Skip to main content

Terahertz Time-of-Flight Tomography Beyond the Axial Resolution Limit: Autoregressive Spectral Estimation Based on the Modified Covariance Method


We present a time-of-flight tomography method for exceeding the naïve axial (i.e., depth) resolution limit of terahertz (THz) deconvolution by autoregressive spectral extrapolation (AR) based on the modified covariance method (AR/MCM). In contrast to Wiener filtering combined with wavelet denoising, AR/MCM does not discard any frequency components in the low signal-to-noise (SNR) regions of the measured data, and unlike the AR approach based on the Burg method (AR/BM), no peak splitting (single peaks in the impulse response function appearing as double peaks) as well as frequency bias (spectral peaks shifted with respect to their correct positions) is observed after deconvolution. After verifying the advantages of AR/MCM over Wiener filtering in conjunction with wavelet denoising as well as over AR/BM, using synthetic data, AR/MCM is employed to reconstruct a single layer of mill scale on a steel coupon from experimental THz time-of-flight tomography data. The reconstruction shows good agreement with the film thickness obtained from destructive cross-sectional measurements. In addition, unlike AR/BM, optimizing the parameters to obtain stable reconstruction is straightforward relying of Akaike’s information criterion suggesting that AR/MCM can be an easier to implement for THz nondestructive characterization of stratigraphy under noisy conditions, particularly when estimates of the stratigraphy may not a priori be available.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. 1.

    J. Dong, A. Locquet, and D.S. Citrin, IEEE J. Sel. Top. Quantum Electron. 23, 1-7 (2017).

    Article  Google Scholar 

  2. 2.

    W. Tu, S. Zhong, A. Incecik, and X. Fu, Ocean Eng.155, 382-91 (2018).

    Article  Google Scholar 

  3. 3.

    T. Yasui, T. Yasuda, K. Sawanaka, and T. Araki, Appl. Opt. 44, 6849-56 (2005).

    Article  Google Scholar 

  4. 4.

    J. Dong, J. B Jackson, M. Melis, D. Giovanacci, G.C. Walker, A. Locquet, J.W. Bowen, and D.S. Citrin, Opt. Express 24, 26972-85 (2016).

    Article  Google Scholar 

  5. 5.

    J. B. Jackson, J. Bowen, G. Walker, J. Labaune, G. Mourou, M. Menu, and K. Fukunaga, IEEE. Trans. Terahertz Sci. Technol. 1, 220-31 (2016).

    Article  Google Scholar 

  6. 6.

    C.L. Koch-Dandolo, T. Filtenborg, K. Fukunaga, J. Skou-Hansen, and P.U. Jepsen, Appl. Opt. 54, 5123–9 (2015).

  7. 7.

    A.Y. Pawar, D.D. Sonawane, K.B. Erande, and D.V. Derle, Drug Invent. Today 5,157-63 (2013).

    Article  Google Scholar 

  8. 8.

    M. Naftaly and R.E. Miles, Proc. IEEE. 95, 1658-65 (2007).

    Article  Google Scholar 

  9. 9.

    R. Piesiewicz, T. Kleine-Ostmann, N. Krumbholz, D. Mittleman, M. Koch, and T. Kurner, Electron. Lett. 41, 1002-4 (2005).

    Article  Google Scholar 

  10. 10.

    A. Taschin, P. Bartolini, J. Tasseva, J. Striova, R. Fontana, C. Riminesi, and R. Torre, arXiv preprint arXiv: 1703. 01770 (2017).

  11. 11.

    A.A. Gowen, C.O Sullivan, and C.P. O’Donnell, Trends Food Sci. Technol. 25, 40-6 (2012).

    Article  Google Scholar 

  12. 12.

    F. Rutz, M. Koch, S. Khare, M. Moneke, H. Richter, and U. Ewert, J. Infrared Millim. Terahertz. Waves 25, 547-56 (2006).

    Google Scholar 

  13. 13.

    Y. Chen, Y. Sun, E.P. Macpherson, Fluct. Noise. Lett, 09, 387-94 (2010).

    Article  Google Scholar 

  14. 14.

    R.M. Woodward, B.E. Cole, V.P. Wallace, R.J. Pye, D.D. Arnone, E.H. Linfield, and M. Pepper, Phys. Med. Biol. 47, 3853 (2002).

    Article  Google Scholar 

  15. 15.

    R.K.H. Galvao, S. Hadjiloucas, A. Zafiropoulos, G.C. Walker, J.W. Bowen, and R. Dudley, Opt. Lett. 32, 3008-10 (2007).

    Article  Google Scholar 

  16. 16.

    Y. Chen, S. Huang, and E. Pickwell-MacPherson, Opt. Express 18, 1177-90 (2010).

    Article  Google Scholar 

  17. 17.

    D.M. Mittleman, R.H. Jacobsen, and M.C. Nuss, IEEE J. Sel. Top. Quantum Electron. 2, 679-92 (1996).

    Article  Google Scholar 

  18. 18.

    D.M. Mittleman, R.H. Jacobsen, R. Neelamani, R.G. Baraniuk, M.C. Nuss, Appl. Phys. B. 67, 379-90 (1998).

    Article  Google Scholar 

  19. 19.

    B. Ferguson and D. Abbott, Microelectron. J. 32, 943-53 (2001).

    Article  Google Scholar 

  20. 20.

