Exponential-spline-based Features for Ultrasound Signal Characterization

Transcription

1 1 / 29 Exponential-spline-based Features for Ultrasound Signal Characterization Simona Maggio Department of Electronics, Computer Science and Systems (DEIS) University of Bologna Scuola di Dottorato: Scienze e Ingegneria dell Informazione Corso di Dottorato: Ingegneria Elettronica, Informatica e delle Telecomunicazioni 19/10/2009

2 2 / 29 Outline 1 Tissue Characterization for Ultrasound Diagnostic 2 Ultrasound Signals Modeling Predictive Deconvolution Continuous/Discrete Approach 3 Exponential Splines 4 From Continuous to Discrete E-spline-based Whitening Model Non Stationary Discrete Whitening 5 Conclusions Analysis Results Further Developments

3 2 / 29 Outline 1 Tissue Characterization for Ultrasound Diagnostic 2 Ultrasound Signals Modeling Predictive Deconvolution Continuous/Discrete Approach 3 Exponential Splines 4 From Continuous to Discrete E-spline-based Whitening Model Non Stationary Discrete Whitening 5 Conclusions Analysis Results Further Developments

4 2 / 29 Outline 1 Tissue Characterization for Ultrasound Diagnostic 2 Ultrasound Signals Modeling Predictive Deconvolution Continuous/Discrete Approach 3 Exponential Splines 4 From Continuous to Discrete E-spline-based Whitening Model Non Stationary Discrete Whitening 5 Conclusions Analysis Results Further Developments

5 2 / 29 Outline 1 Tissue Characterization for Ultrasound Diagnostic 2 Ultrasound Signals Modeling Predictive Deconvolution Continuous/Discrete Approach 3 Exponential Splines 4 From Continuous to Discrete E-spline-based Whitening Model Non Stationary Discrete Whitening 5 Conclusions Analysis Results Further Developments

6 2 / 29 Outline 1 Tissue Characterization for Ultrasound Diagnostic 2 Ultrasound Signals Modeling Predictive Deconvolution Continuous/Discrete Approach 3 Exponential Splines 4 From Continuous to Discrete E-spline-based Whitening Model Non Stationary Discrete Whitening 5 Conclusions Analysis Results Further Developments

7 3 / 29 Ultrasound Imaging Pros: real time, non invasive. Limits: low resolution. Tissue characterization: Highligthing features invisible by visual inspection Cancer detection and staging Avoiding unnecessary biopsy Specific application: Prostate cancer computer-aided detection

8 3 / 29 Ultrasound Imaging Pros: real time, non invasive. Limits: low resolution. Tissue characterization: Highligthing features invisible by visual inspection Cancer detection and staging Avoiding unnecessary biopsy Specific application: Prostate cancer computer-aided detection

9 4 / 29 Procedure for Tissue Characterization Figure: Feature extraction for ultrasound analysis

10 4 / 29 Procedure for Tissue Characterization Figure: Feature extraction for ultrasound analysis

11 4 / 29 Procedure for Tissue Characterization Figure: Feature extraction for ultrasound analysis

12 5 / 29 From Tissue to Echo Discrete model: z = HΣs + n = Hx + n Σ: coherent reflection, macroscopic interactions, mean value of diffused field. s: incoherent reflections, interactions smaller than wavelength, random fluctuations of diffused field. Scattering as generalized gaussian noise: p(s µ, b) = a e s µ b 2

13 5 / 29 From Tissue to Echo Discrete model: z = HΣs + n = Hx + n Σ: coherent reflection, macroscopic interactions, mean value of diffused field. s: incoherent reflections, interactions smaller than wavelength, random fluctuations of diffused field. Scattering as generalized gaussian noise: p(s µ, b) = a e s µ b 2

14 5 / 29 From Tissue to Echo Discrete model: z = HΣs + n = Hx + n Σ: coherent reflection, macroscopic interactions, mean value of diffused field. s: incoherent reflections, interactions smaller than wavelength, random fluctuations of diffused field. Scattering as generalized gaussian noise: p(s µ, b) = a e s µ b 2

15 5 / 29 From Tissue to Echo Discrete model: z = HΣs + n = Hx + n Σ: coherent reflection, macroscopic interactions, mean value of diffused field. s: incoherent reflections, interactions smaller than wavelength, random fluctuations of diffused field. Scattering as generalized gaussian noise: p(s µ, b) = a e s µ b 2

