

( )ng 0

~2

2

( )ng 1

~

( )ng 0

~2

2

( )ng 1

~

( )ng 0

~2

2

Real Tree

Coefficients

Real Tree

Coefficients

Imaginary Tree

Coefficients

Imaginary Tree

Coefficients

Level 3

Level 2

Level 1

sr(n)

sf(n)

-

+

Results

• The proposed methods success is measured in two ways. Firstly, the directional signals are obtained by using PFT from a quadrature signal to compare with. Then the same quadrature signal was decomposed to five levels and then reconstructed by using the MDTCWT resulting in the forward and reverse signals.

• Finally both results are compared statistically by using the percent root mean square difference (PRD) for both forward and reverse signals

58

100)(

2

2

×−

=∑

∑j

ji

x

xxPRD

In PRD formula, xj is the resulting directional signal

obtained by the PFT and xi is the resulting directional signal obtained by the MDTCWT.

Results

59

For an objective evaluation of the PRD, PRD values of 50 embolic Doppler signals with length of 4096

samples each were calculated and the final PRD value was obtained by averaging the 50 PRD values.

Error(PRD)

Forward output signal: 5.69×10−8

Reverse output signal: 2.65×10−8

Results

60

The signals representing forward (blue line) and reverse (red line) flow components of the embolic

Doppler signal, which are obtained by using the MDTCWT and the PFT

are shown in Figure (a) and (b) respectively. The error signals

obtained by subtracting the signals in Figure (b) from the signals in

Figure (a) are illustrated in Figure (c) and (d) respectively. It is

remarkable that the difference signals for both forward and reverse

flow signals are around -80 dB, indicating that the algorithm works

as exactly intended.

11

Results

• Secondly, the computational complexity of the algorithm was also compared with the PFT followed by two real DWTs, and the PFT followed by two DTCWTs on a PC with Pentium M 1.86 GHz processor and 1 GB RAM.

• The algorithms were implemented in Matlab and tested using a quadrature Doppler signal having 1024 samples. In order to minimize effect of any computational time used by any program, which might be running at the background, each algorithm was run 1000 times and average execution time of the algorithms were calculated.

61

Results

Method: Processing time(ms):

PFT with DWT: 9.0

PFT with DTCWT: 18.1

MDTCWT: 9.1

62

Computational cost of the proposed algorithm (9.1 ms) is almostsame as the PFT algorithm followed by two DWTs (9.0 ms) andhalf of the PFT algorithm followed by two DTCWTs (18.1 ms).

Results

• In conclusion, the MDTCWT algorithm is computationally efficient, inherently offers advantages provided by the conventional DTCWT, and additionally maps directional signals at the end of the reconstruction stage. In the future, it may be possible to design new complex wavelet filters that will have properties similar to that of a Hilbert transformer for further reducing the computational complexity.

63 64

advanced digital signal processingyildiz.edu.tr/~naydin/adp/lectures/pdf/adp_10.pdf · advanced...

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