Comparisons between fast algorithms for the continuous wavelet transform and applications in cosmology: The one-dimensional case

The continuous wavelet transform (CWT) is very useful for processing signals with intricate and irregular structures in astrophysics and cosmology. It is crucial to propose precise and fast algorithms for the CWT. In this work, we review and compare four different fast CWT algorithms for the 1D signals, including the FFTCWT, the V97CWT, the M02CWT, and the A19CWT. The FFTCWT algorithm implements the CWT using the Fast Fourier Transform (FFT) with a computational complexity of $\mathcal{O}(N\log_2N)$ per scale. The rest algorithms achieve the complexity of $\mathcal{O}(N)$ per scale by simplifying the CWT into some smaller convolutions. We illustrate explicitly how to set the parameters as well as the boundary conditions for them. To examine the actual performance of these algorithms, we use them to perform the CWT of signals with different wavelets. From the aspect of accuracy, we find that the FFTCWT is the most accurate algorithm, though its accuracy degrades a lot when processing the non-periodic signal with zero boundaries. The accuracy of $\mathcal{O}(N)$ algorithms is robust to signals with different boundaries, and the M02CWT is more accurate than the V97CWT and A19CWT. From the aspect of speed, the $\mathcal{O}(N)$ algorithms do not show an overall speed superiority over the FFTCWT at sampling numbers of $N\lesssim10^6$, which is due to their large leading constants. Only the speed of the V97CWT with real wavelets is comparable to that of the FFTCWT. However, both the FFTCWT and V97CWT are substantially less efficient in processing the non-periodic signal because of zero padding. Finally, we conduct wavelet analysis of the 1D density fields, which demonstrate the convenience and power of techniques based on the CWT. We publicly release our CWT codes as resources for the community.