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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme
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Smaggio Pres Jan2010

  1. 1. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Exponential-spline-based identification of ultrasound signal Autoregressive models 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 14/01/2010 1 / 33
  2. 2. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Outline 1 Tissue Characterization for Ultrasound Diagnostic 2 Ultrasound Signals Modeling Predictive Deconvolution Continuous/Discrete Approach: CAR systems 3 Exponential Splines 4 From Continuous to Discrete Representation in Exponential spline basis Standard CAR System identification Novel CAR System identification 5 Results Further Developments 2 / 33
  3. 3. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Outline 1 Tissue Characterization for Ultrasound Diagnostic 2 Ultrasound Signals Modeling Predictive Deconvolution Continuous/Discrete Approach: CAR systems 3 Exponential Splines 4 From Continuous to Discrete Representation in Exponential spline basis Standard CAR System identification Novel CAR System identification 5 Results Further Developments 2 / 33
  4. 4. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Outline 1 Tissue Characterization for Ultrasound Diagnostic 2 Ultrasound Signals Modeling Predictive Deconvolution Continuous/Discrete Approach: CAR systems 3 Exponential Splines 4 From Continuous to Discrete Representation in Exponential spline basis Standard CAR System identification Novel CAR System identification 5 Results Further Developments 2 / 33
  5. 5. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Outline 1 Tissue Characterization for Ultrasound Diagnostic 2 Ultrasound Signals Modeling Predictive Deconvolution Continuous/Discrete Approach: CAR systems 3 Exponential Splines 4 From Continuous to Discrete Representation in Exponential spline basis Standard CAR System identification Novel CAR System identification 5 Results Further Developments 2 / 33
  6. 6. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Outline 1 Tissue Characterization for Ultrasound Diagnostic 2 Ultrasound Signals Modeling Predictive Deconvolution Continuous/Discrete Approach: CAR systems 3 Exponential Splines 4 From Continuous to Discrete Representation in Exponential spline basis Standard CAR System identification Novel CAR System identification 5 Results Further Developments 2 / 33
  7. 7. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Ultrasound Imaging • Pros: real time, non invasive. • Limits: low resolution. • Tissue characterization: • Highlighting features invisible by visual inspection • Cancer detection and staging • Avoiding unnecessary biopsy • Specific application: Prostate cancer computer-aided detection 3 / 33
  8. 8. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Ultrasound Imaging • Pros: real time, non invasive. • Limits: low resolution. • Tissue characterization: • Highlighting features invisible by visual inspection • Cancer detection and staging • Avoiding unnecessary biopsy • Specific application: Prostate cancer computer-aided detection 3 / 33
  9. 9. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Procedure for Tissue Characterization Figure: Feature extraction for ultrasound analysis 4 / 33
  10. 10. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Procedure for Tissue Characterization Figure: Feature extraction for ultrasound analysis 4 / 33
  11. 11. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Procedure for Tissue Characterization Figure: Feature extraction for ultrasound analysis 4 / 33
