Editing Haar wavelet

{{Short description|First known wavelet basis}}
{{Use dmy dates|date=March 2024}}
[[Image:Haar wavelet.svg|thumb|right|The 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 basis]]. The Haar sequence is now recognised as the first known wavelet basis and is extensively used as a teaching example.

The '''Haar sequence''' was proposed in 1909 by [[Alfréd Haar]].<ref>see p.&nbsp;361 in {{harvtxt|Haar|1910}}.</ref> 
Haar used these functions to give an example of an orthonormal system for the space of [[square-integrable function]]s on the [[unit interval]]&nbsp;[0,&nbsp;1]. 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 '''Db1'''.

The Haar wavelet is also the simplest possible wavelet. The technical disadvantage of the Haar wavelet is that it is not [[continuous function|continuous]], and therefore not [[derivative|differentiable]]. This property can, however, be an advantage for the analysis of signals with sudden transitions ([[Digital signal (signal processing)|discrete signals]]), such as monitoring of tool failure in machines.<ref>{{cite journal |first1=B. |last1=Lee |first2=Y. S. |last2=Tarng |title=Application of the discrete wavelet transform to the monitoring of tool failure in end milling using the spindle motor current |journal=International Journal of Advanced Manufacturing Technology |year=1999 |volume=15 |issue=4 |pages=238–243 |doi=10.1007/s001700050062 |s2cid=109908427 }}</ref>

The Haar wavelet's mother wavelet function <math>\psi(t)</math> can be described as
: <math>\psi(t) = \begin{cases}
  1 \quad & 0 \leq  t < \frac{1}{2},\\
 -1 & \frac{1}{2} \leq t < 1,\\
  0 &\mbox{otherwise.}
\end{cases}</math>

Its [[Father wavelets|scaling function]] <math>\varphi(t)</math> can be described as
: <math>\varphi(t) = \begin{cases}1 \quad & 0 \leq  t < 1,\\0 &\mbox{otherwise.}\end{cases}</math>

== Haar functions and Haar system ==
For every pair ''n'', ''k'' of integers in <math>\mathbb{Z}</math>, the '''Haar function''' ''ψ''<sub>''n'',''k''</sub> is defined on the [[real line]] <math>\mathbb{R}</math> by the formula
:<math> \psi_{n,k}(t) = 2^{n / 2} \psi(2^n t-k), \quad t \in \mathbb{R}.</math>
This function is supported on the [[Semi-open interval|right-open interval]] {{nowrap| ''I''<sub>''n'',''k''</sub> {{=}}}} {{nowrap|[ ''k''2<sup>&minus;''n''</sup>, (''k''+1)2<sup>&minus;''n''</sup>)}}, ''i.e.'', it [[Zero of a function|vanishes]] outside that interval. It has integral 0 and norm&nbsp;1 in the [[Hilbert space]]&nbsp;[[Lp space|''L''<sup>2</sup>(<math>\mathbb{R}</math>)]],
:<math> \int_{\mathbb{R}} \psi_{n, k}(t) \, d t = 0, \quad \|\psi_{n, k}\|^2_{L^2(\mathbb{R})} = \int_{\mathbb{R}} \psi_{n, k}(t)^2 \, d t = 1.</math>
The Haar functions are pairwise [[Orthogonality#Orthogonal functions|orthogonal]],
:<math> \int_{\mathbb{R}} \psi_{n_1, k_1}(t) \psi_{n_2, k_2}(t) \, d t = \delta_{n_1n_2} \delta_{k_1k_2}, </math>
where <math>\delta_{ij}</math> represents the [[Kronecker delta]]. Here is the reason for orthogonality: when the two supporting intervals <math>I_{n_1, k_1}</math> and <math>I_{n_2, k_2}</math> are not equal, then they are either disjoint, or else the smaller of the two supports, say <math>I_{n_1, k_1}</math>, is contained in the lower or in the upper half of the other interval, on which the function <math>\psi_{n_2, k_2}</math> remains constant. It follows in this case that the product of these two Haar functions is a multiple of the first Haar function, hence the product has integral&nbsp;0.

