Open Access
August 2007 Nonparametric estimation of a point-spread function in multivariate problems
Peter Hall, Peihua Qiu
Ann. Statist. 35(4): 1512-1534 (August 2007). DOI: 10.1214/009053606000001442


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.


Download Citation

Peter Hall. Peihua Qiu. "Nonparametric estimation of a point-spread function in multivariate problems." Ann. Statist. 35 (4) 1512 - 1534, August 2007.


Published: August 2007
First available in Project Euclid: 29 August 2007

zbMATH: 1209.62057
MathSciNet: MR2351095
Digital Object Identifier: 10.1214/009053606000001442

Primary: 62G07
Secondary: 62P30

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

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 4 • August 2007
Back to Top