  12. 12. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 5 / 33
  13. 13. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 5 / 33
  14. 14. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 5 / 33
  15. 15. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 5 / 33
  16. 16. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 / 33
  17. 17. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 7 / 33
  18. 18. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 7 / 33
  19. 19. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Improvement due to Predictive Deconvolution Prostate Images No Preprocessing RLS Deconvolution SE 0.69 ± 0.06 0.75 ± 0.09 SP 0.94 ± 0.02 0.93 ± 0.01 Acc 0.93 ± 0.02 0.93 ± 0.02 Az 0.92 ± 0.02 0.95 ± 0.02 8 / 33
  20. 20. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Improvement due to Predictive Deconvolution Benignant case No Preprocessing RLS Deconvolution SE 0.69 ± 0.06 0.75 ± 0.09 SP 0.94 ± 0.02 0.93 ± 0.01 Acc 0.93 ± 0.02 0.93 ± 0.02 Az 0.92 ± 0.02 0.95 ± 0.02 8 / 33
  21. 21. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Improvement due to Predictive Deconvolution Malignant case No Preprocessing RLS Deconvolution SE 0.69 ± 0.06 0.75 ± 0.09 SP 0.94 ± 0.02 0.93 ± 0.01 Acc 0.93 ± 0.02 0.93 ± 0.02 Az 0.92 ± 0.02 0.95 ± 0.02 8 / 33
  22. 22. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Continuous/Discrete Approach for PD • Continuous Auto-Regressive (CAR) system • g(t) continuous RF signal in a stationary case • w(t) continuous uncorrelated unpredictable (scattering) • Link with discrete: autocorrelation represented in a spline basis • Exponential splines: multiresolution version of Lα • Identification of continuous model from sampled data • Shift-variant multiresolution description of scattering signal 9 / 33
  23. 23. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Continuous/Discrete Approach for PD • Continuous Auto-Regressive (CAR) system • g(t) continuous RF signal in a stationary case • w(t) continuous uncorrelated unpredictable (scattering) • Link with discrete: autocorrelation represented in a spline basis • Exponential splines: multiresolution version of Lα • Identification of continuous model from sampled data • Shift-variant multiresolution description of scattering signal 9 / 33
  24. 24. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Continuous/Discrete Approach for PD • Continuous Auto-Regressive (CAR) system • g(t) continuous RF signal in a stationary case • w(t) continuous uncorrelated unpredictable (scattering) • Link with discrete: autocorrelation represented in a spline basis • Exponential splines: multiresolution version of Lα • Identification of continuous model from sampled data • Shift-variant multiresolution description of scattering signal 9 / 33
  25. 25. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Exponential Splines −500 500 −1 0 1 Wavelet scale 4−500 500 −1 0 1 Wavelet scale 3−500 500 −1 0 1 Wavelet scale 2−500 500 −1 0 1 Wavelet scale 1 −500 500 −1 0 1 LP • CAR autocorrelation represented in exponential B-spline basis a : ϕα(t) = k∈Z c[k]β−α,α,T(t − k) • E-B-splines: βα,T(t) = F−1 n i=1 1−eαi−jω jω−αi • Non conventional Wavelet Analysis • Main property: wavelet coefficients of a signal f(t) are samples of smoothed version of Lα(f) a [Khalidov and Unser, 2006], [Unser and Blu, 2005a], [Unser and Blu, 2005b] 10/ 33
