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Estimation of Time-Scaling Factor for Ultrasound Medical Images Using the Hilbert Transform

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A new formulation for the estimation of the time-scaling factor between two ultrasound signals is presented. The estimator is derived under the assumptions of a small time-scaling factor and signals with constant spectrum over its bandwidth. Under these conditions, we show that the proposed approach leads to a simple analytic formulation of the time-scaling factor estimator. The influences of an increase of the time-scaling factor and of signal-to-noise ratio (SNR) are studied. The mathematical developments of the expected mean and bias of the estimator are presented. An iterative version is also proposed to reduce the bias. The variance is calculated and compared to the Cramer-Rao lower bound (CRLB). The estimator characteristics are measured on flat-spectra simulated signals and experimental ultrasound scanner signals and are compared to the theoretical mean and variance. Results show that the estimator is unbiased and that variance tends towards the CRLB for SNR higher than 20 dB. This is in agreement with typical ultrasound signals used in the medical field, as shown on typical examples. Effects of the signal spectrum shape and of the bandwidth size are evaluated. Finally, the iterative version of the estimator improves the quality of the estimation for SNR between 0 and 20 dB as well as the time-scaling factor estimation validity range (up to).


  1. 1.

    Doisy Y, Deruaz L, Beerens SP, Been R: Target Doppler estimation using wideband frequency modulated signals. IEEE Transactions on Signal Processing 2000,48(5):1213–1224. 10.1109/78.839970

  2. 2.

    Foster FS, Burns PN, Simpson DH, Wilson SR, Christopher DA, Goertz DE: Ultrasound for the visualization and quantification of tumor microcirculation. Cancer and Metastasis Reviews 2000,19(1–2):131–138.

  3. 3.

    Brusseau E, Fromageau J, Rognin NG, Delachartre P, Vray D: Investigating elastic properties of soft biological tissues. IEEE Engineering in Medicine and Biology Magazine 2002,21(4):86–94.

  4. 4.

    Pesavento A, Perrey C, Krueger M, Ermert H: A time-efficient and accurate strain estimation concept for ultrasonic elastography using iterative phase zero estimation. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 1999,46(5):1057–1067. 10.1109/58.796111

  5. 5.

    Sharif MdR, Abeysekera SS: Efficient wideband signal parameter estimation using combined narrowband and wideband ambiguity functions. Proceedings of IEEE Pacific RIM Conference on Communications, Computers, and Signal Processing (PACRIM '03), August 2003, Victoria, BC, Canada 1: 426–429.

  6. 6.

    Jin Q, Wong KM, Luo Z-Q: The estimation of time delay and Doppler stretch of wideband signals. IEEE Transactions on Signal Processing 1995,43(4):904–916. 10.1109/78.376843

  7. 7.

    Giunta G: Fast estimators of time delay and doppler stretch based on discrete-time methods. IEEE Transactions on Signal Processing 1998,46(7):1785–1797. 10.1109/78.700948

  8. 8.

    Wong KM, Luo Z-Q, Jin Q: Design of optimum signals for the simultaneous estimation of time delay and Doppler shift. IEEE Transactions on Signal Processing 1993,41(6):2141–2154. 10.1109/78.218142

  9. 9.

    Dooley SR, Nandi AK: Adaptive time delay and Doppler shift estimation for narrowband signals. IEE Proceedings: Radar, Sonar and Navigation 1999,146(5):243–250. 10.1049/ip-rsn:19990601

  10. 10.

    Bracewell RN: The Fourier Transform and Its Application. 2nd edition. McGraw-Hill, New York, NY, USA; 1986.

  11. 11.

    Grennberg A, Sandell M: Estimation of subsample time delay differences in narrowband ultrasonic echoes using the Hilbert transform correlation. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 1994,41(5):588–595. 10.1109/58.308493

  12. 12.

    Arfken G: Mathematical Methods for Physicists. 3rd edition. Academic Press, Orlando, Fla, USA; 1985.

  13. 13.

    Weiss AJ, Weinstein E: Fundamental limitations in passive time delay estimation- part 1: narrow-band systems. IEEE Transactions on Acoustics, Speech, and Signal Processing 1983,31(2):472–486. 10.1109/TASSP.1983.1164061

  14. 14.

    Friedlander B: On the Cramer-Rao bound for time delay and Doppler estimation. IEEE Transactions on Information Theory 1984,30(3):575–580. 10.1109/TIT.1984.1056901

  15. 15.

    Dogandžić A, Nehorai A: Cramer-Rao bounds for estimating range, velocity, and direction with an active array. IEEE Transactions on Signal Processing 2001,49(6):1122–1137. 10.1109/78.923295

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Correspondence to Jérémie Fromageau.

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  • Ultrasound Scanner
  • Signal Spectrum
  • Iterative Version
  • Factor Estimation
  • Simulated Signal