The '''Haar system''' on the real line is the set of functions
:<math> \{ \psi_{n,k}(t) \; : \; n \in \mathbb{Z}, \; k \in \mathbb{Z} \}.</math>
It is [[Orthonormal basis|complete]] in ''L''<sup>2</sup>(<math>\mathbb{R}</math>): ''The Haar system on the line is an orthonormal basis in'' ''L''<sup>2</sup>(<math>\mathbb{R}</math>).

==Haar wavelet properties==
The Haar wavelet has several notable properties:

{{ordered list

| Any continuous real function with compact support can be approximated uniformly by [[linear combination]]s of <math>\varphi(t),\varphi(2t),\varphi(4t),\dots,\varphi(2^n t),\dots</math> and their shifted functions. This extends to those function spaces where any function therein can be approximated by continuous functions.

| Any continuous real function on [0,&nbsp;1] can be approximated uniformly on [0,&nbsp;1] by linear combinations of the constant function&nbsp;'''1''', <math>\psi(t),\psi(2t),\psi(4t),\dots,\psi(2^n t),\dots</math> and their shifted functions.<ref>As opposed to the preceding statement, this fact is not obvious: see p.&nbsp;363 in {{harvtxt|Haar|1910}}.</ref>

| [[Orthogonality]] in the form
: <math>
 \int_{-\infty}^{\infty}2^{(n+n_1)/2}\psi(2^n t-k)\psi(2^{n_1} t - k_1)\, dt = \delta_{nn_1}\delta_{kk_1}.
 </math>
Here, <math>\delta_{ij}</math> represents the [[Kronecker delta]]. The [[dual function]] of ψ(''t'') is ψ(''t'') itself.

| Wavelet/scaling functions with different scale ''n'' have a functional relationship:<ref>{{cite book |last=Vidakovic |first=Brani |title=Statistical Modeling by Wavelets |series=Wiley Series in Probability and Statistics |year=2010 |edition=2 |doi=10.1002/9780470317020 |pages=60, 63|isbn=9780470317020 }}</ref> since
:<math>
\begin{align}
  \varphi(t) &= \varphi(2t)+\varphi(2t-1)\\[.2em]
  \psi(t) &= \varphi(2t)-\varphi(2t-1),
\end{align}</math>
it follows that coefficients of scale ''n'' can be calculated by coefficients of scale ''n+1'':<br/>
If <math> \chi_w(k, n)= 2^{n/2}\int_{-\infty}^\infty x(t)\varphi(2^nt-k)\, dt</math><br/>
and <math> \Chi_w(k, n)= 2^{n/2}\int_{-\infty}^\infty x(t)\psi(2^nt-k)\, dt</math><br/>
then
: <math> \chi_w(k,n)= 2^{-1/2} \bigl( \chi_w(2k,n+1)+\chi_w(2k+1,n+1) \bigr)</math>
: <math> \Chi_w(k,n)= 2^{-1/2} \bigl( \chi_w(2k,n+1)-\chi_w(2k+1,n+1) \bigr).</math>

}}
<!--The structure of [[multiresolution analysis]] (MRA):
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== Haar system on the unit interval and related systems ==
In this section, the discussion is restricted to the [[unit interval]] [0,&nbsp;1] and to the Haar functions that are supported on [0,&nbsp;1]. The system of functions considered by Haar in 1910,<ref>p.&nbsp;361 in {{harvtxt|Haar|1910}}</ref>
called the '''Haar system on [0,&nbsp;1]''' in this article, consists of the subset of Haar wavelets defined as
:<math>\{ t \in [0, 1] \mapsto \psi_{n,k}(t) \; : \; n, k \in \N \cup \{0\}, \; 0 \leq k < 2^n\},</math>
with the addition of the constant function '''1''' on [0,&nbsp;1].

In [[Hilbert space]] terms, this Haar system on [0,&nbsp;1] is a complete orthonormal system, ''i.e.'', an [[orthonormal basis]], for the space ''L''<sup>2</sup>([0,&nbsp;1]) of square integrable functions on the unit interval.

