Completeness and stability of partial dyadic wavelet domain signal representations

The authors present a necessary and sufficient condition for the completeness of any partial dyadic wavelet transform domain representation (PDWTDR) of discrete finite data length signals (including dyadic wavelet transform extrema and zero-crossing representations). It is shown that completeness depends only on the locations of the retained samples of the dyadic wavelet transform. The present completeness test is more convenient and easier to verify than previously derived tests. It is also shown how to ensure the completeness of the representation by adding additional information in those cases where the PDWTDR is incomplete. The numerical stability of such a representation is also discussed. A fast-Fourier-transform (FFT)-based reconstruction algorithm from such a signal representation is also described.<>