16 Deconvolution to recover Tissue Response Convolutional model: z[k] = x[k] h[k] + n[k] Deconvolution to restore tissue response: x[k] Point Spread Function (PSF) h[n] not known Blind adaptive deconvolution approach 1 Advantages: simplicity, low computational cost, variable PSF. 1 [Ng et al., 2007], [Michailovich and Adam, 2005], [Jensen, 1994],[Rasmussen, 1994] 6 / 29

17 7 / 29 Predictive Deconvolution (PD) W(z) predictive filter to remove predictable features of z[n] Ideal reconstruction if the transducer is an AR system and x[n] is white Gaussian whitening Recursive Least Squares solution for adaptive whitening z[n] as non stationary AR process and x[n] generalized Gaussian Recovering unpredictable part of RF signal: scattering

18 7 / 29 Predictive Deconvolution (PD) W(z) predictive filter to remove predictable features of z[n] Ideal reconstruction if the transducer is an AR system and x[n] is white Gaussian whitening Recursive Least Squares solution for adaptive whitening z[n] as non stationary AR process and x[n] generalized Gaussian Recovering unpredictable part of RF signal: scattering

19 Improvement due to Predictive Deconvolution Prostate Images No Preprocessing RLS Deconvolution SE 0.69 ± ± 0.09 SP 0.94 ± ± 0.01 Acc 0.93 ± ± 0.02 Az 0.92 ± ± / 29

20 Improvement due to Predictive Deconvolution Benignant case No Preprocessing RLS Deconvolution SE 0.69 ± ± 0.09 SP 0.94 ± ± 0.01 Acc 0.93 ± ± 0.02 Az 0.92 ± ± / 29

21 Improvement due to Predictive Deconvolution Malignant case No Preprocessing RLS Deconvolution SE 0.69 ± ± 0.09 SP 0.94 ± ± 0.01 Acc 0.93 ± ± 0.02 Az 0.92 ± ± / 29

22 Continuous/Discrete Approach for PD Continuous whitening g(t) continuous RF signal in a stationary case w(t) continuous uncorrelated unpredictable (scattering) Link with discrete: discretization in a spline basis Exponential splines: multiresolution version of L α Identification of continuous model from sampled data Shift-variant multiresolution description of scattering signal 9 / 29

23 Continuous/Discrete Approach for PD Continuous whitening g(t) continuous RF signal in a stationary case w(t) continuous uncorrelated unpredictable (scattering) Link with discrete: discretization in a spline basis Exponential splines: multiresolution version of L α Identification of continuous model from sampled data Shift-variant multiresolution description of scattering signal 9 / 29

24 Continuous/Discrete Approach for PD Continuous whitening g(t) continuous RF signal in a stationary case w(t) continuous uncorrelated unpredictable (scattering) Link with discrete: discretization in a spline basis Exponential splines: multiresolution version of L α Identification of continuous model from sampled data Shift-variant multiresolution description of scattering signal 9 / 29

25 Exponential Splines E-splines 2 : s(t) = k Z a kρ α (t t k ) + p 0 (t) ρ α green function of L α s(t) reconstructed in exponential B-spline basis: s(t) = k Z c[k]β α(t k) β α (t) = α {ρ α (t)} Scale T for generic sampling time: β αt (t) = αt {ρ α (t)} 2 [Khalidov and Unser, 2006], [Unser and Blu, 2005a], [Unser and Blu, 2005b] 10 / 29

26 Exponential Splines E-splines 2 : s(t) = k Z a kρ α (t t k ) + p 0 (t) ρ α green function of L α s(t) reconstructed in exponential B-spline basis: s(t) = k Z c[k]β α(t k) β α (t) = α {ρ α (t)} Scale T for generic sampling time: β αt (t) = αt {ρ α (t)} 2 [Khalidov and Unser, 2006], [Unser and Blu, 2005a], [Unser and Blu, 2005b] 10 / 29