The Haar system on [0,&nbsp;1] &mdash;with the constant function '''1''' as first element, followed with the Haar functions ordered according to the [[Lexicographical order|lexicographic]] ordering of couples {{nowrap|(''n'', ''k'')}}&mdash; is further a [[Schauder basis#Properties|monotone]] [[Schauder basis]] for the space [[Lp space|''L''<sup>''p''</sup>([0,&nbsp;1])]] when {{nowrap|1 ≤ ''p'' &lt; ∞}}.<ref name="L. Tzafriri, 1977">see p.&nbsp;3 in [[Joram Lindenstrauss|J. Lindenstrauss]], L. Tzafriri, (1977), "Classical Banach Spaces I, Sequence Spaces", Ergebnisse der Mathematik und ihrer Grenzgebiete '''92''', Berlin: Springer-Verlag, {{ISBN|3-540-08072-4}}.</ref> 
This basis is [[Schauder basis#Unconditionality|unconditional]] when {{nowrap|1 &lt; ''p'' &lt; ∞}}.<ref>The result is due to [[Raymond Paley|R. E. Paley]], ''A remarkable series of orthogonal functions (I)'', Proc. London Math. Soc. '''34''' (1931) pp. 241-264. See also p.&nbsp;155 in J. Lindenstrauss, L. Tzafriri, (1979), "Classical Banach spaces II, Function spaces". Ergebnisse der Mathematik und ihrer Grenzgebiete '''97''', Berlin: Springer-Verlag, {{ISBN|3-540-08888-1}}.</ref>

There is a related [[Rademacher system]] consisting of sums of Haar functions, 
:<math>r_n(t) = 2^{-n/2} \sum_{k=0}^{2^n - 1} \psi_{n, k}(t), \quad t \in [0, 1], \ n \ge 0.</math>
Notice that |''r''<sub>''n''</sub>(''t'')|&nbsp;= 1 on [0,&nbsp;1). This is an orthonormal system but it is not complete.<ref>{{SpringerEOM |title=Orthogonal system}}</ref><ref>{{cite book |first1=Gilbert G. |last1=Walter |first2=Xiaoping |last2=Shen |title=Wavelets and Other Orthogonal Systems |year=2001 |location=Boca Raton |publisher=Chapman |isbn=1-58488-227-1 }}</ref>
In the language of [[probability theory]], the Rademacher sequence is an instance of a sequence of [[Independence (probability theory)|independent]] [[Bernoulli distribution|Bernoulli]] [[random variables]] with [[mean]]&nbsp;0. The [[Khintchine inequality]] expresses the fact that in all the spaces ''L''<sup>''p''</sup>([0,&nbsp;1]), {{nowrap|1 ≤ ''p'' &lt; ∞}}, the Rademacher sequence is [[Schauder basis#Definitions|equivalent]] to the unit vector basis in ℓ<sup>''2''</sup>.<ref>see for example p.&nbsp;66 in [[Joram Lindenstrauss|J. Lindenstrauss]], L. Tzafriri, (1977), "Classical Banach Spaces I, Sequence Spaces", Ergebnisse der Mathematik und ihrer Grenzgebiete '''92''', Berlin: Springer-Verlag, {{ISBN|3-540-08072-4}}.</ref> In particular, the [[Linear span#Closed linear span|closed linear span]] of the Rademacher sequence in ''L''<sup>''p''</sup>([0,&nbsp;1]), {{nowrap|1 ≤ ''p'' &lt; ∞}}, is [[Isomorphic normed spaces|isomorphic]] to ℓ<sup>''2''</sup>.

