Haar wavelet

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The Haar wavelet
The Haar wavelet

The Haar wavelet is the first known wavelet and was proposed in 1909 by Alfred Haar. Note that the term wavelet was coined 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 can also be described as a step function f(x) with

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

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

H_2 = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}

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 H2, 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 & 0 \\ 1 & 1 & -1 & 0 \\ 1 & -1 & 0 & 1\\ 1 & -1 & 0 & -1 \end{bmatrix},

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

  • 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
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