Generalized cross validation for wavelet thresholding
Signal Processing
On the computation of wavelet coefficients
Journal of Approximation Theory
The lifting scheme: a construction of second generation wavelets
SIAM Journal on Mathematical Analysis
Wavelet regression for random or irregular design
Computational Statistics & Data Analysis
Wavelet shrinkage for unequally spaced data
Statistics and Computing
Adaptive Wavelet Transforms for Image Coding Using Lifting
DCC '98 Proceedings of the Conference on Data Compression
De-noising by soft-thresholding
IEEE Transactions on Information Theory
Nonlinear wavelet transforms for image coding via lifting
IEEE Transactions on Image Processing
Adaptive lifting for nonparametric regression
Statistics and Computing
Adaptive splitting for stabilizing 1-D wavelet decompositions on irregular grids
Signal Processing - Signal processing in UWB communications
Journal of Signal Processing Systems
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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 the so-called second generation wavelets, based on the lifting scheme. The lifting scheme itself leads to a grid-adaptive 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.