27 Exponential Splines E-splines 2 : s(t) = k Z a kρ α (t t k ) + p 0 (t) ρ α green function of L α s(t) reconstructed in exponential B-spline basis: s(t) = k Z c[k]β α(t k) β α (t) = α {ρ α (t)} Scale T for generic sampling time: β αt (t) = αt {ρ α (t)} 2 [Khalidov and Unser, 2006], [Unser and Blu, 2005a], [Unser and Blu, 2005b] 10 / 29

28 E-Spline Multi-Resolution Analysis Dyadic scale for multiresolution analysis: T = 2 i Scaling function: ϕ i (t) = β 2 i(t)/ β 2 i L2 At scale i it is not a dilated version of the scaling function at scale 0. Mallat s filter bank algorithms but precomputation of filters for each scale Main property: wavelet coefficients of a signal f(t) are samples of smoothed version of L α (f) 11 / 29

29 E-Spline Multi-Resolution Analysis Dyadic scale for multiresolution analysis: T = 2 i Scaling function: ϕ i (t) = β 2 i(t)/ β 2 i L2 At scale i it is not a dilated version of the scaling function at scale 0. Mallat s filter bank algorithms but precomputation of filters for each scale Main property: wavelet coefficients of a signal f(t) are samples of smoothed version of L α (f) 11 / 29

30 12 / 29 E-spline-based Whitening Model Continuous: R gg (t) = (ρ α1 (t) ρ α1 ( t)) ( ρ αp (t) ρ αp ( t) ) Discrete: R(z) = z p B ( α: α) (z) p i=1 e αi z(1 e αi z)(1 e αi z 1 ) = b[1] p 1 i=1 1 ζ i (1 ζ i z)(1 ζ i z 1 ) p i=1 (1 e e αi αi z)(1 e αi z 1 ) Inter-dependence: ζ i = f( α) Whitening filter: W(z 1 ) = p 1 Y (1 ζ i z 1 ) i=1 py (1 e αi z 1 ) i=1 ARMA(p, p 1) discrete model (stable)

31 12 / 29 E-spline-based Whitening Model Continuous: R gg (t) = (ρ α1 (t) ρ α1 ( t)) ( ρ αp (t) ρ αp ( t) ) Discrete: R(z) = z p B ( α: α) (z) p i=1 e αi z(1 e αi z)(1 e αi z 1 ) = b[1] p 1 i=1 1 ζ i (1 ζ i z)(1 ζ i z 1 ) p i=1 (1 e e αi αi z)(1 e αi z 1 ) Inter-dependence: ζ i = f( α) Whitening filter: W(z 1 ) = p 1 Y (1 ζ i z 1 ) i=1 py (1 e αi z 1 ) i=1 ARMA(p, p 1) discrete model (stable)

32 12 / 29 E-spline-based Whitening Model Continuous: R gg (t) = (ρ α1 (t) ρ α1 ( t)) ( ρ αp (t) ρ αp ( t) ) Discrete: R(z) = z p B ( α: α) (z) p i=1 e αi z(1 e αi z)(1 e αi z 1 ) = b[1] p 1 i=1 1 ζ i (1 ζ i z)(1 ζ i z 1 ) p i=1 (1 e e αi αi z)(1 e αi z 1 ) Inter-dependence: ζ i = f( α) Whitening filter: W(z 1 ) = p 1 Y (1 ζ i z 1 ) i=1 py (1 e αi z 1 ) i=1 ARMA(p, p 1) discrete model (stable)

33 12 / 29 E-spline-based Whitening Model Continuous: R gg (t) = (ρ α1 (t) ρ α1 ( t)) ( ρ αp (t) ρ αp ( t) ) Discrete: R(z) = z p B ( α: α) (z) p i=1 e αi z(1 e αi z)(1 e αi z 1 ) = b[1] p 1 i=1 1 ζ i (1 ζ i z)(1 ζ i z 1 ) p i=1 (1 e e αi αi z)(1 e αi z 1 ) Inter-dependence: ζ i = f( α) Whitening filter: W(z 1 ) = p 1 Y (1 ζ i z 1 ) i=1 py (1 e αi z 1 ) i=1 ARMA(p, p 1) discrete model (stable)

