Advanced Digital Signal Processing

1. Advanced Digital Signal Processing Prof. Nizamettin AYDIN naydin@yildiz.edu.tr http://www.yildiz.edu.tr/~naydin

2. Identification and Detection of Embolic Signals Using DWT and Fuzzy Logic

3. 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.

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

5. Typical Doppler System for Detecting Emboli sin cos Gated transmiter Master osc. Demodulator Sample & hold Band-pass filter si Further processing Logic unit Receiver amplifier RF filter Demodulator Sample & hold Band-pass filter sq Transducer V • Transcranial Doppler (TCD) ultrasound • Operating frequency=1-2 MHz

6. Detection System DWT (n scale) Forward Coeff. filtering IDWT (n scale) [.]2 (n scale) sf(k) Detection and Classifi-cation Quadrature to directional conversion si(k) sr(k) sq(k) DWT (n scale) Reverse Coeff. filtering IDWT (n scale) [.]2 (n scale)

7. DWT of ES time (ms) time (ms) time (ms) time (ms)

8. 0 1 • th1th2th3th4x(n) Fuzzy Membership Function and Detection Rules • 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 Trapezoidal membership function used for the derivation of membership values

9. Some Parameters Used in Detection power Apk Ath ton tpk toff time A sketch of an instantaneous power. 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

10. 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)

11. 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

12. Membership Values for Parameters

13. Mean and Standard Deviations of Parameters

14. Threshold Values of Parameters

15. Detection Results • When it was tested on a third data set, 198 ES out of 202 were detected as ES

16. Denoising of Embolic Doppler Signals • Wavelet denoising involves: • Taking DWT • Estimating noise level and an appropriate threshold • Shrinking the coefficients • Taking inverse DWT

17. DWT and Denoising N N/2 HPF 2 D1 N 2 HPF S(k) S(k) N N N N/2 LPF 2 A1 N 2 LPF • Scaling and positions are dyadic • Fast algorithms exist • Scaling function and wavelet must satisfy the following conditions • Wavelet denoising involves; • Taking DWT • Estimating noise level and an appropriate threshold • Shrinking the coefficients • Taking inverse DWT

18. 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.

19. 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.

20. DWT of ES time (ms) time (ms) time (ms) time (ms)

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

22. Improvement on EBR

23. Improvement on HWM

24. Improvement on ESO

25. Denoising Examples Red: WFT,green: WFT_d,blue: WT,yellow: WT_d time (ms)

26. Denoising Examples Red: WFT,green: WFT_d,blue: WT,yellow: WT_d time (ms)

27. Denoising Examples Red: WFT,green: WFT_d,blue: WT,yellow: WT_d time (ms)

28. Denoising Examples Red: WFT,green: WFT_d,blue: WT,yellow: WT_d time (ms)

29. Denoising Examples Red: WFT,green: WFT_d,blue: WT,yellow: WT_d time (ms)

30. Denoising Examples Red: WFT,green: WFT_d,blue: WT,yellow: WT_d time (ms)

31. Denoising Examples Red: WFT,green: WFT_d,blue: WT,yellow: WT_d time (ms)

32. 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

34. Denoising Embolic Doppler Ultrasound Signals Using Dual Tree Complex Discrete Wavelet Transform

35. 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.

36. Dual tree complex wavelet transform 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.

37. Dual tree complex wavelet transform • 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.

38. Dual tree complex wavelet transform Decomposition

39. Dual tree complex wavelet transform Reconstruction

40. Dual tree complex wavelet transform

41. Dual tree complex wavelet transform

42. 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.

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

44. DWT of ES time (ms) time (ms) time (ms) time (ms)

45. Example of denoising using DTCWT Example of denoising using DWT

46. Example of denoising using DTCWT Example of denoising using DWT

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

48. 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

49. Processing Complex Quadrature Signals Using Modified Complex Discrete Wavelet Transform

50. 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.