  26. 26. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme E-spline-based Whitening Model • Φ(jω; α/θ) = σ2 n i=1 1 jω−αi 2 = σ2 1 (jω)n+a1(jω)n−1+···+an 2 • Theorem 1: Let ϕα(t) = F−1 {Φ(jω; α)} be the autocorrelation function associated with the CAR process with poles α. Then Φ(z; α, T) = σ2 · zn · B(−α:α),T(z) · n i=1 eαiT (1 − eαiTz)(1 − eαiTz−1) (1) • Proposition 1: The z-transform of the discrete ACF is the product of a causal and an anti-causal filter: Φ(z; α, T) = λ2 Hd(z; θ, T)Hd z−1 ; θ, T Hd(z; θ, T) = n−1 i=1 (1 − ζiz−1 )/ n i=1 (1 − eαiT z−1 ) (2) where ζi are the zeros of B(−α:α),T(z) inside the unit circle. • λ2 variance of AWGN in input to the discretized CAR model 11/ 33
  27. 27. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme E-spline-based Whitening Model • Φ(jω; α/θ) = σ2 n i=1 1 jω−αi 2 = σ2 1 (jω)n+a1(jω)n−1+···+an 2 • Theorem 1: Let ϕα(t) = F−1 {Φ(jω; α)} be the autocorrelation function associated with the CAR process with poles α. Then Φ(z; α, T) = σ2 · zn · B(−α:α),T(z) · n i=1 eαiT (1 − eαiTz)(1 − eαiTz−1) (1) • Proposition 1: The z-transform of the discrete ACF is the product of a causal and an anti-causal filter: Φ(z; α, T) = λ2 Hd(z; θ, T)Hd z−1 ; θ, T Hd(z; θ, T) = n−1 i=1 (1 − ζiz−1 )/ n i=1 (1 − eαiT z−1 ) (2) where ζi are the zeros of B(−α:α),T(z) inside the unit circle. • λ2 variance of AWGN in input to the discretized CAR model 11/ 33
  28. 28. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme E-spline-based Whitening Model • Φ(jω; α/θ) = σ2 n i=1 1 jω−αi 2 = σ2 1 (jω)n+a1(jω)n−1+···+an 2 • Theorem 1: Let ϕα(t) = F−1 {Φ(jω; α)} be the autocorrelation function associated with the CAR process with poles α. Then Φ(z; α, T) = σ2 · zn · B(−α:α),T(z) · n i=1 eαiT (1 − eαiTz)(1 − eαiTz−1) (1) • Proposition 1: The z-transform of the discrete ACF is the product of a causal and an anti-causal filter: Φ(z; α, T) = λ2 Hd(z; θ, T)Hd z−1 ; θ, T Hd(z; θ, T) = n−1 i=1 (1 − ζiz−1 )/ n i=1 (1 − eαiT z−1 ) (2) where ζi are the zeros of B(−α:α),T(z) inside the unit circle. • λ2 variance of AWGN in input to the discretized CAR model 11/ 33
  29. 29. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme E-spline-based Whitening Model • Φ(jω; α/θ) = σ2 n i=1 1 jω−αi 2 = σ2 1 (jω)n+a1(jω)n−1+···+an 2 • Theorem 1: Let ϕα(t) = F−1 {Φ(jω; α)} be the autocorrelation function associated with the CAR process with poles α. Then Φ(z; α, T) = σ2 · zn · B(−α:α),T(z) · n i=1 eαiT (1 − eαiTz)(1 − eαiTz−1) (1) • Proposition 1: The z-transform of the discrete ACF is the product of a causal and an anti-causal filter: Φ(z; α, T) = λ2 Hd(z; θ, T)Hd z−1 ; θ, T Hd(z; θ, T) = n−1 i=1 (1 − ζiz−1 )/ n i=1 (1 − eαiT z−1 ) (2) where ζi are the zeros of B(−α:α),T(z) inside the unit circle. • λ2 variance of AWGN in input to the discretized CAR model 11/ 33
  30. 30. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme System identification: first attempt • Annihilating Polynomial: coeff of AP of ϕα[k] are functions of α: n i=1 1 − aiz−1 ϕα[k] = 0 • 1. Backscattering feature: • Recursive Annihilating polynomial for shift-variant α[k]. • Segment-wise E-B-spline wavelet multiresolution analysis • Better than predictive deconvolution but most info in energy (16% improvement for real data) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1−SP SE ROC curves: cancer detection ESW − 1 feat PD 12/ 33
  31. 31. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme System identification: first attempt • Annihilating Polynomial: coeff of AP of ϕα[k] are functions of α: n i=1 1 − aiz−1 ϕα[k] = 0 • 1. Backscattering feature: • Recursive Annihilating polynomial for shift-variant α[k]. • Segment-wise E-B-spline wavelet multiresolution analysis • Better than predictive deconvolution but most info in energy (16% improvement for real data) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1−SP SE ROC curves: cancer detection Nakagami ESW − 1 feat PD 12/ 33
