WAVELET DECOMPOSITION APPROACHES TO STATISTICAL INVERSE PROBLEMS

A wide variety of scientific settings involve indirect noisy measureme nts where one faces a linear inverse problem in the presence of noise. Primary interest is in some function f(t) but data are accessible onl y about some linear transform corrupted by noise; The usual linear met hods for such inverse problems do not perform satisfactorily when f(t) is spatially inhomogeneous. One existing nonlinear alternative is the wavelet-vaguelette decomposition method, based on the expansion of th e unknown f(t) in wavelet series. In the vaguelette-wavelet decomposit ion method proposed here, the observed data are expanded directly in w avelet series. The performances of various methods are compared throug h exact risk calculations, in the context of the estimation of the der ivative of a function observed subject to noise. A result is proved de monstrating that, with a suitable universal threshold somewhat larger than that used for standard denoising problems, both the wavelet-based approaches have an ideal spatial adaptivity property.