=== The Faber&ndash;Schauder system ===
The '''Faber&ndash;Schauder system'''<ref name="Faber">Faber, Georg (1910), "Über die Orthogonalfunktionen des Herrn Haar", ''Deutsche Math.-Ver'' (in German) '''19''': 104&ndash;112. {{issn|0012-0456}}; 
http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X ; http://resolver.sub.uni-goettingen.de/purl?GDZPPN002122553</ref><ref>Schauder, Juliusz (1928), "Eine Eigenschaft des Haarschen Orthogonalsystems", ''Mathematische Zeitschrift'' '''28''': 317&ndash;320.</ref><ref>{{eom|id=f/f038020
 |title=Faber–Schauder system|first=B.I.|last= Golubov}}</ref> 
is the family of continuous functions on [0,&nbsp;1] consisting of the constant function&nbsp;'''1''', and of multiples of [[Antiderivative|indefinite integrals]] of the functions in the Haar system on&nbsp;[0,&nbsp;1], chosen to have norm&nbsp;1 in the [[Uniform norm|maximum norm]]. This system begins with ''s''<sub>0</sub>&nbsp;=&nbsp;'''1''', then {{nowrap| ''s''<sub>1</sub>(''t'') {{=}} ''t''}} is the indefinite integral vanishing at&nbsp;0 of the function&nbsp;'''1''', first element of the Haar system on [0,&nbsp;1]. Next, for every integer {{nowrap|''n'' ≥ 0}}, functions {{nowrap| ''s''<sub>''n'',''k''</sub>}} are defined by the formula
:<math>
 s_{n, k}(t) = 2^{1 + n/2} \int_0^t \psi_{n, k}(u) \, d u, \quad t \in [0, 1], \ 0 \le k < 2^n.</math>
These functions {{nowrap| ''s''<sub>''n'',''k''</sub>}} are continuous, [[Piecewise linear function|piecewise linear]], supported by the interval {{nowrap| ''I''<sub>''n'',''k''</sub>}} that also supports {{nowrap| ψ<sub>''n'',''k''</sub>}}. The function {{nowrap| ''s''<sub>''n'',''k''</sub>}} is equal to&nbsp;1 at the midpoint {{nowrap| ''x''<sub>''n'',''k''</sub>}} of the interval&nbsp;{{nowrap| ''I''<sub>''n'',''k''</sub>}}, linear on both halves of that interval. It takes values between&nbsp;0 and&nbsp;1 everywhere.

The Faber&ndash;Schauder system is a [[Schauder basis]] for the space ''C''([0,&nbsp;1]) of continuous functions on [0,&nbsp;1].<ref name="L. Tzafriri, 1977"/> 
For every&nbsp;''f'' in ''C''([0,&nbsp;1]), the partial sum
:<math> 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])</math>
of the [[series expansion]] of ''f'' in the Faber&ndash;Schauder system is the continuous piecewise linear function that agrees with&nbsp;''f'' at the {{nowrap|2<sup>''n''</sup> + 1}} points {{nowrap|''k''2<sup>&minus;''n''</sup>}}, where {{nowrap| 0 ≤ ''k'' ≤ 2<sup>''n''</sup>}}. Next, the formula
:<math> 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} </math>
gives a way to compute the expansion of ''f'' step by step. Since ''f'' is [[Heine–Borel theorem|uniformly continuous]], the sequence {''f''<sub>''n''</sub>} converges uniformly to ''f''. It follows that the Faber&ndash;Schauder series expansion of ''f'' converges in ''C''([0,&nbsp;1]), and the sum of this series is equal to&nbsp;''f''.

=== The Franklin system ===
The '''Franklin system''' is obtained from the Faber&ndash;Schauder system by the [[Gram–Schmidt process|Gram&ndash;Schmidt orthonormalization procedure]].<ref>see Z. Ciesielski, ''Properties of the orthonormal Franklin system''. Studia Math. 23 1963 141–157.</ref><ref>Franklin system. B.I. Golubov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Franklin_system&oldid=16655</ref>
Since the Franklin system has the same [[linear span]] as that of the Faber&ndash;Schauder system, this span is dense in ''C''([0,&nbsp;1]), hence in ''L''<sup>2</sup>([0,&nbsp;1]). The Franklin system is therefore an orthonormal basis for ''L''<sup>2</sup>([0,&nbsp;1]), consisting of continuous piecewise linear functions. P. Franklin proved in 1928 that this system is a Schauder basis for ''C''([0,&nbsp;1]).<ref>Philip Franklin, ''A set of continuous orthogonal functions'', Math. Ann. 100 (1928), 522-529. {{doi|10.1007/BF01448860}}</ref> 
The Franklin system is also an unconditional Schauder basis for the space ''L''<sup>''p''</sup>([0,&nbsp;1]) when {{nowrap|1 &lt; ''p'' &lt; ∞}}.<ref name=Bo>S. V. Bočkarev, ''Existence of a basis in the space of functions analytic in the disc, and some properties of Franklin's system''. Mat. Sb. '''95''' (1974), 3–18 (Russian). Translated in Math. USSR-Sb. '''24''' (1974), 1–16.</ref>
The Franklin system provides a Schauder basis in the [[disk algebra]] ''A''(''D'').<ref name=Bo />
This was proved in 1974 by Bočkarev, after the existence of a basis for the disk algebra had remained open for more than forty years.<ref>The question appears p.&nbsp;238, §3 in Banach's book, {{citation|first=Stefan|last=Banach|author-link=Stefan Banach|url=http://matwbn.icm.edu.pl/kstresc.php?tom=1&wyd=10|title=Théorie des opérations linéaires|publication-place=Warszawa|publisher=Subwencji Funduszu Kultury Narodowej|year=1932|series=Monografie Matematyczne|volume=1|zbl=0005.20901}}.  The disk algebra ''A''(''D'') appears as Example&nbsp;10, p.&nbsp;12 in Banach's book.</ref>