  32. 32. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme System identification: first attempt • Annihilating Polynomial: coeff of AP of ϕα[k] are functions of α: n i=1 1 − aiz−1 ϕα[k] = 0 • 2. Texural feature: • Annihilating polynomial for shift-invariant parameters, tuned on RF signal • shift-invariant E-B-spline wavelet multiresolution analysis • No difference with traditional Haar wavelet 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1−Specificity Sensitivity ROC curves: cancer detection Nakagami Naka + Variance ESW 12/ 33
  33. 33. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme System identification: first attempt • Annihilating Polynomial: coeff of AP of ϕα[k] are functions of α: n i=1 1 − aiz−1 ϕα[k] = 0 • 2. Texural feature: • Annihilating polynomial for shift-invariant parameters, tuned on RF signal • shift-invariant E-B-spline wavelet multiresolution analysis • No difference with traditional Haar wavelet 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1−SP SE ROC curves: cancer detection (whole prostate) Nakagami Naka + Variance ESW Naka + Variance WT 12/ 33
  34. 34. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme System identification: novel algorithm • Discretized CAR process y[k] is multivariate Gaussian • Novel Maximum-likelihood estimator: ˆθ = arg minθ VN(θ) • Cost function: VN(θ) = log |Σ| + 1 2 y · Σ−1 · y • Asintotic expression: VN(θ) ∼= N 2 log λ2 + 1 2 y ∗ g 2 ℓ2 (3) • Frequency domain interpretation: VN(θ) = N k=1 |Y[k]|2 2λ2|Hd(ejωk ; θ, T)|2 + 1 2 log λ2 |Hd(ejωk ; θ, T)|2 (4) • No apriori assumptions on digital data • Link continuous-discrete through E-B-splines • valid for any sampling interval 13/ 33
  35. 35. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme System identification: novel algorithm • Discretized CAR process y[k] is multivariate Gaussian • Novel Maximum-likelihood estimator: ˆθ = arg minθ VN(θ) • Cost function: VN(θ) = log |Σ| + 1 2 y · Σ−1 · y • Asintotic expression: VN(θ) ∼= N 2 log λ2 + 1 2 y ∗ g 2 ℓ2 (3) • Frequency domain interpretation: VN(θ) = N k=1 |Y[k]|2 2λ2|Hd(ejωk ; θ, T)|2 + 1 2 log λ2 |Hd(ejωk ; θ, T)|2 (4) • No apriori assumptions on digital data • Link continuous-discrete through E-B-splines • valid for any sampling interval 13/ 33
  36. 36. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme System identification: novel algorithm • Discretized CAR process y[k] is multivariate Gaussian • Novel Maximum-likelihood estimator: ˆθ = arg minθ VN(θ) • Cost function: VN(θ) = log |Σ| + 1 2 y · Σ−1 · y • Asintotic expression: VN(θ) ∼= N 2 log λ2 + 1 2 y ∗ g 2 ℓ2 (3) • Frequency domain interpretation: VN(θ) = N k=1 |Y[k]|2 2λ2|Hd(ejωk ; θ, T)|2 + 1 2 log λ2 |Hd(ejωk ; θ, T)|2 (4) • No apriori assumptions on digital data • Link continuous-discrete through E-B-splines • valid for any sampling interval 13/ 33
  37. 37. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme System identification: novel algorithm • Discretized CAR process y[k] is multivariate Gaussian • Novel Maximum-likelihood estimator: ˆθ = arg minθ VN(θ) • Cost function: VN(θ) = log |Σ| + 1 2 y · Σ−1 · y • Asintotic expression: VN(θ) ∼= N 2 log λ2 + 1 2 y ∗ g 2 ℓ2 (3) • Frequency domain interpretation: VN(θ) = N k=1 |Y[k]|2 2λ2|Hd(ejωk ; θ, T)|2 + 1 2 log λ2 |Hd(ejωk ; θ, T)|2 (4) • No apriori assumptions on digital data • Link continuous-discrete through E-B-splines • valid for any sampling interval 13/ 33
