A Trimmed Translation-Invariant Denoising Estimator

A popular wavelet method for estimating jumps in functions is through the use of the translation invariant (TI) estimator. The TI estimator addresses a particular problem, the susceptibility of wavelet estimates to the location of features in a function with respect to the support of the wavelet basis functions. The TI estimator reduces this reliance by cycling the data through a set of shifts, thus changing the relation between the wavelet support and the jump location. However, a drawback to TI is that it includes every shifted analysis in the reconstruction, even those that may reduce, rather than improve, the effectiveness of the method. In this paper, we propose a method that modifies the TI to improve jump reconstruction in terms of mean square error of the reconstructions and visual performance. Information from the set of shifted data sets is used to mimic the performance of an oracle which knows exactly which are the best TI shifts to retain in the reconstruction. The TI estimate is a special case of the proposed method. A simulation study comparing this proposed method to existing wavelet estimators and the oracle is provided.