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# Haar wavelet

(Redirected from Haar transform)

In mathematics, the Haar wavelet is a certain sequence of functions. It is now recognised as the first known wavelet.

This sequence was proposed in 1909 by Alfréd Haar.[1] 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, and even 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 technical 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

$\{ t \mapsto \psi_{n,k}(t)=\psi(2^n t-k) ; n \in \N, 0 \leq k < 2^n\}.$

In Hilbert space terms, this constitutes a complete orthogonal system for the functions on the unit interval. There is a related Rademacher system, of sums of Haar functions, which is an orthogonal system but not complete.[2][3]

The Haar system (with the natural ordering) is further 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 notable properties:

1. Any continuous real function can be approximated by linear combinations of $\phi(t),\phi(2t),\phi(4t),\dots,\phi(2^k t),\dots$ and their shifted functions. This extends to those function spaces where any function therein can be approximated by continuous functions.
2. Any continuous real 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 in the form
$\int_{-\infty}^{\infty}2^m\psi(2^mt-n)\psi(2^{m_1}t-n_1)\, dt=\delta_{m,m_1}\delta_{n,n_1}.$
Here δi,j represents the Kronecker 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 $\chi_w(n,m)=2^{m/2}\int_{-\infty}^{\infty}x(t)\phi(2^mt-n)\, dt$
and $\Chi_w(n,m)=2^{m/2}\int_{-\infty}^{\infty}x(t)\psi(2^mt-n)\, dt$
then
$\chi_w(n,m)=\sqrt{\frac{1}{2}}(\chi_w(2n,m+1)+\chi_w(2n+1,m+1))$
$\Chi_w(n,m)=\sqrt{\frac{1}{2}}(\chi_w(2n,m+1)-\chi_w(2n+1,m+1)).$

## 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.

Compare with a Walsh matrix, which is a non-localized 1/–1 matrix.

## Haar transform

The Haar transform is the simplest of the wavelet transforms. This transform cross-multiplies a function against the Haar wavelet with various shifts and stretches, like the Fourier transform cross-multiplies a function against a sine wave with two phases and many stretches.[4]

The Haar transform is derived from the Haar matrix. An example of a 4x4 Haar matrix is shown below.

$H_4 = \frac{1}{\sqrt{4}}\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ \sqrt{2} & -\sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{2} & -\sqrt{2}\end{bmatrix}$

The Haar transform can be thought of as a sampling process in which rows of the transform matrix act as samples of finer and finer resolution.

Compare with the Walsh transform, which is also 1/–1, but is non-localized.

## Notes

1. ^ Haar, Alfred; Zur Theorie der orthogonalen Funktionensysteme. (German) Mathematische Annalen 69 (1910), no. 3, 331–371.
2. ^ http://eom.springer.de/O/o070380.htm
3. ^ Gilbert G. Walter, Xiaoping Shen, Wavelets and Other Orthogonal Systems (2001)
4. ^ The Haar 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