34 12 / 29 E-spline-based Whitening Model Continuous: R gg (t) = (ρ α1 (t) ρ α1 ( t)) ( ρ αp (t) ρ αp ( t) ) Discrete: R(z) = z p B ( α: α) (z) p i=1 e αi z(1 e αi z)(1 e αi z 1 ) = b[1] p 1 i=1 1 ζ i (1 ζ i z)(1 ζ i z 1 ) p i=1 (1 e e αi αi z)(1 e αi z 1 ) Inter-dependence: ζ i = f( α) Whitening filter: W(z 1 ) = p 1 Y (1 ζ i z 1 ) i=1 py (1 e αi z 1 ) i=1 ARMA(p, p 1) discrete model (stable)

35 13 / 29 Exponential Parameters Estimation MA correction: AR(p): Yule-Walker equations. Only poles a i = e α i ARMA(p,p-1): No pole-zero inter-dependence. Annihilating Polynomial: coeff of AP of R gg (k) are functions { of α: p ( i=1 1 ai z 1)} R gg (k) = 0 AR to compute poles pole-zero inter-dependence to estimate zeros 1 MA(z) filtering to cancel zeros new AR computation for better pole estimation possibly iterations

36 13 / 29 Exponential Parameters Estimation MA correction: AR(p): Yule-Walker equations. Only poles a i = e α i ARMA(p,p-1): No pole-zero inter-dependence. Annihilating Polynomial: coeff of AP of R gg (k) are functions { of α: p ( i=1 1 ai z 1)} R gg (k) = 0 AR to compute poles pole-zero inter-dependence to estimate zeros 1 MA(z) filtering to cancel zeros new AR computation for better pole estimation possibly iterations

37 13 / 29 Exponential Parameters Estimation MA correction: AR(p): Yule-Walker equations. Only poles a i = e α i ARMA(p,p-1): No pole-zero inter-dependence. Annihilating Polynomial: coeff of AP of R gg (k) are functions { of α: p ( i=1 1 ai z 1)} R gg (k) = 0 AR to compute poles pole-zero inter-dependence to estimate zeros 1 MA(z) filtering to cancel zeros new AR computation for better pole estimation possibly iterations

38 13 / 29 Exponential Parameters Estimation MA correction: AR(p): Yule-Walker equations. Only poles a i = e α i ARMA(p,p-1): No pole-zero inter-dependence. Annihilating Polynomial: coeff of AP of R gg (k) are functions { of α: p ( i=1 1 ai z 1)} R gg (k) = 0 AR to compute poles pole-zero inter-dependence to estimate zeros 1 MA(z) filtering to cancel zeros new AR computation for better pole estimation possibly iterations

39 13 / 29 Exponential Parameters Estimation MA correction: AR(p): Yule-Walker equations. Only poles a i = e α i ARMA(p,p-1): No pole-zero inter-dependence. Annihilating Polynomial: coeff of AP of R gg (k) are functions { of α: p ( i=1 1 ai z 1)} R gg (k) = 0 AR to compute poles pole-zero inter-dependence to estimate zeros 1 MA(z) filtering to cancel zeros new AR computation for better pole estimation possibly iterations

40 14 / 29 Restoring Non Stationarity Non stationary RF signal Shift-variant whitening to get the non stationary scattering Two possibilities: Sliding window Recursive parameter estimation Recursive Annihilating Polynomial vs LS solution Time dependent parameters: α[n] Shift-variant E-spline wavelet Shift-variant multiresolution analysis (sliding window) Piece-wise multiresolution description of scattering

41 14 / 29 Restoring Non Stationarity Non stationary RF signal Shift-variant whitening to get the non stationary scattering Two possibilities: Sliding window Recursive parameter estimation Recursive Annihilating Polynomial vs LS solution Time dependent parameters: α[n] Shift-variant E-spline wavelet Shift-variant multiresolution analysis (sliding window) Piece-wise multiresolution description of scattering

42 14 / 29 Restoring Non Stationarity Non stationary RF signal Shift-variant whitening to get the non stationary scattering Two possibilities: Sliding window Recursive parameter estimation Recursive Annihilating Polynomial vs LS solution Time dependent parameters: α[n] Shift-variant E-spline wavelet Shift-variant multiresolution analysis (sliding window) Piece-wise multiresolution description of scattering

