Adaptive splitting for stabilizing 1-D wavelet decompositions on irregular grids

This paper proposes a scheme to improve the stability of wavelet decompositions on 1-D irregular grids. Wavelet transforms on irregular grids are constructed using the lifting scheme. The filters in this scheme take the structure of the grid into account. Nice as it is, however, we undoubtfully bump into numerical stability issues directly related to the irregularity of the grid. Existing stabilizing methods concentrate on the filters used in the lifting scheme itself. While this may be effective in reducing the instability, they are inadequate when a highly irregular grid is involved. The approach presented in this paper is different, as it concentrates on the subsampling or subdivision. Grid locations in the multiscale transform are inserted in such an order that the irregularity of the grid at coarse scales is kept under control. This way, the proposed algorithm prevents instability at coarse scales, rather than healing it. Simulations illustrate that the proposed multiscale decomposition scheme is much more stable than the currently available transforms, especially at coarse scales, where effects of instability have a wide range.