The Vaguelette-wavelet Decomposition Approach to Statistical Inverse Problems

A wide variety of scientiic settings have to do with indirect noisy measurements. We are interesting in some object 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. Recovering f(t) from indirect observations, one faces a Statistical Linear Inverse Problem (SLIP). The usual linear methods for SLIPs, like windowed SVD, do not perform satisfactorily when the original function f(t) is \spatially inhomogeneous". As an alternative, Donoho (1995) suggested the use of wavelet techniques for SLIPs and proposed a method, named wavelet-vaguelette decomposition (WVD), based on the expansion of the unknown f(t) in wavelet series. In this paper we consider an alternative vaguelette-wavelet decomposition (VWD) which is also based on wavelet expansion. However, in contrast to wavelet-vaguelette decomposition, we expand the observed signal in wavelet series. We discuss theoretical properties of vaguelette-wavelet decomposition and compare its performance with wavelet-vaguelette decomposition and windowed SVD in the context of numerical differentiation of several test functions.