Stabilised wavelet transforms for non-equispaced data smoothing

This paper discusses wavelet thresholding in smoothing from non-equispaced, noisy data in one dimension. To deal with the irregularity of the grid we use so called second generation wavelets, based on the lifting scheme. The lifting scheme itself leads to a gridadaptive wavelet transform. We explain that a good numerical condition is an absolute requisite for successful thresholding. If this condition is not satisfied the output signal can show an arbitrary bias. We examine the nature and origin of stability problems in second generation wavelet transforms. The investigation concentrates on lifting with interpolating prediction, but the conclusions are extendible. The stability problem is a cumulated effect of the three successive steps in a lifting scheme: split, predict and update. The paper proposes three ways to stabilise the second generation wavelet transform. The first is a change in update and reduces the influence of the previous steps. The second is a change in prediction and operates on the interval boundaries. The third is a change in splitting procedure and concentrates on the irregularity of the data points. Illustrations show that reconstruction from thresholded coefficients with this stabilised second generation wavelet transform leads to smooth and close fits.