Multiresolution Analysis

Definition

Multiresolution analysis decomposes a signal into a coarse approximation and a series of details at different scales. $f(t)=\sum _{k=-\infty }^{\infty }{c}_k{\varphi }_k(t)+\sum _{j=-\infty }^{\infty }\sum _{k=-\infty }^{\infty }{d}_{j,k}{\psi }_{j,k}(t)$ where the first term $\sum _{k=-\infty }^{\infty }{c}_k{\varphi }_k(t)$ represents the coarse approximation and the second term $\sum _{j=-\infty }^{\infty }\sum _{k=-\infty }^{\infty }{d}_{j,k}{\psi }_{j,k}(t)$ represents the sum of details. Here, ${\varphi }_k(t)$ is a scaling function (father wavelet), ${\psi }_{j,k}(t)$ is a wavelet function (mother wavelet), $c_k$ is approximation coefficient, $d_{j,k}$ is a detail coefficient.

The set $\{{\varphi }_k(x),{\psi }_{j,k}(x),j,k\in \mathbb{Z}\}$ is an orthonormal basis of the $L^2$ function space.