43 14 / 29 Restoring Non Stationarity Non stationary RF signal Shift-variant whitening to get the non stationary scattering Two possibilities: Sliding window Recursive parameter estimation Recursive Annihilating Polynomial vs LS solution Time dependent parameters: α[n] Shift-variant E-spline wavelet Shift-variant multiresolution analysis (sliding window) Piece-wise multiresolution description of scattering

44 15 / 29 Phantoms Targets Hypo and hyper echoic target detection Linear classifier: learning on ±9 db, testing on ±6 db Comparison with traditional predictive Deconvolution: improvement 40.3% 1 ROC curves: hypoechoic target detection ESW 1 feat PD Sensitivity Specificity

45 15 / 29 Phantoms Targets Hypo and hyper echoic target detection Linear classifier: learning on ±9 db, testing on ±6 db Comparison with traditional predictive Deconvolution: improvement 2.1% 1 ROC curves: hyperechoic target detection ESW 1 feat PD Sensitivity Specificity

46 16 / 29 Real Data Real data cancer detection: prostate trans-rectal Dataset: 15 benignant cases, 22 malignant cases Linear classifier: training 18 cases, testing 19 unknown img Comparison with traditional predictive Deconvolution: improvement 16.9%

47 17 / 29 Conclusions Predictive Deconvolution to restore US tissue response Continuous approach to predictive deconvolution Discretization in Exponential spline basis Identifications of E-splines tuned on US signal Multiresolution description of tissue response Improvement in detection performance: 16.9% Further Developments Improve parameter estitmation Better MA correction algorithm E-spline for texture analysis

48 17 / 29 Conclusions Predictive Deconvolution to restore US tissue response Continuous approach to predictive deconvolution Discretization in Exponential spline basis Identifications of E-splines tuned on US signal Multiresolution description of tissue response Improvement in detection performance: 16.9% Further Developments Improve parameter estitmation Better MA correction algorithm E-spline for texture analysis

49 18 / 29 Thank you for your attention!

50 Bibliography Jensen, J. (1994). Estimation of in vivo pulses in medical ultrasound. Ultrasonic Imaging, 16: Khalidov, I. and Unser, M. (2006). From differential equations to the constructio of new wavelet-like bases. IEEE Transactions on Signal processing, 54(4). Michailovich, O. V. and Adam, D. (2005). A novel approach to the 2-d blind deconvolution problem in medical ultrasound. IEEE Transactions on Medical Imaging, 24(1): Ng, J., Prager, R., Kingsbury, N., Treece, G., and Gee, A. (2007). Wavelet restoration of medical pulse-echo ultrasound images in an em framework. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 54(3): Rasmussen, K. (1994). Maximum likelihood estimation of the attenuated uktrasound pulse. IEEE Transactions on Signal Processing, 42: Unser, M. and Blu, T. (2005a). Cardinal exponential splines: Part i - theory and filtering algorithms. IEEE Transactions on Signal Processing, 5373(4). Unser, M. and Blu, T. (2005b). Cardinal exponential splines: Part ii - think analog, act digital. IEEE Transactions on Signal Processing, 53(4). 19 / 29

51 20 / 29 Publications 2009 (to be published in next issue) IEEE Transactions on Medical Imaging S. Maggio, A. Palladini, L. De Marchi, M. Alessandrini, N. Speciale, G. Masetti Predictive deconvolution and hybrid feature Selection for Computer-Aided Detection of prostate cancer 2009 March Proceedings International Symposium on Acoustical Imaging M. Scebran, A. Palladini, S. Maggio, L. De Marchi, N. Speciale Automatic regions of interests segmentation for computer aided classification of prostate TRUS images 2008 November Proceedings IEEE IUS2008 S. Maggio, L. De Marchi, M. Alessandrini, N. Speciale Computer aided detection of prostate cancer based on GDA and predictive deconvolution 2005 November WSEAS Transactions on Systems S. Maggio, N. Testoni, L. De Marchi, N. Speciale, G. Masetti Ultrasound Images Enhancement by means of Deconvolution Algorithms in the Wavelet Domain 2005 September WSEAS Int. Conf. on Signal Processing, Computational Geometry and Artificial Vision (ISCGAV) S. Maggio, N. Testoni, L. De Marchi, N. Speciale, G. Masetti Wavelet-based Deconvolution Algorithms Applied to Ultrasound Images

