Noise reduction using wavelet cycle spinning: analysis of useful periodicities in the z-transform domain

Cycle spinning (CS) and a’trous algorithms are different implementations of the undecimated wavelet transform (UWT). Both algorithms can be used for UWT and even though the resulting wavelet coefficients are different, they keep a correspondence. This paper describes an analysis of the CS algorithm performed in the z-transform domain, showing the similarities and differences with the a’trous implementation. CS generates more wavelet coefficients than a’trous, but the number of significative and different coefficients is the same in both cases because of the occurrence of a periodic repetition in CS coefficients. Mathematical expressions for the relationship between CS and a’trous coefficients and for CS coefficient periodicities are provided in the z-transform domain. In some wavelet denoising applications, periodicities (present in the coefficients of the CS procedure) can also be found in the performance measure of the processed signals. In particular, in ultrasonic CS denoising applications, periodicities have been appreciated in the signal-to-noise ratio (SNR) of the ultrasonic denoised signals. These periodicities can be used to optimize the number of CS coefficients for an efficient implementation. Two examples showing the periodicities in the SNR are included. A selection of several reduced sets of CS wavelet coefficients has been utilized in the examples, and the SNRs resulting after denoising are analyzed.