Haar wavelet

For every pair n, k of integers in $\backslash mathbb\{Z\}$, the Haar function ψ_n,k is defined on the real line $\backslash mathbb\{R\}$ by the formula

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

This function is supported on the right-open interval {{nowrap| I_n,k {{=}}}} {{nowrap|[ k2⁻ⁿ, (k+1)2⁻ⁿ)}}, i.e., it vanishes outside that interval. It has integral 0 and norm 1 in the Hilbert space L²($\backslash mathbb\{R\}$),

:$\backslash int\_\{\backslash mathbb\{R\}\}\; \backslash psi\_\{n,\; k\}(t)\; \backslash ,\; d\; t\; =\; 0,\; \backslash quad\; \backslash |\backslash psi\_\{n,\; k\}\backslash |^2\_\{L^2(\backslash mathbb\{R\})\}\; =\; \backslash int\_\{\backslash mathbb\{R\}\}\; \backslash psi\_\{n,\; k\}(t)^2\; \backslash ,\; d\; t\; =\; 1.$

The Haar functions are pairwise orthogonal,

:$\backslash int\_\{\backslash mathbb\{R\}\}\; \backslash psi\_\{n\_1,\; k\_1\}(t)\; \backslash psi\_\{n\_2,\; k\_2\}(t)\; \backslash ,\; d\; t\; =\; \backslash delta\_\{n\_1n\_2\}\; \backslash delta\_\{k\_1k\_2\},$

where $\backslash delta\_\{ij\}$ 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 $\backslash psi\_\{n\_2,\; k\_2\}$ 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 0.

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

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

It is complete in L²($\backslash mathbb\{R\}$): The Haar system on the line is an orthonormal basis in L²($\backslash mathbb\{R\}$).

The Haar wavelet has several notable properties:

{{ordered list

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

| Orthogonality in the form

: $$

\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}.

Here, $\backslash delta\_\{ij\}$ 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:{{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 }} since

:$$

\begin{align}

\varphi(t) &= \varphi(2t)+\varphi(2t-1)\\[.2em]

\psi(t) &= \varphi(2t)-\varphi(2t-1),

\end{align}

it follows that coefficients of scale n can be calculated by coefficients of scale n+1:

If $\backslash chi\_w(k,\; n)=\; 2^\{n/2\}\backslash int\_\{-\backslash infty\}^\backslash infty\; x(t)\backslash varphi(2^nt-k)\backslash ,\; dt$

and $\backslash Chi\_w(k,\; n)=\; 2^\{n/2\}\backslash int\_\{-\backslash infty\}^\backslash infty\; x(t)\backslash psi(2^nt-k)\backslash ,\; dt$

then

: $\backslash chi\_w(k,n)=\; 2^\{-1/2\}\; \backslash bigl(\; \backslash chi\_w(2k,n+1)+\backslash chi\_w(2k+1,n+1)\; \backslash bigr)$

: $\backslash Chi\_w(k,n)=\; 2^\{-1/2\}\; \backslash bigl(\; \backslash chi\_w(2k,n+1)-\backslash chi\_w(2k+1,n+1)\; \backslash bigr).$

}}

In this section, the discussion is restricted to the unit interval [0, 1] and to the Haar functions that are supported on [0, 1]. The system of functions considered by Haar in 1910,p. 361 in {{harvtxt|Haar|1910}}

called the Haar system on [0, 1] in this article, consists of the subset of Haar wavelets defined as

:

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

In Hilbert space terms, this Haar system on [0, 1] is a complete orthonormal system, i.e., an orthonormal basis, 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 {{nowrap|(n, k)}}— is further a monotone Schauder basis for the space L^p([0, 1]) when {{nowrap|1 ≤ p < ∞}}.see p. 3 in 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}}.

This basis is unconditional when {{nowrap|1 < p < ∞}}.The result is due to R. E. Paley, A remarkable series of orthogonal functions (I), Proc. London Math. Soc. 34 (1931) pp. 241-264. See also p. 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}}.