  38. 38. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme System identification: novel algorithm • Discretized CAR process y[k] is multivariate Gaussian • Novel Maximum-likelihood estimator: ˆθ = arg minθ VN(θ) • Cost function: VN(θ) = log |Σ| + 1 2 y · Σ−1 · y • Asintotic expression: VN(θ) ∼= N 2 log λ2 + 1 2 y ∗ g 2 ℓ2 (3) • Frequency domain interpretation: VN(θ) = N k=1 |Y[k]|2 2λ2|Hd(ejωk ; θ, T)|2 + 1 2 log λ2 |Hd(ejωk ; θ, T)|2 (4) • No apriori assumptions on digital data • Link continuous-discrete through E-B-splines • valid for any sampling interval 13/ 33
  39. 39. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme System identification: novel algorithm • Discretized CAR process y[k] is multivariate Gaussian • Novel Maximum-likelihood estimator: ˆθ = arg minθ VN(θ) • Cost function: VN(θ) = log |Σ| + 1 2 y · Σ−1 · y • Asintotic expression: VN(θ) ∼= N 2 log λ2 + 1 2 y ∗ g 2 ℓ2 (3) • Frequency domain interpretation: VN(θ) = N k=1 |Y[k]|2 2λ2|Hd(ejωk ; θ, T)|2 + 1 2 log λ2 |Hd(ejωk ; θ, T)|2 (4) • No apriori assumptions on digital data • Link continuous-discrete through E-B-splines • valid for any sampling interval 13/ 33
  40. 40. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Exponential B-spline based ML estimator • Exact discretization → exact link discrete/continuous data • Proposition 2: The Lagrange representation in an exponential spline basis of the autocorrelation of a CAR system is ϕα(t) = k∈Z ϕα,T[k]ηα,T(t − kT) (5) where the fundamental spline interpolator ηα,T(t) is defined in the Fourier domain as ˆηα,T(ω) = ˆβ(−α:α),T(ω) B(−α:α),T(ejω) (6) • Exact spectral weight to restore the continuous autocorrelation from the discrete samples: Φ(jω; α) = ˆηα,T(ω)Φ(ejω ; α, T). 14/ 33
  41. 41. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Exponential B-spline based ML estimator • Exact discretization → exact link discrete/continuous data • Proposition 2: The Lagrange representation in an exponential spline basis of the autocorrelation of a CAR system is ϕα(t) = k∈Z ϕα,T[k]ηα,T(t − kT) (5) where the fundamental spline interpolator ηα,T(t) is defined in the Fourier domain as ˆηα,T(ω) = ˆβ(−α:α),T(ω) B(−α:α),T(ejω) (6) • Exact spectral weight to restore the continuous autocorrelation from the discrete samples: Φ(jω; α) = ˆηα,T(ω)Φ(ejω ; α, T). 14/ 33
  42. 42. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Exponential B-spline based ML estimator • Exact discretization → exact link discrete/continuous data • Proposition 2: The Lagrange representation in an exponential spline basis of the autocorrelation of a CAR system is ϕα(t) = k∈Z ϕα,T[k]ηα,T(t − kT) (5) where the fundamental spline interpolator ηα,T(t) is defined in the Fourier domain as ˆηα,T(ω) = ˆβ(−α:α),T(ω) B(−α:α),T(ejω) (6) • Exact spectral weight to restore the continuous autocorrelation from the discrete samples: Φ(jω; α) = ˆηα,T(ω)Φ(ejω ; α, T). 14/ 33
  43. 43. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Characteristic of the proposed MLE • Example CAR(2): 1 s2+a1s+a2 . (α = −1 ± 5j) • Several minima: one in any Bk = [kπ T , (k + 1)π T ] • Correct model in the global minimum • Random starting points in Bk provide as solution the minimum in that band 15/ 33
  44. 44. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Experimental Results: MSE −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 −30 −25 −20 −15 −10 −5 0 5 Normalized Sampling Interval (logaritmic scale) relativeMSE(dB) ARMA MLE exp MLE poly CRB −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 −40 −35 −30 −25 −20 −15 −10 −5 0 Normalized Sampling Interval (logaritmic scale) relativeMSE(dB) ARMA MLE exp MLE poly CRB • Example CAR(2): 1 s2+a1s+a2 . (α − 1 ± 10j) • Comparison with MLE based on polynomial splines2 and standard MATLAB ARMA estimator. • The proposed MLEexp follows the Cramér-Rao Bound • Correct solution even in strong aliasing conditions • E-B-spline based MLE outperforms standard methods. 2 [Gillberg and Ljung, 2009] 16/ 33
