The Annals of Statistics

Nonparametric estimation of a point-spread function in multivariate problems

Peter Hall and Peihua Qiu

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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

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

First available in Project Euclid: 29 August 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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