The Annals of Statistics

Consistent estimates of deformed isotropic Gaussian random fields on the plane

Ethan Anderes and Sourav Chatterjee

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This paper proves fixed domain asymptotic results for estimating a smooth invertible transformation f: ℝ2→ℝ2 when observing the deformed random field Zf on a dense grid in a bounded, simply connected domain Ω, where Z is assumed to be an isotropic Gaussian random field on ℝ2. The estimate is constructed on a simply connected domain U, such that ⊂Ω and is defined using kernel smoothed quadratic variations, Bergman projections and results from quasiconformal theory. We show, under mild assumptions on the random field Z and the deformation f, that Rθf+c uniformly on compact subsets of U with probability one as the grid spacing goes to zero, where Rθ is an unidentifiable rotation and c is an unidentifiable translation.

Article information

Ann. Statist., Volume 37, Number 5A (2009), 2324-2350.

First available in Project Euclid: 15 July 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G60: Random fields 62M30: Spatial processes 62M40: Random fields; image analysis
Secondary: 62G05: Estimation

Deformation quasiconformal maps nonstationary random fields Bergman space


Anderes, Ethan; Chatterjee, Sourav. Consistent estimates of deformed isotropic Gaussian random fields on the plane. Ann. Statist. 37 (2009), no. 5A, 2324--2350. doi:10.1214/08-AOS647.