Bočkarev's construction of a Schauder basis in ''A''(''D'') goes as follows: let&nbsp;''f'' be a complex valued [[Lipschitz continuity|Lipschitz function]] on [0,&nbsp;π]; then&nbsp;''f'' is the sum of a [[Fourier series|cosine series]] with [[Absolute convergence|absolutely summable]] coefficients. Let&nbsp;''T''(''f'') be the element of ''A''(''D'') defined by the complex [[power series]] with the same coefficients,

:<math> \left\{ f : x \in [0, \pi] \rightarrow \sum_{n=0}^\infty a_n \cos(n x) \right\} \longrightarrow \left\{ T(f) : z \rightarrow \sum_{n=0}^\infty a_n z^n, \quad |z| \le 1 \right\}.</math>
Bočkarev's basis for ''A''(''D'') is formed by the images under&nbsp;''T'' of the functions in the Franklin system on&nbsp;[0,&nbsp;π]. Bočkarev's equivalent description for the mapping&nbsp;''T'' starts by extending ''f'' to an [[Even and odd functions|even]] Lipschitz function&nbsp;''g''<sub>1</sub> on [&minus;π,&nbsp;π], identified with a Lipschitz function on the [[unit circle]]&nbsp;'''T'''. Next, let ''g''<sub>2</sub> be the [[Hardy space conjugate function|conjugate function]] of&nbsp;''g''<sub>1</sub>, and define ''T''(''f'') to be the function in&nbsp;''A''(''D'') whose value on the boundary '''T''' of&nbsp;''D'' is equal to&nbsp;{{nowrap|''g''<sub>1</sub> + i''g''<sub>2</sub>}}.

When dealing with 1-periodic continuous functions, or rather with continuous functions ''f'' on [0,&nbsp;1] such that {{nowrap|''f''(0) {{=}} ''f''(1)}}, one removes the function {{nowrap| ''s''<sub>1</sub>(''t'') {{=}} ''t''}} from the Faber&ndash;Schauder system, in order to obtain the '''periodic Faber&ndash;Schauder system'''. The '''periodic Franklin system''' is obtained by orthonormalization from the periodic Faber&ndash;-Schauder system.<ref name="Prz">See p.&nbsp;161, III.D.20 and p.&nbsp;192, III.E.17 in {{citation
 | last=Wojtaszczyk | first= Przemysław
 | title = Banach spaces for analysts
 | series = Cambridge Studies in Advanced Mathematics
 | volume = 25
 | publisher = Cambridge University Press 
 | location = Cambridge
 | year= 1991
 | pages = xiv+382
 | isbn = 0-521-35618-0 
}}</ref>
One can prove Bočkarev's result on ''A''(''D'') by proving that the periodic Franklin system on [0,&nbsp;2π] is a basis for a Banach space ''A''<sub>''r''</sub> isomorphic to ''A''(''D'').<ref name="Prz" /> 
The space ''A''<sub>''r''</sub> consists of complex continuous functions on the unit circle '''T''' whose [[Harmonic conjugate|conjugate function]] is also continuous.