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

:$r\_n(t)\; =\; 2^\{-n/2\}\; \backslash sum\_\{k=0\}^\{2^n\; -\; 1\}\; \backslash psi\_\{n,\; k\}(t),\; \backslash quad\; t\; \backslash in\; [0,\; 1],\; \backslash \; n\; \backslash ge\; 0.$

Notice that |r_n(t)| = 1 on [0, 1). This is an orthonormal system but it is not complete.{{SpringerEOM |title=Orthogonal system}}{{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 }}

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]), {{nowrap|1 ≤ p < ∞}}, the Rademacher sequence is equivalent to the unit vector basis in ℓ².see for example p. 66 in 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}}. In particular, the closed linear span of the Rademacher sequence in L^p([0, 1]), {{nowrap|1 ≤ p < ∞}}, is isomorphic to ℓ².

The Faber–Schauder systemFaber, Georg (1910), "Über die Orthogonalfunktionen des Herrn Haar", Deutsche Math.-Ver (in German) 19: 104–112. {{issn|0012-0456}};

http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X ; http://resolver.sub.uni-goettingen.de/purl?GDZPPN002122553Schauder, Juliusz (1928), "Eine Eigenschaft des Haarschen Orthogonalsystems", Mathematische Zeitschrift 28: 317–320.{{eom|id=f/f038020

|title=Faber–Schauder system|first=B.I.|last= Golubov}}

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 {{nowrap| 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 {{nowrap|n ≥ 0}}, functions {{nowrap| s_n,k}} are defined by the formula

:$$

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.

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

The Faber–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

:$f\_\{n+1\}\; =\; a\_0\; s\_0\; +\; a\_1\; s\_1\; +\; \backslash sum\_\{m\; =\; 0\}^\{n-1\}\; \backslash Bigl(\; \backslash sum\_\{k=0\}^\{2^m\; -\; 1\}\; a\_\{m,k\}\; s\_\{m,\; k\}\; \backslash Bigr)\; \backslash in\; C([0,\; 1])$

of the series expansion of f in the Faber–Schauder system is the continuous piecewise linear function that agrees with f at the {{nowrap|2ⁿ + 1}} points {{nowrap|k2⁻ⁿ}}, where {{nowrap| 0 ≤ k ≤ 2ⁿ}}. Next, the formula

:$f\_\{n+2\}\; -\; f\_\{n+1\}\; =\; \backslash sum\_\{k=0\}^\{2^n\; -\; 1\}\; \backslash bigl(\; f(x\_\{n,k\})\; -\; f\_\{n+1\}(x\_\{n,\; k\})\; \backslash bigr)\; s\_\{n,\; k\}\; =\; \backslash sum\_\{k=0\}^\{2^n\; -\; 1\}\; a\_\{n,\; k\}\; s\_\{n,\; k\}$

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. It follows that the Faber–Schauder series expansion of f converges in C([0, 1]), and the sum of this series is equal to f.

The Franklin system is obtained from the Faber–Schauder system by the Gram–Schmidt orthonormalization procedure.see Z. Ciesielski, Properties of the orthonormal Franklin system. Studia Math. 23 1963 141–157.Franklin system. B.I. Golubov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Franklin_system&oldid=16655

Since the Franklin system has the same linear span as that of the Faber–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]).Philip Franklin, A set of continuous orthogonal functions, Math. Ann. 100 (1928), 522-529. {{doi|10.1007/BF01448860}}

The Franklin system is also an unconditional Schauder basis for the space L^p([0, 1]) when {{nowrap|1 < p < ∞}}.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.

The Franklin system provides 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 open for more than forty years.The question appears p. 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 10, p. 12 in Banach's book.

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

:$\backslash left\backslash \{\; f\; :\; x\; \backslash in\; [0,\; \backslash pi]\; \backslash rightarrow\; \backslash sum\_\{n=0\}^\backslash infty\; a\_n\; \backslash cos(n\; x)\; \backslash right\backslash \}\; \backslash longrightarrow\; \backslash left\backslash \{\; T(f)\; :\; z\; \backslash rightarrow\; \backslash sum\_\{n=0\}^\backslash infty\; a\_n\; z^n,\; \backslash quad\; |z|\; \backslash le\; 1\; \backslash right\backslash \}.$

Bočkarev's basis for A(D) is formed by the images under T of the functions in the Franklin system on [0, π]. Bočkarev's equivalent description for the mapping T starts by extending f to an even Lipschitz function g₁ on [−π, π], identified with a Lipschitz function on the unit circle T. Next, let g₂ be the conjugate function of g₁, and define T(f) to be the function in A(D) whose value on the boundary T of D is equal to {{nowrap|g₁ + ig₂}}.

