The Annals of Statistics

Estimating deformations of stationary processes

Maureen Clerc and Stéphane Mallat

Full-text: Open access


This paper studies classes of nonstationary processes, such as warped processes and frequency-modulated processes, that result from the deformation of stationary processes. Estimating deformations can often provide important information about an underlying physical phenomenon. A computational harmonic analysis viewpoint shows that the deformed autocovariance satisfies a transport equation at small scales, with a velocity proportional to a deformation gradient. We derive an estimator of the deformation from a single realization of the deformed process, with a proof of consistency under appropriate assumptions.

Article information

Ann. Statist., Volume 31, Number 6 (2003), 1772-1821.

First available in Project Euclid: 16 January 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 60G12: General second-order processes
Secondary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]

Nonstationary processes inverse problem wavelets warping frequency modulation scalogram spectrogram


Clerc, Maureen; Mallat, Stéphane. Estimating deformations of stationary processes. Ann. Statist. 31 (2003), no. 6, 1772--1821. doi:10.1214/aos/1074290327.

Export citation


  • Abry, P., Gonçalves, P. and Flandrin, P. (1995). Wavelets, spectrum analysis and $1/f$ processes. Wavelets and Statistics. Lecture Notes in Statist. 103 15--29. Springer, New York.
  • Adler, R. J. (1981). The Geometry of Random Fields. Wiley, New York.
  • Bakirov, N. K. (1989). Extrema of the distributions of quadratic forms of Gaussian variables. Theory Probab. Appl. 34 207--215.
  • Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
  • Clerc, M. and Mallat, S. (1999). Shape from texture through deformations. In Proc. Seventh International Conference on Computer Vision 1 405--410.
  • Clerc, M. and Mallat, S. (2002). The texture gradient equation for recovering shape from texture. IEEE Trans. Pattern Anal. Machine Intelligence 24 536--549.
  • Flandrin, P. (1989). On the spectrum of fractional Brownian motions. IEEE Trans. Inform. Theory 35 197--199.
  • Gao, H.-Y. (1993). Wavelet estimation of spectral densities in times series analysis. Ph.D. dissertation, Dept. Statistics, Univ. California, Berkeley.
  • Gårding, J. (1992). Shape from texture for smooth curved surfaces in perspective projection. J. Math. Imaging Vision 2 327--350.
  • Malik, J. and Rosenholtz, R. (1997). Computing local surface orientation and shape from texture for curved surfaces. Internat. J. Comput. Vision 23 149--168.
  • Mallat, S. (1999). A Wavelet Tour of Signal Processing, 2nd ed. Academic Press, New York.
  • Patil, G. P. and Rao, C. R. (1994). Handbook of Statistics 12. Environmental Statistics. 661--689. North-Holland, Amsterdam.
  • Perrin, O. and Senoussi, R. (1999). Reducing non-stationary stochastic processes to stationarity by a time deformation. Statist. Probab. Lett. 43 393--397.
  • Priestley, M. B. (1965). Evolutionary spectra and non-stationary processes (with discussion). J. Roy. Statist. Soc. Ser. B 27 204--237.
  • Sampson, P. D. and Guttorp, P. (1992). Nonparametric estimation of nonstationary spatial covariance structure. J. Amer. Statist. Assoc. 87 108--119.
  • Yaglom, A. M. (1987). Correlation Theory of Stationary and Related Random Functions 1. Springer, Berlin.