Smoothing irregularly sampled signals using wavelets and cross validation

Coefficient thresholding is a popular method in wavelet based noise reduction. A wavelet decomposition is typically a sparse representation of noise-free signals: the essential information is captured by a limited number of large, important coefficients, while the main part of coefficients is close to zero. Replacing these small coefficients by zero is a straightforward way to reduce noise variance without affecting the noise-free signal too much. Recently, algorithms have been developed for wavelet decompositions of non-equidistant samples, using the so called lifting scheme and second generation wavelets. We investigate how to apply these algorithms to reduce noise in signals on a non-equidistant grid. The paper also illustrates that the method of generalized cross validation in these settings still succeeds in finding good thresholds. Unlike other methods, we do not `precondition' the input, but use the lifting scheme to deal with the irregularity of the grid. This approach seems more natural and shows promising results. However, instability problems arise from the actual scheme. This article describes the method and explains in what cases problems may occur.nrpages: 17status: publishe