    B. Ferguson and D. Abbott, Fluct. Noise Lett. 1, L65-L69 (2001).

    Article  Google Scholar 

  21. 21.

    J. Dong, P. Pomarede, L. Chehami, A. Locquet, F. Meraghni, N.F. Declercq, and D.S. Citrin, NDT&E Int. 99, 72-9 (2018).

    Article  Google Scholar 

  22. 22.

    J. Dong, X. Wu, A. Locquet, and D.S. Citrin, IEEE Trans. Terahertz Sci. Technol. 7, 260-7 (2017).

    Article  Google Scholar 

  23. 23.

    EM. Stübling, A. Rehn, T. Siebrecht, Y. Bauckhage, L. Öhrström, P. Eppenberger, JC. Balzer, F. Rühli, M. Koch. Sci. Rep. 9, 1-8 (2019).

    Article  Google Scholar 

  24. 24.

    J. Dong, A. Locquet, M. Melis, and D.S. Citrin, Sci. Rep. 7, 1-13 (2017).

    Article  Google Scholar 

  25. 25.

    H. Karsli, J. Appl. Geophys. 59, 324-36 (2006).

    Google Scholar 

  26. 26.

    T. Miyashita and T. Itaya, High resolution acoustic impulse response in air with spectral extrapolation by linear prediction, 1998. IEEE Untrasonics Symposium Proceedings 1, 873-6 (1998).

  27. 27.

    C.A. Zala, I. Barrodale, and K.I. MaRae, In Signal processing and pattern recognition in nondestructive evaluation of materials, ed. C.H. Chen (Springer, Berlin, Heidelberg, 1988), 44, 101-8.

  28. 28.

    J. Dong, A. Locquet, and D.S. Citrin, Opt. Lett. 42, 1828-31 (2017).

    Article  Google Scholar 

  29. 29.

    S. Kay and L. Marple, Sources and remedies for spectral line splitting in autoregressive spectrum analysis, ICASSP’79 IEEE International Conference on Acoustics, Speech, and Signal Processing, 4, 151-4 (1979).

    Google Scholar 

  30. 30.

    B.M. Bell, IEEE Trans. Signal Process 39, 185-9 (1991).

  31. 31.

    M. Ortigueira, and J. Tribolet, A framework for the evaluation of spectral analysis techniques, ICASSP’84. IEEE International Conference on Acoustics, Speech, and Signal Processing, 9, 247-50 (1984).

    Article  Google Scholar 

  32. 32.

    M. Zhai, A. Locquet, and D.S. Citrin, Surf. Coat. Technol. 393, 125765 (2020).

  33. 33.

    H. Akaike, IEEE Trans. Automat Contr. 19, 716 (1974).

    Article  Google Scholar 

  34. 34.

    S.M. Alessio, Digital signal processing and spectral analysis for scientists: Concepts and application. (Springer, Turin, 2015).

    Google Scholar 

  35. 35.

    P.J. Brockwell, R. Dahlhaus, J. Econom., 118, 129-49 (2004).

    Article  Google Scholar 

  36. 36.

    R. Shibata, Biometrika, 63, 117-26 (1976).

    MathSciNet  Article  Google Scholar 

  37. 37.

    M. Kazubek, IEEE Signal Proc. Lett. 10, 324-6 (2003).

  38. 38.

    J. Pei, P. Ye, and W. Xie, Optimal wavelet analysis for THz-TDS pulse signals, Proc. SPIE 7277, Photonics and Optoelectronics Meetings (POEM) 2008: Terahertz Science and Technology, 727708 (2009).

  39. 39.

    E.C. Ifeachor, and B.W. Jervis, Digital signal processing: a practical approach. (Pearson Education, 2002).

  40. 40.

    S. Beheshti, IEEE Proceeding, ICASSP, 3, 520-3 (2006).

    Google Scholar 

  41. 41.

    P. Sheel, R. Mehra, P. Singh. International Journal of Engineering Trends and Technology, 29, 19-22 (2015).

    Article  Google Scholar 

  42. 42.

    I. Gupta, IEEE Trans. Antennnas. Propag. 42, 1540-1545 (1994).

  43. 43.

    Th. Henning, B. Begemann, H. Mutschke, and J. Dorschner, Astron. Astrophys. Suppl. Ser. 112, 143-9 (1995).

    Google Scholar 

  44. 44.

    .L.M. Van Mechelen, A.B. Kuzmenko, and H. Merbold, Opt. Lett. 39(13), 3853-6 (2014).

  45. 45.

    Q. Cassar, A. Chopard, F. Fauquet, J.P. Guillet, M. Pan, J.B. Perraud, and P. Mounaix, IEEE Trans. Terahertz Sci. Technol., 9(6), 684-94 (2019).

Download references


This work was supported in part by of by Conseil Régional Grand Est, ArcelorMittal Maizières Research, and CPER SusChemProc.

Author information



Corresponding author

Correspondence to D. S. Citrin.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhai, M., Locquet, A., Roquelet, C. et al. Terahertz Time-of-Flight Tomography Beyond the Axial Resolution Limit: Autoregressive Spectral Estimation Based on the Modified Covariance Method. J Infrared Milli Terahz Waves 41, 926–939 (2020).

Download citation


  • Terahertz imaging
  • Autoregressive spectral extrapolation
  • Burg method
  • Modified covariance method
  • Deconvolution
  • Pulsed terahertz tomography