Wavelet Decomposition Approaches to Statistical

A wide variety of scientiic settings involve indirect noisy measurements where one faces a linear inverse problem in the presence of noise. Primary interest is in some function f(t) but the data is accessible only about some transform (Kf)(t), where K is some linear operator, and (Kf)(t) is in addition corrupted by noise. The usual linear methods for such inverse problems, for example those based on singular value decompositions, do not perform satisfactorily when the original function f(t) is spatially inhomogeneous. One alternative that has been suggested is the wavelet{ vaguelette decomposition method, based on the expansion of the unknown f(t) in wavelet series. The vaguelette{wavelet decomposition method proposed in this paper is also based on wavelet expansion. In contrast to wavelet{vaguelette decomposition, the observed data are expanded directly in wavelet series. Using exact risk calculations, the performances of the two wavelet-based methods are compared with one another and with singular value decomposition methods, in the context of the estimation of the derivative of a function observed subject to noise. A result is proved demonstrating that, with a suitable universal threshold somewhat larger than that used for standard denoising problems, both wavelet-based approaches have an ideal spatial adaptivity property.