Continuous wavelet transform

Summary

The continuous wavelet transform compares a function $f$ with all rescaled and shifted copies $\abs{a}^{-1/2} \psi((t - b)/a)$ of one analyzing function $\psi$. It gives a function of scale $a$ and position $b$, which shows where $f$ has features of each size. When $\psi$ satisfies the admissibility condition, the transform preserves energy up to a constant, and $f$ can be rebuilt from it. Unlike the short-time Fourier transform, its time resolution adapts to the scale.