Haar wavelet

The Haar wavelet is the first known wavelet and was proposed in 1909 by Alfréd Haar. Haar used these functions to give an example of a countable orthonormal system for the space of square-integrable functions on the real line. The study of wavelets, as well as the term "wavelet", did not come until much later. As a special case of the Daubechies wavelet, it is also known as D2.

The Haar wavelet is also the simplest possible wavelet. The disadvantage of the Haar wavelet is that it is not continuous and therefore not differentiable.

The Haar wavelet

The Haar wavelet's mother wavelet function $\psi (t)$ can be described as

$\psi (t)={\begin{cases}1\quad &0\leq t<1/2,\\-1&1/2\leq t<1,\\0&{\mbox{otherwise.}}\end{cases}}$

and its scaling function $\phi (t)$ can be described as

$\phi (t)={\begin{cases}1\quad &0\leq t<1,\\0&{\mbox{otherwise.}}\end{cases}}$

Haar system

In functional analysis, the Haar systems denotes the set of Haar wavelets $\{t\mapsto \psi (2^{n}t-k);n\in \mathbb {N} ,0\leq k<2^{n}\}$. The Haar system (with the natural ordering) is a Schauder basis for the space $L^{p}[0,1]$ for $1\leq p<+\infty$. This basis is unconditional for p>1.

Haar wavelet properties

The Haar wavelet has several properties:

(1) Any function can be approximated by linear combinations of $\phi (t),\phi (2t),\phi (4t),\dots ,\phi (2^{k}t),\dots$ and their shifted functions.

(2) Any function can be approximated by linear combinations of the constant function, $\psi (t),\psi (2t),\psi (4t),\dots ,\psi (2^{k}t),\dots$ and their shifted functions.

(3) Orthogonality

$\int _{-\infty }^{\infty }2^{m}\psi (2^{m}t-n)\psi (2^{m_{1}}t-n_{1})\,dt=\delta _{m,m_{1}}\delta _{n,n_{1}}$

Here δ_i,j represents the Kroneker delta. The dual function of $\psi (t)$ is $\psi (t)$ itself.

(4) Wavelet/scaling functions with different scale m have a functional relationship:

$\phi (t)=\phi (2t)+\phi (2t-1)$

$\psi (t)=\phi (2t)-\phi (2t-1)$

(5) Coefficients of scale m can be calculated by coefficients of scale m+1:

If $\chi _{w}(n,m)=2^{m/2}\int _{-\infty }^{\infty }x(t)\phi (2^{m}t-n)\,dt$
and $\mathrm {X} _{w}(n,m)=2^{m/2}\int _{-\infty }^{\infty }x(t)\psi (2^{m}t-n)\,dt$

$\chi _{w}(n,m)={\sqrt {\frac {1}{2}}}(\chi _{w}(2n,m+1)+\chi _{w}(2n+1,m+1))$

$\mathrm {X} _{w}(n,m)={\sqrt {\frac {1}{2}}}(\chi _{w}(2n,m+1)-\chi _{w}(2n+1,m+1))$

Haar matrix

The 2×2 Haar matrix that is associated with the Haar wavelet is

$H_{2}={\begin{bmatrix}1&1\\1&-1\end{bmatrix}}.$

Using the discrete wavelet transform, one can transform any sequence $(a_{0},a_{1},\dots ,a_{2n},a_{2n+1})$ of even length into a sequence of two-component-vectors $\left(\left(a_{0},a_{1}\right),\dots ,\left(a_{2n},a_{2n+1}\right)\right)$. If one right-multiplies each vector with the matrix $H_{2}$, one gets the result $\left(\left(s_{0},d_{0}\right),\dots ,\left(s_{n},d_{n}\right)\right)$ of one stage of the fast Haar-wavelet transform. Usually one separates the sequences s and d and continues with transforming the sequence s.

If one has a sequence of length a multiple of four, one can build blocks of 4 elements and transform them in a similar manner with the 4×4 Haar matrix

$H_{4}={\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\1&-1&0&0\\0&0&1&-1\end{bmatrix}}$,

which combines two stages of the fast Haar-wavelet transform.

References

• Haar A. Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen, 69, pp 331-371, 1910.
• Charles K. Chui, An Introduction to Wavelets, (1992), Academic Press, San Diego, ISBN 0585470901

External links

• Free Haar wavelet filtering implementation and interactive demo