The Annals of Statistics

Nonparametric estimation of a point-spread function in multivariate problems

Peter Hall and Peihua Qiu

Full-text: Open access

Abstract

The removal of blur from a signal, in the presence of noise, is readily accomplished if the blur can be described in precise mathematical terms. However, there is growing interest in problems where the extent of blur is known only approximately, for example in terms of a blur function which depends on unknown parameters that must be computed from data. More challenging still is the case where no parametric assumptions are made about the blur function. There has been a limited amount of work in this setting, but it invariably relies on iterative methods, sometimes under assumptions that are mathematically convenient but physically unrealistic (e.g., that the operator defined by the blur function has an integrable inverse). In this paper we suggest a direct, noniterative approach to nonparametric, blind restoration of a signal. Our method is based on a new, ridge-based method for deconvolution, and requires only mild restrictions on the blur function. We show that the convergence rate of the method is close to optimal, from some viewpoints, and demonstrate its practical performance by applying it to real images.

Article information

Source
Ann. Statist., Volume 35, Number 4 (2007), 1512-1534.

Dates
First available in Project Euclid: 29 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1188405620

Digital Object Identifier
doi:10.1214/009053606000001442

Mathematical Reviews number (MathSciNet)
MR2351095

Zentralblatt MATH identifier
1209.62057

Subjects
Primary: 62G07: Density estimation
Secondary: 62P30: Applications in engineering and industry

Keywords
blind signal restoration blur convergence rate deconvolution Fourier inversion Fourier transform ill-posed problem image restoration inverse problem minimax optimality noise point degradation ridge test pattern

Citation

Hall, Peter; Qiu, Peihua. Nonparametric estimation of a point-spread function in multivariate problems. Ann. Statist. 35 (2007), no. 4, 1512--1534. doi:10.1214/009053606000001442. https://projecteuclid.org/euclid.aos/1188405620


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References

  • Andrews, H. C. and Hunt, B. R. (1977). Digital Image Restoration. Prentice-Hall, Englewood Cliffs, NJ.
  • Bates, R. H. T. and McDonnell, M. J. (1986). Image Restoration and Reconstruction. Clarendon Press, Oxford.
  • Cannon, M. (1976). Blind deconvolution of spatially invariant image blurs with phase. IEEE Trans. Acoust. Speech Signal Process. 24 58–63.
  • Carasso, A. S. (1999). Linear and nonlinear image deblurring–-a documented study. SIAM J. Numer. Anal. 36 1659–1689.
  • Carasso, A. S. (2001). Direct blind deconvolution. SIAM J. Appl. Math. 61 1980–2007.
  • Donoho, D. L. (1994). Statistical estimation and optimal recovery. Ann. Statist. 22 238–270.
  • Donoho, D. L. and Low, M. G. (1992). Renormalization exponents and optimal pointwise rates of convergence. Ann. Statist. 20 944–970.
  • Ermakov, M. (2003). Asymptotically minimax and Bayes estimation in a deconvolution problem. Inverse Problems 19 1339–1359.
  • Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272.
  • Figueiredo, M. A. T. and Nowak, R. D. (2003). An EM algorithm for wavelet-based image restoration. IEEE Trans. Image Process. 12 906–916.
  • Gonzalez, R. C. and Woods, R. E. (1992). Digital Image Processing. Addison–Wesley, Reading, MA.
  • Hall, P. (1990). Optimal convergence rates in signal recovery. Ann. Probab. 18 887–900.
  • Johnstone, I. M. and Silverman, B. W. (1990). Speed of estimation in positron emission tomography and related inverse problems. Ann. Statist. 18 251–280.
  • Joshi, M. V. and Chaudhuri, S. (2005). Joint blind restoration and surface recovery in photometric stereo. J. Optical Soc. Amer. Ser. A 22 1066–1076.
  • Katsaggelos, A. K. and Lay, K.-T. (1990). Image identification and image restoration based on the expectation–maximization algorithm. Optical Engineering 29 436–445.
  • Kundur, D. and Hatzinakos, D. (1998). A novel blind deconvolution scheme for image restoration using recursive filtering. IEEE Trans. Signal Processing 46 375–390.
  • Marron, J. S. and Tsybakov, A. B. (1995). Visual error criteria for qualitative smoothing. J. Amer. Statist. Assoc. 90 499–507.
  • Qiu, P. (2005). Image Processing and Jump Regression Analysis. Wiley, Hoboken, NJ.
  • Rajagopalan, A. N. and Chaudhuri, S. (1999). MRF model-based identification of shift-variant point spread function for a class of imaging systems. Signal Processing 76 285–299.
  • Skilling, J., ed. (1989). Maximum Entropy and Bayesian Methods. Kluwer, Dordrecht.
  • Van Rooij, A. C. M., Ruymgaart, F. H. and van Zwet, W. R. (1999). Asymptotic efficiency of inverse estimators. Theory Probab. Appl. 44 722–738.
  • Yang, Y., Galatsanos, N. P. and Stark, H. (1994). Projection-based blind deconvolution. J. Optical Soc. Amer. Ser. A 11 2401–2409.