Wavelet Deconvolution With Noisy Eigenvalues

Over the last decade, there has been a lot of interest in wavelet-vaguelette methods for the recovery of noisy signals or images in motion blur. Nonlinear wavelet estimators are known to have good adaptive properties and to outperform linear approximations over a wide range of signals and images, see e.g., the recent WaveD method of Johnstone, Kerkyacharian, Picard, and Raimondo (2004) or the ForWarD method of Neelamani, Choi, and Baraniuk (2004) and also Fan and Koo (2002) in the density setting. In the deblurring setting, wavelet-vaguelette methods rely on the complete knowledge of a convolution operator's eigenvalues. This is an unlikely situation in practice, however. A more realistic scenario, such as would arise when passing the Fourier basis as an input signal through a linear-time-invariant system, is to imagine that one also observes a set of noisy eigenvalues. In this paper, we define a version of the WaveD estimator which is near-optimal when used with noisy eigenvalues. A key feature of our method includes a data-driven method for choosing the fine resolution level in WaveD estimation. Asymptotic theory is illustrated with a wide range of finite sample examples