Thresholding estimators for linear inverse problems and deconvolutions

• 成果类型: Article
• 署名作者: Kalifa, K; Mallat, S
• 署名单位: Institut Polytechnique de Paris; ENSTA Paris; Ecole Polytechnique; New York University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 2003
• 页码: 58-109
• 关键词: wavelet shrinkage; transform

摘要

Thresholding algorithms in an orthonormal basis are studied to estimate noisy discrete signals degraded by a linear operator whose inverse is not bounded. For signals in a set Theta, sufficient conditions are established on the basis to obtain a maximum risk with minimax rates of convergence. Deconvolutions with kernels having a Fourier transform which vanishes at high frequencies are examples of unstable inverse problems, where a thresholding in a wavelet basis is a suboptimal estimator. A new mirror wavelet basis is constructed to obtain a deconvolution risk which is proved to be asymptotically equivalent to the minimax risk over bounded variation signals. This thresholding estimator is used to restore blurred satellite images.