Morlet wavelet transform using attenuated sliding Fourier transform and kernel integral for graphic processing unit

Morlet or Gabor wavelet transforms as well as Gaussian smoothing, are widely used in signal processing and image processing. However, the computational complexity of their direct calculations is proportional not only to the number of data points in a signal but also to the smoothing size, which is the standard deviation in the Gaussian function in their transform functions. Thus, when the standard deviation is large, its considerable computation time diminishes the advantages of aforementioned transforms. Therefore, it is important to formulate an algorithm to reduce the calculation time of the transformations. In this paper, we first review calculation methods of Gaussian smoothing by using the sliding Fourier transform (SFT) and our proposed attenuated SFT (ASFT) \cite{YamashitaICPR2020}. Based on these methods, we propose two types of calculation methods for Morlet wavelet transforms. We also propose an algorithm to calculate SFT using the kernel integral on graphic processing unit (GPU). When the number of calculation cores in GPU is not less than the number of data points, the order of its calculation time is the logarithm of the smoothing size and does not depend on the number of data points. Using experiments, we compare the two methods for calculating the Morlet wavelet transform and evaluate the calculation time of the proposed algorithm using a kernel integral on GPU. For example, when the number of data points and the standard deviation are 102400 and 8192.0, respectively, the calculation time of the Morlet wavelet transform by the proposed method is 0.545 ms, which 413.6 times faster than a conventional method. (In this version, mistakes in fitures are corrected.)