==Haar matrix==
The 2×2 Haar matrix that is associated with the Haar wavelet is
: <math> H_2 = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}.</math>
Using the [[discrete wavelet transform]], one can transform any sequence <math>(a_0,a_1,\dots,a_{2n},a_{2n+1})</math> of even length into a sequence of two-component-vectors <math> \left(\left(a_0,a_1\right),\left(a_2,a_3\right),\dots,\left(a_{2n},a_{2n+1}\right)\right) </math>. If one right-multiplies each vector with the matrix <math> H_2 </math>, one gets the result <math>\left(\left(s_0,d_0\right),\dots,\left(s_n,d_n\right)\right)</math> 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.<ref>{{cite book |first1=David K. |last1=Ruch |first2=Patrick J. |last2=Van Fleet |title=Wavelet Theory: An Elementary Approach with Applications |year=2009 |publisher=John Wiley & Sons|isbn=978-0-470-38840-2 }}</ref>

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
: <math> H_4 = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 0 & 0\\ 0 & 0 & 1 & -1 \end{bmatrix},</math>
which combines two stages of the fast Haar-wavelet transform.

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

Generally, the 2N×2N Haar matrix can be derived by the following equation.

: <math> H_{2N} = \begin{bmatrix} H_{N} \otimes [1, 1] \\ I_{N} \otimes [1, -1] \end{bmatrix}</math>
:where <math>I_{N} = \begin{bmatrix} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 1 \end{bmatrix}</math> and <math>\otimes</math> is the [[Kronecker product]].

The [[Kronecker product]] of <math>A \otimes B</math>, where <math>A</math> is an m×n matrix and <math>B</math> is a p×q matrix, is expressed as

: <math>A \otimes B = \begin{bmatrix} a_{11}B & \dots & a_{1n}B \\ \vdots & \ddots & \vdots \\ a_{m1}B & \dots & a_{mn}B\end{bmatrix}.</math>

An un-normalized 8-point Haar matrix <math>H_8</math> is shown below

: <math>H_{8} = \begin{bmatrix} 1&1&1&1&1&1&1&1 \\ 1&1&1&1&-1&-1&-1&-1 \\ 1&1&-1&-1&0&0&0&0& \\ 0&0&0&0&1&1&-1&-1 \\ 1&-1&0&0&0&0&0&0& \\ 0&0&1&-1&0&0&0&0 \\ 0&0&0&0&1&-1&0&0& \\ 0&0&0&0&0&0&1&-1 \end{bmatrix}.</math>

Note that, the above matrix is an un-normalized Haar matrix. The Haar matrix required by the Haar transform should be normalized.

From the definition of the Haar matrix <math>H</math>, one can observe that, unlike the [[Fourier transform]], <math>H</math> has only real elements (i.e., 1, -1 or 0) and is non-symmetric.

Take the 8-point Haar matrix <math>H_8</math> as an example. The first row of <math>H_8</math> measures the average value, and the second row of <math>H_8</math> measures a low frequency component of the input vector. The next two rows are sensitive to the first and second half of the input vector respectively, which corresponds to moderate frequency components. The remaining four rows are sensitive to the four section of the input vector, which corresponds to high frequency components.<ref>{{cite web |url=http://fourier.eng.hmc.edu/e161/lectures/Haar/index.html |title=haar |publisher=Fourier.eng.hmc.edu |date=30 October 2013 |access-date=23 November 2013 |archive-date=21 August 2012 |archive-url=https://web.archive.org/web/20120821004423/http://fourier.eng.hmc.edu/e161/lectures/Haar/index.html |url-status=dead }}</ref>

==Haar transform==
The '''Haar transform''' is the simplest of the [[wavelet transform]]s. 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.<ref>[http://sepwww.stanford.edu/public/docs/sep75/ray2/paper_html/node4.html The Haar Transform<!-- Bot generated title -->]</ref>{{clarify|Is this comparing the kernels being integrated over, and decomposing exponentials into sine and cosine to treat the Fourier kernel as a space of sines, changing the parametrization accordingly? If so, we can give more specific, linkable language than "cross-multiplies", talk about inner products or projections and integrating them, and then lucidly compare that to a convolutional treatment.|date=June 2018}}

=== Introduction ===
The Haar transform is one of the oldest transform functions, proposed in 1910 by the Hungarian mathematician [[Alfréd Haar]]. It is found effective in applications such as signal and image compression in electrical and computer engineering as it provides a simple and computationally efficient approach for analysing the local aspects of a signal.

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

:<math>H_4 = \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}
</math>

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.

=== Property ===
The Haar transform has the following properties

# No need for multiplications. It requires only additions and there are many elements with zero value in the Haar matrix, so the computation time is short. It is faster than [[Walsh transform]], whose matrix is composed of +1 and −1.
# Input and output length are the same. However, the length should be a power of 2, i.e. <math>N = 2^k,  k\in \mathbb{N}</math>.
# It can be used to analyse the localized feature of signals. Due to the [[orthogonal]] property of the Haar function, the frequency components of input signal can be analyzed.