  45. 45. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Why does E-B-spline MLE work with aliasing? 0 2 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω (rad) DFT Ideal MLE exp MLE poly η poly (ω) ηexp (ω) • Exponential spectral weight: ˆηexp(ω) = ˆβ(−α:α),T(ω) B(−α:α),T (ejω) • Polynomial spectral weight: ˆηpoly(ω) = ejωT −1 jωT 2n ejωT 2n−1(ejωT) (2n−1)! 17/ 33
  46. 46. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Why does E-B-spline MLE work with aliasing? 17/ 33
  47. 47. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Conclusions • Predictive Deconvolution to restore US tissue response • Continuous approach to predictive deconvolution • CAR systems • ACF representation in Exponential spline basis • Novel MLE algorithm for CAR identification • Accurate identification even in the presence of aliasing Further Developments • MLE for CARMA systems • Non ideal sampling • Any applications? 18/ 33
  48. 48. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Conclusions • Predictive Deconvolution to restore US tissue response • Continuous approach to predictive deconvolution • CAR systems • ACF representation in Exponential spline basis • Novel MLE algorithm for CAR identification • Accurate identification even in the presence of aliasing Further Developments • MLE for CARMA systems • Non ideal sampling • Any applications? 18/ 33
  49. 49. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Conclusions • Predictive Deconvolution to restore US tissue response • Continuous approach to predictive deconvolution • CAR systems • ACF representation in Exponential spline basis • Novel MLE algorithm for CAR identification • Accurate identification even in the presence of aliasing Further Developments • MLE for CARMA systems • Non ideal sampling • Any applications? 18/ 33
  50. 50. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Conclusions • Predictive Deconvolution to restore US tissue response • Continuous approach to predictive deconvolution • CAR systems • ACF representation in Exponential spline basis • Novel MLE algorithm for CAR identification • Accurate identification even in the presence of aliasing Further Developments • MLE for CARMA systems • Non ideal sampling • Any applications? 18/ 33
  51. 51. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Thank you for your attention! http://mas.deis.unibo.it/ 19/ 33
  52. 52. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Bibliography Gillberg, J. and Ljung, L. (2009). Frequency-domain identification of continuous-time ARMA models from sampled data. Automatica, 45:1371–1378. Jensen, J. (1994). Estimation of in vivo pulses in medical ultrasound. Ultrasonic Imaging, 16:190–203. 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):86–104. 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):550–568. Rasmussen, K. (1994). Maximum likelihood estimation of the attenuated uktrasound pulse. IEEE Transactions on Signal Processing, 42:220–222. 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. 20/ 33
  53. 53. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 21/ 33
  54. 54. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1−SP SE ROC curves: cancer detection (whole prostate) Nakagami Naka + Variance ESW Naka + Variance WT 22/ 33
