Editing Haar wavelet (section)

=== Haar transform and Inverse Haar transform ===
The Haar transform ''y''<sub>''n''</sub> of an n-input function ''x''<sub>''n''</sub> is

: <math> y_n = H_n x_n</math>

The Haar transform matrix is real and orthogonal. Thus, the inverse Haar transform can be derived by the following equations.

: <math> H = H^*, H^{-1} = H^T, \text{ i.e. } HH^T = I </math>

: where <math>I</math> is the identity matrix. For example, when n = 4

: <math> H_4^{T}H_4 = \frac{1}{2}\begin{bmatrix} 1&1&\sqrt{2}&0 \\ 1&1&-\sqrt{2}&0 \\ 1&-1&0&\sqrt{2} \\ 1&-1&0&-\sqrt{2}\end{bmatrix}
\cdot\; \frac{1}{2}\begin{bmatrix} 1&1&1&1 \\ 1&1&-1&-1 \\ \sqrt{2}&-\sqrt{2}&0&0 \\ 0&0&\sqrt{2}&-\sqrt{2}\end{bmatrix}
= \begin{bmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix}
</math>

Thus, the inverse Haar transform is

: <math> x_{n} = H^{T}y_{n}</math>