Statistica Sinica 14(2004), 103-125

$\mbf\Gamma$-MINIMAX WAVELET SHRINKAGE: A ROBUST

INCORPORATION OF INFORMATION ABOUT ENERGY

OF A SIGNAL IN DENOISING APPLICATIONS

Claudia Angelini and Brani Vidakovic

Istituto per le Applicazioni del Calcolo -Sezione di Napoli

and Georgia Institute of Technology

Abstract: In this paper we propose a method for wavelet-filtering of noisy signals when prior information about the $L^2$-energy of the signal of interest is available. Assuming the independence model, according to which the wavelet coefficients are treated individually, we propose a level dependent shrinkage rule that turns out to be the $\Gamma$-minimax rule for a suitable class, say $\Gamma$, of realistic priors on the wavelet coefficients.

The proposed methodology is particularly well suited for denoising tasks where signal-to-noise ratio is low, and it is illustrated on a battery of standard test functions. Performance comparisons with some others methods existing in the literature are provided. An example in atomic force microscopy (AFM) is also discussed.

Key words and phrases: Atomic force microscopy, bounded normal mean, γ-minimaxity, shrinkage, wavelet regression.