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  • [1] Adler, R. J. and Pyke, R. (1993). Uniform quadratic variation for Gaussian processes. Stochastic Process. Appl. 48 191–209.
  • [2] Ahlfors, L. V. (2006). Lectures on Quasiconformal Mappings. University Lecture Series 38 Amer. Math. Soc., Providence, RI.
  • [3] Anderes, E. B. (2005). Estimating deformations of isotropic Gaussian random fields. Ph.D. thesis, Univ. Chicago.
  • [4] Anderes, E. B. and Chatterjee, S. (2008). Consistent estimates of deformed isotropic Gaussian random fields on the plane. Technical Report 739, Statistics Dept., Univ. California at Berkeley. Available at
  • [5] Anderes, E. B. and Stein, M. L. (2008). Estimating deformations of isotropic Gaussian random fields on the plane. Ann. Statist. 36 719–741.
  • [6] Baxter, G. (1956). A strong limit theorem for Gaussian processes. Proc. Amer. Math. Soc. 7 522–527.
  • [7] Benassi, A., Cohen, S., Istas, J. and Jaffard, S. (1998). Identification of filtered white noises. Stochastic Process. Appl. 75 31–49.
  • [8] Berman, S. M. (1967). A version of the Lévy–Baxter theorem for the increments of Brownian motion of several parameters. Proc. Amer. Math. Soc. 18 1051–1055.
  • [9] Clerc, M. and Mallat, S. (2002). The texture gradient equation for recovering shape from texture. IEEE Trans. on Pattern Analysis and Machine Intelligence 24 536–549.
  • [10] Clerc, M. and Mallat, S. (2003). Estimating deformations of stationary processes. Ann. Statist. 31 1772–1821.
  • [11] Cohen, S., Guyon, X., Perrin, O. and Pontier, M. (2006). Identification of an isometric transformation of the standard Brownian sheet. J. Statist. Plann. Inference 136 1317–1330.
  • [12] Cohen, S., Guyon, X., Perrin, O. and Pontier, M. (2006). Singularity functions for fractional processes: Application to the fractional Brownian sheet. Ann. Inst. H. Poincaré. Probab. Statist. 42 187–205.
  • [13] Damian, D., Sampson, P. and Guttorp, P. (2001). Bayesian estimation of semi-parametric non-stationary spatial covariance structures. Environmetrics 12 161–178.
  • [14] Dieudonné, J. (1960). Foundations of Modern Analysis. Academic Press, New York.
  • [15] Dudley, R. M. (1973). Sample functions of the Gaussian process. Ann. Probab. 1 66–103.
  • [16] Duren, P. and Schuster, A. (2004). Bergman Spaces. Mathematical Surveys and Monographs 100. Amer. Math. Soc., Providence, RI.
  • [17] Gårding, J. (1992). Shape from texture for smooth curved surfaces in perspective projection. J. Math. Imaging Vision 2 327–350.
  • [18] Gladyshev, E. G. (1961). A new limit theorem for stochastic processes with Gaussian increments. Theory Probab. Appl. 6 52–61.
  • [19] Guyon, X. and Leon, G. (1989). Convergence en loi des h-variations d’un processus gaussien stationnaire. Ann. Inst. H. Poincaré 25 265–282.
  • [20] Guyon, X. and Perrin, O. (2000). Identification of space deformation using linear and superficial quadratic variations. Statist. Probab. Lett. 47 307–316.
  • [21] Hanson, D. L. and Wright, F. T. (1971). A bound on tail probabilities for quadratic form in independent random variables. Ann. Math. Statist. 42 1079–1083.
  • [22] Hu, W. (2001). Dark synergy: Gravitational lensing and the cmb. Phys. Rev. D 65.
  • [23] Iovleff, S. and Perrin, O. (2004). Estimating a nonstationary spatial structure using simulated annealing. J. Comput. Graph. Statist. 13 90–105.
  • [24] Istas, J. and Lang, G. (1997). Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist. 33 407–436.
  • [25] Klein, R. and Gine, E. (1975). On quadratic variation of processes with Gaussian increments. Ann. Probab. 3 716–721.
  • [26] Krushkal’, S. L. (1979). Quasiconformal Mappings and Riemann Surfaces. V. H. Winston & Sons, Washington, DC.
  • [27] Ławrynowicz, J. (1983). Quasiconformal Mappings in the Plane: Parametrical Methods. Lecture Notes in Mathematics 978. Springer, Berlin.
  • [28] Lehto, O. and Virtanen, K. I. (1965). Quasiconformal Mappings in the Plane. Springer, New York.
  • [29] Leon, J. and Ortega, J. (1989). Weak convergence of different types of variation for biparametric Gaussian processes. In Limit Theorems in Probability and Statistics. Colloqnia Mathematica Societatis János Bolyali 57 349–364.
  • [30] Lévy, P. (1940). Le mouvement brownien plan. Amer. J. Math. 62 487–550.
  • [31] Loh, W. (2005). Fixed-domain asymptotics for a subclass of matern-type Gaussian random fields. Ann. Statist. 33 2344–2394.
  • [32] Malik, J. and Rosenholtz, R. (1997). Computing local surface orientation and shape from texture for curved surfaces. Int. J. Comp. Vision 23 149–168.
  • [33] Perrin, O. (1998). Functional convergence in distribution of quadratic variations for a large class of Gaussian processes: Application to a time deformation model. Technical report, Unit of Biometrics at Avignon.
  • [34] Perrin, O. and Meiring, W. (1999). Identifiability for non-stationary spatial structure. J. Appl. Probab. 36 1244–1250.
  • [35] Perrin, O. and Senoussi, R. (2000). Reducing non-stationary random fields to stationarity and isotropy using a space deformation. Statist. Probab. Lett. 48 23–32.
  • [36] Pommerenke, C. (1975). Univalent Functions. Vandenhoeck & Ruprecht, Göttingen, Germany.
  • [37] Sampson, P. and Guttorp, P. (1992). Nonparametric estimation of nonstationary spatial covariance structure. J. Amer. Statist. Assoc. 87 108–119.
  • [38] Schmidt, A. and O’Hagan, A. (2003). Bayesian inference for nonstationary spatial covariance structure via spatial deformations. J. Roy. Statist. Soc. Ser. B 65 745–758.
  • [39] Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.
  • [40] Stompor, R. and Efstathiou, G. (1999). Gravitational lensing of the cosmic microwave background anisotropies and cosmological parameter estimation. Monthly Notices of the Royal Astronomical Society 302 735–747.
  • [41] Strait, P. T. (1969). On Berman’s version of the Lévy–Baxter theorem. Proc. Amer. Math. Soc. 23 91–93.
  • [42] Zhang, H. (2004). Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics. J. Amer. Statist. Assoc. 99 250–261.