Discrete wavelet transform

Remarks

The map $a \mapsto (c, d)$ is a unitary operator from $\ell^2(\Z)$ onto $\ell^2(\Z) \oplus \ell^2(\Z)$, so the transform is inverted by its adjoint. If $a_n = \ip{f}{\varphi_{j+1, n}}$ for $f \in L^2(\R)$, where $\varphi_{j,k}(x) = 2^{j/2} \varphi(2^j x - k)$, then $c_k = \ip{f}{\varphi_{j,k}}$ and $d_k = \ip{f}{\psi_{j,k}}$ for the associated orthonormal wavelet $\psi$. The same name is used for the coefficient family $(\ip{f}{\psi_{j,k}})_{j,k}$ of a function, and for finite signals with periodic or other boundary rules.

Example: for the Haar filter $h_0 = h_1 = 1/\sqrt2$ one gets $g_0 = 1/\sqrt2$ and $g_1 = -1/\sqrt2$. Then $c_k = (a_{2k} + a_{2k+1})/\sqrt2$ and $d_k = (a_{2k} - a_{2k+1})/\sqrt2$. For $a = (4, 2, 5, 5)$ (and zeros elsewhere) one level gives $c = (6/\sqrt2, 10/\sqrt2)$ and $d = (2/\sqrt2, 0)$. Energy is preserved: $16 + 4 + 25 + 25 = 70 = 18 + 50 + 2 + 0$.

Non-example: the pairwise means $(a_{2k} + a_{2k+1})/2$ alone discard information, since $(1, 0)$ and $(0, 1)$ have the same mean; the transform keeps the details $d_k$ as well.