Wavelet deconvolution in a periodic setting

• 成果类型: Article
• 署名作者: Johnstone, IM; Kerkyacharian, G; Picard, D; Raimondo, M
• 署名单位: Stanford University; Centre National de la Recherche Scientifique (CNRS); Universite Paris Nanterre; Universite Paris Cite; University of Sydney
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1111/j.1467-9868.2004.02056.x
• 发表日期: 2004
• 页码: 547-573

摘要

Deconvolution problems are naturally represented in the Fourier domain, whereas thresholding in wavelet bases is known to have broad adaptivity properties. We study a method which combines both fast Fourier and fast wavelet transforms and can recover a blurred function observed in white noise with O{n log(n)2} steps. In the periodic setting, the method applies to most deconvolution problems, including certain 'boxcar' kernels, which are important as a model of motion blur, but having poor Fourier characteristics. Asymptotic theory informs the choice of tuning parameters and yields adaptivity properties for the method over a wide class of measures of error and classes of function. The method is tested on simulated light detection and ranging data suggested by underwater remote sensing. Both visual and numerical results show an improvement over competing approaches. Finally, the theory behind our estimation paradigm gives a complete characterization of the 'maxiset' of the method: the set of functions where the method attains a near optimal rate of convergence for a variety of L-p loss functions.