52 21 / 29 Texture analysis through E-splines Global learning of exponential parameters E-spline wavelet tuned on ultrasound signal E-spline wavelet transform for texture typing Comparison with traditional wavelets Improvement with respect to energy information: 14.9% 1 ROC curves: cancer detection Nakagami Naka + Variance ESW Sensitivity Specificity

53 Comparison with previous works Table: Published methods for ultrasound-based prostate tissue characterization Work Ground Truth Technique Results % # ROIs Features SE SP Acc Az Basset 37 Textural Huynen - Textural Houston 25 Textural Schmitz 3405 Multi Scheipers Multi Feleppa 1019 Spectral Mohamed 96 Textural Llobet 4944 Textural Mohamed 108 Multi Han 2000 Multi Maggio Multi on RLS / 29

54 Autocorrelation Function Discretization 1/4 In the first order case the ACF of g(t) can be obtained as: R gg (τ) = F 1 {Φ g } = F 1 { 1 jω α } F 1 { } 1 jω α = e αt u(t) e αt u( t) = ρ α (t) ρ α ( t) (1) + Green function of L α : ρ α (t) = e αt u(t) = p α [k]β α (t k) R gg (t) = = = + k=0 + k=0 + k=0 p α [k]β α (t k) + p α [k] p α [k] k =0 + k =0 + k =0 k=0 p α [k ]β α ( t k ) p α [k ]β α (t k) β α ( t k ) p α [k ]e α β α (t k) β α (t + k + 1) (2) 23 / 29

55 24 / 29 Autocorrelation Function Discretization 2/4 The expression for ACF simplifies to R gg (t) = + k=0 p α [k] p α [k k 1]e α β ( α:α) (t k ) k Z = (p α p α)[k + 1]e α β ( α:α) (t k ) (3) k Z where p α [k] = p α[ k]. It turns out that the z-transform of the discrete signal ACF is given as: R(z) = e α zp α (z)p α (z)b ( α:α)(z) e α zb ( α:α) (z) = (1 e α z 1 )(1 e α z) (4)

56 25 / 29 Autocorrelation Function Discretization 3/4 The extension to higher-order AR models follows naturally as shown below: { R gg (t) = F 1 1 } { 2 jω α 1 F 1 1 } 2 jω α p = (ρ α1 (t) ρ α1 ( t)) ( ρ αp (t) ρ αp ( t) ) = k1 Z(p α1 p α 1 )[k 1 + 1]e α 1 β ( α1 :α 1 )(t k 1 ) kp Z(p αp p α p )[k p + 1]e αp β ( αp:α p)(t k p ) (5)

57 26 / 29 Autocorrelation Function Discretization 4/4 The final expression of the z-transform of R gg (k) is: R(z) = = p e α i zp αi (z)p α i (z) B ( α: α) (z) i=1 p i=1 e α i z (1 e α i z)(1 e α iz 1 ) zp B ( α: α) (z) (6)

58 27 / 29 Whitening Filter Stability 1/3 The proof that the z-transform of B-spline β ( α: α), B ( α, α) (z), doesn t have zeros on unit circle is shown by the following considerations: ˆβ ( α, α) (ω) = = = F [β α β α (t)] p e αi F [β α β α (t p)] i=1 p e αi e jωp ˆβ α (ω) 2 (7) i=1

59 28 / 29 Whitening Filter Stability 2/3 B ( α, α) (z) is linked to the Fourier transform of samples of β ( α, α) (t), ˆβ s,( α, α) (ω): ˆβ s,( α, α) (ω) = k ˆβ ( α, α) (ω + 2πk) = = p e αi e jωp ˆβ α (ω + 2πk) 2 k p e αi e jωp A α (e jω ) (8) i=1 i=1

60 29 / 29 Whitening Filter Stability 3/3 where A α (e jω ) is the discrete Fourier transform of the Gram sequence of B-splines, and, for the Riesz basis property, it is always grater than zero: 0 < r 2 α < A α(e jω ) < R 2 α < + (9) As a consequence ˆβ s,( α, α) (ω) > 0 for every ω and, since B ( α, α) (z) = ˆβ s,( α, α) (ω) e jω =z, B ( α, α) (z) doesn t have any zeros on the unit circle. This proof and the consideration about reciprocal roots of B ( α, α) (z) guarantee the whitening filter stability.