Haar wavelet

In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal function basis. The Haar sequence is now recognised as the first known wavelet basis and extensively used as a teaching example.

The Haar sequence 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, and even the term "wavelet", did not come until much later. As a special case of the Daubechies wavelet, the Haar wavelet 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. This property can, however, be an advantage for the analysis of signals with sudden transitions, such as monitoring of tool failure in machines.

The Haar wavelet

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

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

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 functions and Haar system

For every pair n, k of integers in Z, the Haar function ψ_n, k is defined on the real line R by the formula

$\psi _{n,k}(t)=2^{n/2}\psi (2^{n}t-k),\quad t\in \mathbf {R} .$

This function is supported on the right-open interval I_{n,  k} = [ k 2⁻ⁿ, (k+1) 2⁻ⁿ), i.e., it vanishes outside that interval. It has integral 0 and norm 1 in the Hilbert space L²(R),

$\int _{\mathbf {R} }\psi _{n,k}(t)\,dt=0,\quad \|\psi _{n,k}\|_{L^{2}(\mathbf {R} )}^{2}=\int _{\mathbf {R} }\psi _{n,k}(t)^{2}\,dt=1.$

The Haar functions are pairwise orthogonal,

$\int _{\mathbf {R} }\psi _{n_{1},k_{1}}(t)\psi _{n_{2},k_{2}}(t)\,dt=\delta _{n_{1},n_{2}}\delta _{k_{1},k_{2}},$

where δ_i,j represents the Kronecker delta. Here is the reason for orthogonality: when the two supporting intervals $I_{n_{1},k_{1}}$ and $I_{n_{2},k_{2}}$ are not equal, then they are either disjoint, or else, the smaller of the two supports, say $I_{n_{1},k_{1}}$, is contained in the lower or in the upper half of the other interval, on which the function $\psi _{n_{2},k_{2}}$ remains constant.

The Haar system on the real line is the set of functions

$\{\psi _{n,k}(t)\;;\;n\in \mathbf {Z} ,\;k\in \mathbf {Z} \}.$

It is complete in L²(R): The Haar system on the line is an orthonormal basis in L²(R).

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

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

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

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

${\begin{aligned}\phi (t)&=\phi (2t)+\phi (2t-1)\\[.2em]\psi (t)&=\phi (2t)-\phi (2t-1)\end{aligned}}$

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

then

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

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

Haar system on [0, 1] and related systems

In this section, the discussion is restricted to the unit interval [0, 1] and to the Haar functions that are supported on [0, 1]. In functional analysis, the Haar system on [0, 1] consists of the subset of Haar wavelets defined as

$\{t\in [0,1]\mapsto \psi _{n,k}(t)\;;\;n\in \mathbb {N} ,\;0\leq k<2^{n}\},$

with the addition of the constant function 1 on [0, 1].

In Hilbert space terms, this Haar system on [0, 1] is a complete orthogonal system for the space L²([0, 1]) of square integrable functions on the unit interval.

The Haar system on [0, 1] (with the constant function 1 as first element, followed with the Haar functions ordered according to the lexicographic ordering of couples (n, k)) is further a monotone Schauder basis for the space L^p([0, 1]) when 1 ≤ p < ∞. This basis is unconditional for p > 1.

There is a related Rademacher system consisting of sums of Haar functions,

$r_{n}(t)=\sum _{k=0}^{2^{n}-1}\psi _{n,k}(t),\quad t\in [0,1],\ n\geq 0.$

This is an orthogonal system but it is not complete. In the language of probability theory, the Rademacher sequence is an instance of a sequence of independent Bernoulli random variables with mean 0. The Khintchine inequality expresses the fact that in all the spaces L^p([0, 1]), 1 ≤ p < ∞, this sequence is equivalent to the unit vector basis in ℓ². In particular, the closed linear span of the Rademacher sequence in L^p([0, 1]), 1 ≤ p < ∞, is isomorphic to ℓ².

The Schauder system and the Franklin system

The Schauder system is the family of continuous functions on [0, 1] consisting of the constant function 1, and of multiples of indefinite integrals of the functions in the Haar system on [0, 1], chosen to have norm 1 in the maximum norm. This system begins with s₀ = 1, then s₁(t) = t is the indefinite integral vanishing at 0 of the function 1, first element of the Haar system on [0, 1]. Next, for every integer n ≥ 0, functions s_n, k are defined by the formula

$s_{n,k}(t)=2^{1+n/2}\int _{0}^{t}\psi _{n,k}(u)\,du,\quad t\in [0,1],\ 0\leq k<2^{n}.$

These functions s_n, k are continuous, piecewise linear, supported by the interval I_n, k that also supports ψ_n, k. The function s_n, k is equal to 1 at the midpoint x_n, k of the interval  I_n, k, linear on both halves of that interval. It takes values between 0 and 1 everywhere.

The Schauder system is a Schauder basis for the space C([0, 1]) of continuous functions on [0, 1]. For every f in C([0, 1]), the partial sum

$f_{n+1}=a_{0}s_{0}+a_{1}s_{1}+\sum _{m=0}^{n-1}{\Bigl (}\sum _{k=0}^{2^{m}-1}a_{m,k}s_{m,k}{\Bigr )}\in C([0,1])$

of the series expansion of f in the Schauder system is the continuous piecewise linear function that agrees with f at the 2ⁿ + 1 points k 2⁻ⁿ, where 0 ≤ k ≤ 2ⁿ. Next, the formula

$f_{n+2}-f_{n+1}=\sum _{k=0}^{2^{n}-1}{\bigl (}f(x_{n,k})-f_{n+1}(x_{n,k}){\bigr )}s_{n,k}=\sum _{k=0}^{2^{n}-1}a_{n,k}s_{n,k}$

gives a way to compute the expansion of f step by step. Since f is uniformly continuous, the sequence {f_n} converges uniformly to f, i.e., the sum of the Schauder series expansion of f is equal to f.

The Franklin system is obtained from the Schauder system by the Gram–Schmidt orthonormalization procedure. Since the Franklin system has the same linear span as that of the Schauder system, this span is dense in C([0, 1]), hence in L²([0, 1]). The Franklin system is therefore an orthonormal basis for L²([0, 1]), consisting of continuous piecewise linear functions. P. Franklin proved in 1928 that this system is a Schauder basis for C([0, 1]). The Franklin system is also an unconditional basis for the space L^p([0, 1]) when 1 < p < ∞. It is a Schauder basis in the disk algebra A(D): this was proved in 1974 by Bočkarev, after the existence of a basis for the disk algebra had remained an open question for several years.

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. Sequence s is often referred to as the averages part, whereas d is known as the details part.

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.

The Haar transform is derived from the Haar matrix. An example of a 4x4 Haar transformation 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 transformation matrix act as samples of finer and finer resolution.

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

See also

• Dimension reduction
• Walsh matrix
• Walsh transform
• Wavelet
• signal

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 0-585-47090-1

External links

• "Haar system", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Free Haar wavelet filtering implementation and interactive demo
• Free Haar wavelet denoising and lossy signal compression

Haar transform

• http://cnx.org/content/m11087/latest/
• http://math.hws.edu/eck/math371/applets/Haar.html
• http://online.redwoods.cc.ca.us/instruct/darnold/LAPROJ/Fall2002/ames/paper.pdf
• http://scien.stanford.edu/class/ee368/projects2000/project12/2.html
• http://fourier.eng.hmc.edu/e161/lectures/Haar/index.html