=== 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>

=== Example ===
The Haar transform coefficients of a n=4-point signal <math>x_{4} = [1,2,3,4]^{T}</math> can be found as

: <math> y_{4} = H_4 x_4 = 
\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 \\ 2 \\ 3 \\ 4\end{bmatrix}
= \begin{bmatrix} 5 \\ -2 \\ -1/\sqrt{2} \\ -1/\sqrt{2}\end{bmatrix}
</math>

The input signal can then be perfectly reconstructed by the inverse Haar transform

: <math> \hat{x_{4}} = H_{4}^{T}y_{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} \begin{bmatrix} 5 \\ -2 \\ -1/\sqrt{2} \\ -1/\sqrt{2}\end{bmatrix}
= \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}
</math>

==See also==
* [[Dimension reduction]]
* [[Walsh matrix]]
* [[Walsh transform]]
* [[Wavelet]]
* [[Chirplet]]
* [[Signal (electrical engineering)|Signal]]
* [[Haar-like feature]] 
* [[Strömberg wavelet]]
* [[Dyadic transformation]]

==Notes==
{{Reflist}}

==References==
*{{citation
 | last = Haar |first = Alfréd | author-link = Alfréd Haar
 | title = Zur Theorie der orthogonalen Funktionensysteme
 | journal = [[Mathematische Annalen]] 
 | volume = 69 
 | year = 1910 
 | issue = 3 
 | pages = 331–371 
 | doi=10.1007/BF01456326 | hdl = 2027/uc1.b2619563|s2cid = 120024038 | hdl-access = free}}
* Charles K. Chui, ''An Introduction to Wavelets'', (1992), Academic Press, San Diego, {{ISBN|0-585-47090-1}}
* English Translation of Haar's seminal article: [https://web.archive.org/web/20120123124430/https://www.uni-hohenheim.de/~gzim/Publications/haar.pdf]

==External links==
{{Commons category}}

* {{springer|title=Haar system|id=p/h046070}}
* [https://web.archive.org/web/20071109204233/http://www.tomgibara.com/computer-vision/haar-wavelet Free Haar wavelet filtering implementation and interactive demo]
* [http://packages.debian.org/wzip Free Haar wavelet denoising and lossy signal compression]

===Haar transform===
* {{cite web|url=http://cnx.org/content/m11087/latest/|title=The Haar Transform|archive-url=https://web.archive.org/web/20060419183057/http://cnx.org/content/m11087/latest/|archive-date=19 April 2006|first=Nick|last=Kingsbury}}
* {{cite web|url=http://math.hws.edu/eck/math371/applets/Haar.html|title=Haar Transform Demo Applets|first=David|last=Eck|date=31 January 2006}}
* {{cite web|url=http://online.redwoods.cc.ca.us/instruct/darnold/laproj/Fall2002/ames/paper.pdf|title=Image Compression|archive-url=https://web.archive.org/web/20110125080404/http://online.redwoods.cc.ca.us/instruct/darnold/laproj/Fall2002/ames/paper.pdf|archive-date=25 January 2011|first=Greg|last=Ames|date=7 December 2002}}
* {{cite web|url=http://scien.stanford.edu/class/ee368/projects2000/project12/2.html|title=MOSMAT 500. A photomosaic generator. 2. Theory|archive-url=https://web.archive.org/web/20080318071618/http://scien.stanford.edu/class/ee368/projects2000/project12/2.html|archive-date=18 March 2008|first1=Anne|last1=Aaron|first2=Michael|last2=Hill|first3=Anand|last3=Srivatsa|date=}}
* {{cite web|url=http://fourier.eng.hmc.edu/e161/lectures/Haar/index.html|title=Haar Transform|archive-url=https://web.archive.org/web/20120821004423/http://fourier.eng.hmc.edu/e161/lectures/Haar/index.html |archive-date=21 August 2012 |first1=Ruye|last=Wang|date=4 December 2008}}

{{DEFAULTSORT:Haar Wavelet}}
[[Category:Orthogonal wavelets]]