When dealing with 1-periodic continuous functions, or rather with continuous functions f on [0, 1] such that {{nowrap|f(0) {{=}} f(1)}}, one removes the function {{nowrap| s₁(t) {{=}} t}} from the Faber–Schauder system, in order to obtain the periodic Faber–Schauder system. The periodic Franklin system is obtained by orthonormalization from the periodic Faber–-Schauder system.See p. 161, III.D.20 and p. 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

}}

One can prove Bočkarev's result on A(D) by proving that the periodic Franklin system on [0, 2π] is a basis for a Banach space A_r isomorphic to A(D).

The space A_r consists of complex continuous functions on the unit circle T whose conjugate function is also continuous.

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

:

Using the discrete wavelet transform, one can transform any sequence $(a\_0,a\_1,\backslash dots,a\_\{2n\},a\_\{2n+1\})$ of even length into a sequence of two-component-vectors $\backslash left(\backslash left(a\_0,a\_1\backslash right),\backslash left(a\_2,a\_3\backslash right),\backslash dots,\backslash left(a\_\{2n\},a\_\{2n+1\}\backslash right)\backslash right)$. If one right-multiplies each vector with the matrix $H\_2$, one gets the result $\backslash left(\backslash left(s\_0,d\_0\backslash right),\backslash dots,\backslash left(s\_n,d\_n\backslash right)\backslash right)$ 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.{{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 }}

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

:

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.

: $H\_\{2N\}\; =\; \backslash begin\{bmatrix\}\; H\_\{N\}\; \backslash otimes\; [1,\; 1]\; \backslash \backslash \; I\_\{N\}\; \backslash otimes\; [1,\; -1]\; \backslash end\{bmatrix\}$

:where and $\backslash otimes$ is the Kronecker product.

The Kronecker product of $A\; \backslash otimes\; B$, where $A$ is an m×n matrix and $B$ is a p×q matrix, is expressed as

:

An un-normalized 8-point Haar matrix $H\_8$ is shown below

:

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 $H$, one can observe that, unlike the Fourier transform, $H$ has only real elements (i.e., 1, -1 or 0) and is non-symmetric.

Take the 8-point Haar matrix $H\_8$ as an example. The first row of $H\_8$ measures the average value, and the second row of $H\_8$ 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.{{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 }}

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.[http://sepwww.stanford.edu/public/docs/sep75/ray2/paper_html/node4.html The Haar Transform]{{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}}

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.

:$H\_4\; =\; \backslash frac\{1\}\{2\}$

\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ \sqrt{2} & -\sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{2} & -\sqrt{2}\end{bmatrix}

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.

The Haar transform has the following properties

1. 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.
2. Input and output length are the same. However, the length should be a power of 2, i.e. $N\; =\; 2^k,\; k\backslash in\; \backslash mathbb\{N\}$.
3. 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.

The Haar transform y_n of an n-input function x_n is

: $y\_n\; =\; H\_n\; x\_n$

The Haar transform matrix is real and orthogonal. Thus, the inverse Haar transform can be derived by the following equations.

: $H\; =\; H^*,\; H^\{-1\}\; =\; H^T,\; \backslash text\{\; i.e.\; \}\; HH^T\; =\; I$

: where $I$ is the identity matrix. For example, when n = 4

:

\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}

Thus, the inverse Haar transform is

: $x\_\{n\}\; =\; H^\{T\}y\_\{n\}$

The Haar transform coefficients of a n=4-point signal $x\_\{4\}\; =\; [1,2,3,4]^\{T\}$ can be found as

: $y\_\{4\}\; =\; H\_4\; x\_4\; =$

\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}

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

: $\backslash hat\{x\_\{4\}\}\; =\; H\_\{4\}^\{T\}y\_\{4\}\; =$

\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}

• Dimension reduction
• Walsh matrix
• Walsh transform
• Wavelet
• Chirplet
• Signal
• Haar-like feature
• Strömberg wavelet
• Dyadic transformation

{{Reflist}}

• {{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]

{{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]

• {{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