  55. 55. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 83 71 - - Huynen - Textural 80 88.20 - - Houston 25 Textural 73 86 80 - Schmitz 3405 Multi 82 88 - - Scheipers 170 484 Multi - - 75 86 Feleppa 1019 Spectral - - 80 85 Mohamed 96 Textural 83.3 100 93.75 - Llobet 4944 Textural 68 53 61.6 60.1 Mohamed 108 Multi 83.3 100 94.4 - Han 2000 Multi 92 95.9 - - Maggio 58602 Multi on RLS 75 93 93 95 23/ 33
  56. 56. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 24/ 33
  57. 57. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 24/ 33
  58. 58. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 24/ 33
  59. 59. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 24/ 33
  60. 60. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1−Specificity Sensitivity ROC curves: hypoechoic target detection ESW − 1 feat PD 25/ 33
  61. 61. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1−Specificity Sensitivity ROC curves: hyperechoic target detection ESW − 1 feat PD 25/ 33
  62. 62. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1−SP SE ROC curves: cancer detection Nakagami ESW − 1 feat PD 26/ 33
  63. 63. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Autocorrelation Function Discretization 1/4 In the first order case the ACF of g(t) can be obtained as: Rgg(τ) = F−1 {Φg} = F−1 1 jω − α ∗ F−1 1 −jω − α = eαt u(t) ∗ e−αt u(−t) = ρα(t) ∗ ρα(−t) (7) Green function of Lα: ρα(t) = eαtu(t) = +∞ k=0 pα[k]βα(t − k) Rgg(t) = +∞ k=0 pα[k]βα(t − k) ∗ +∞ k′=0 pα[k′ ]βα(−t − k′ ) = +∞ k=0 pα[k] +∞ k′=0 pα[k′ ]βα(t − k) ∗ βα(−t − k′ ) = +∞ k=0 pα[k] +∞ k′=0 pα[k′ ]eα βα(t − k) ∗ β−α(t + k′ + 1) (8) 27/ 33
  64. 64. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Autocorrelation Function Discretization 2/4 The expression for ACF simplifies to Rgg(t) = +∞ k=0 pα[k] k′′∈Z pα[k − k′′ − 1]eα β(−α:α)(t − k′′ ) = k′′∈Z (pα ∗ p′ α)[k′′ + 1]eα β(−α:α)(t − k′′ ) (9) where p′ α[k] = pα[−k]. It turns out that the z-transform of the discrete signal ACF is given as: R(z) = eα zPα(z)P′ α(z)B(−α:α)(z) = eαzB(−α:α)(z) (1 − eαz−1)(1 − eαz) (10) 28/ 33
  65. 65. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Autocorrelation Function Discretization 3/4 The extension to higher-order AR models follows naturally as shown below: Rgg(t) = F−1 1 jω − α1 2 ∗ · · · ∗ F−1 1 jω − αp 2 = (ρα1 (t) ∗ ρα1 (−t)) ∗ · · · ∗ ραp (t) ∗ ραp (−t) =   k1∈Z (pα1 ∗ p′ α1 )[k1 + 1]eα1 β(−α1:α1)(t − k1)   ∗ ∗ · · · ∗   kp∈Z (pαp ∗ p′ αp )[kp + 1]eαp β(−αp:αp)(t − kp)   (11) 29/ 33
  66. 66. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Autocorrelation Function Discretization 4/4 The final expression of the z-transform of Rgg(k) is: R(z) = p i=1 eαi zPαi (z)P′ αi (z) · B(−α:α)(z) = p i=1 eαi z (1 − eαi z)(1 − eαi z−1) · zp · B(−α:α)(z) (12) 30/ 33
  67. 67. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme 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 [βα ∗ β−α(t)] = p i=1 e−αi F [βα ∗ βα(t − p)] = p i=1 e−αi e−jωp ˆβα(ω) 2 (13) 31/ 33
  68. 68. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Whitening Filter Stability 2/3 B(−α,α)(z) is linked to the Fourier transform of samples of β(−α,α)(t), ˆβs,(−α,α)(ω): ˆβs,(−α,α)(ω) = k ˆβ(−α,α)(ω + 2πk) = p i=1 e−αi e−jωp k ˆβα(ω + 2πk) 2 = p i=1 e−αi e−jωp Aα(ejω ) (14) 32/ 33
  69. 69. Tissue Characterization Ultrasound Signals Modeling Exponential Splines From Continuous to Discrete Results Acknowledgme Whitening Filter Stability 3/3 where Aα(ejω) 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 < r2 α < Aα(ejω ) < R2 α < +∞ (15) As a consequence ˆβs,(−α,α)(ω) > 0 for every ω and, since B(−α,α)(z) = ˆβs,(−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. 33/ 33

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