Fast implementation of the continuous wavelet transform with integer scales

We describe a fast noniterative algorithm for the evaluation of continuous spline wavelet transforms at any integer scale m. In this approach, the input signal and the analyzing wavelet are both represented by polynomial splines. The algorithm uses a combination of moving sum and zero-padded filters, and its complexity per scale is O(N), where N is the signal length. The computation is exact, and the implementation is noniterative across scales. We also present examples of spline wavelets exhibiting properties that are desirable for either singularity detection (first and second derivative operators) or Gabor-like time-frequency signal analysis.>