A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Ten lectures on wavelets
The lifting scheme: a construction of second generation wavelets
SIAM Journal on Mathematical Analysis
Wavelet shrinkage for unequally spaced data
Statistics and Computing
Choice of wavelet smoothness, primary resolution and threshold in wavelet shrinkage
Statistics and Computing
Stabilised wavelet transforms for non-equispaced data smoothing
Signal Processing - Signal processing with heavy-tailed models
A Stabilized Lifting Construction of Wavelets on Irregular Meshes on the Interval
SIAM Journal on Scientific Computing
Adaptive Wavelet Transforms for Image Coding Using Lifting
DCC '98 Proceedings of the Conference on Data Compression
Adaptive lifting schemes with perfect reconstruction
IEEE Transactions on Signal Processing
IEEE Transactions on Image Processing
Nonlinear wavelet transforms for image coding via lifting
IEEE Transactions on Image Processing
A `nondecimated' lifting transform
Statistics and Computing
ICCVG 2008 Proceedings of the International Conference on Computer Vision and Graphics: Revised Papers
Spectral estimation for locally stationary time series with missing observations
Statistics and Computing
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Many wavelet shrinkage methods assume that the data are observed on an equally spaced grid of length of the form 2J for some J. These methods require serious modification or preprocessed data to cope with irregularly spaced data. The lifting scheme is a recent mathematical innovation that obtains a multiscale analysis for irregularly spaced data.A key lifting component is the "predict" step where a prediction of a data point is made. The residual from the prediction is stored and can be thought of as a wavelet coefficient. This article exploits the flexibility of lifting by adaptively choosing the kind of prediction according to a criterion. In this way the smoothness of the underlying `wavelet' can be adapted to the local properties of the function. Multiple observations at a point can readily be handled by lifting through a suitable choice of prediction. We adapt existing shrinkage rules to work with our adaptive lifting methods.We use simulation to demonstrate the improved sparsity of our techniques and improved regression performance when compared to both wavelet and non-wavelet methods suitable for irregular data. We also exhibit the benefits of our adaptive lifting on the real inductance plethysmography and motorcycle data.