Periodic boxcar deconvolution and diophantine approximation

• 成果类型: Article
• 署名作者: Johnstone, IM; Raimondo, M
• 署名单位: Stanford University; University of Sydney
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053604000000391
• 发表日期: 2004
• 页码: 1781-1804
• 关键词: INVERSE PROBLEMS; regularization

摘要

We consider the nonparametric estimation of a periodic function that is observed in additive Gaussian white noise after convolution with a boxcar, the indicator function of an interval. This is an idealized model for the problem of recovery of noisy signals and images observed with motion blur. If the length of the boxcar is rational, then certain frequencies are irretreviably lost in the periodic model. We consider the rate of convergence of estimators when the length of the boxcar is irrational, using classical results on approximation of irrationals by continued fractions. A basic question of interest is whether the minimax rate of convergence is slower than for nonperiodic problems with 1/f-like convolution filters. The answer turns out to depend on the type and smoothness of functions being estimated in a manner not seen with homogeneous filters.