Haar wavelet: Difference between revisions
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Introducing Schauder and Franklin systems, close relatives of the Haar system on [0, 1] |
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== Haar system on [0, 1] == |
== Haar system on [0, 1] and related systems == |
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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 this section, the discussion is restricted to the [[unit interval]] [0, 1] and to the Haar functions that are supported on [0, 1]. |
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In [[functional analysis]], the '''Haar system on [0, 1]''' consists of the subset of Haar wavelets defined as |
In [[functional analysis]], the '''Haar system on [0, 1]''' consists of the subset of Haar wavelets defined as |
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This is an orthogonal system but it is not complete.<ref>{{cite web |url=http://eom.springer.de/O/o070380.htm |title=Orthogonal system |work=Encyclopaedia of Mathematics }}</ref><ref>{{cite book |first=Gilbert G. |last=Walter |first2=Xiaoping |last=Shen |title=Wavelets and Other Orthogonal Systems |year=2001 |location=Boca Raton |publisher=Chapman |isbn=1-58488-227-1 }}</ref> |
This is an orthogonal system but it is not complete.<ref>{{cite web |url=http://eom.springer.de/O/o070380.htm |title=Orthogonal system |work=Encyclopaedia of Mathematics }}</ref><ref>{{cite book |first=Gilbert G. |last=Walter |first2=Xiaoping |last=Shen |title=Wavelets and Other Orthogonal Systems |year=2001 |location=Boca Raton |publisher=Chapman |isbn=1-58488-227-1 }}</ref> |
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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]] 0. The [[Khintchine inequality]] expresses the fact that in all the spaces ''L''<sup>''p''</sup>([0, 1]), {{nowrap|1 ≤ ''p'' < ∞}}, this sequence is [[Schauder basis#Definition|equivalent]] to the unit vector basis in ℓ<sup>''2''</sup>.<ref>see for example p. 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, 1]), {{nowrap|1 ≤ ''p'' < ∞}}, is [[Banach space#Linear operators, isomorphisms|isomorphic]] to ℓ<sup>''2''</sup>. |
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]] 0. The [[Khintchine inequality]] expresses the fact that in all the spaces ''L''<sup>''p''</sup>([0, 1]), {{nowrap|1 ≤ ''p'' < ∞}}, this sequence is [[Schauder basis#Definition|equivalent]] to the unit vector basis in ℓ<sup>''2''</sup>.<ref>see for example p. 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, 1]), {{nowrap|1 ≤ ''p'' < ∞}}, is [[Banach space#Linear operators, isomorphisms|isomorphic]] to ℓ<sup>''2''</sup>. |
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=== The Schauder system and the Franklin system === |
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The '''Schauder system''' is the family of continuous functions on [0, 1] consisting of the constant function '''1''', and of multiples of [[Antiderivative|indefinite integrals]] of the functions in the Haar system on [0, 1], chosen to have norm 1 in the [[Uniform norm|maximum norm]]. This system begins with ''s''<sub>0</sub> = '''1''', then {{nowrap| ''s''<sub>1</sub>(''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''<sub>''n'', ''k''</sub>}} are defined by the formula |
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:<math> |
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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> |
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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 1 at the midpoint {{nowrap| ''x''<sub>''n'', ''k''</sub>}} of the interval {{nowrap| ''I''<sub>''n'', ''k''</sub>}}, linear on both halves of that interval. It takes values between 0 and 1 everywhere. |
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The Schauder system is a Schauder basis for the space ''C''([0, 1]) of continuous functions on [0, 1].<ref>see p. 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> |
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For every ''f'' in ''C''([0, 1]), the partial sum |
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:<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> |
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of the [[series expansion]] of ''f'' in the Schauder system is the continuous piecewise linear function that agrees with ''f'' at the {{nowrap|2<sup>''n''</sup> + 1}} points {{nowrap|''k'' 2<sup>−''n''</sup>}}, where {{nowrap| 0 ≤ ''k'' ≤ 2<sup>''n''</sup>}}. Next, the formula |
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:<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> |
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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'', ''i.e.'', the sum of the Schauder series expansion of ''f'' is equal to ''f''. |
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The '''Franklin system''' is obtained from the Schauder system by the [[Gram–Schmidt process|Gram–Schmidt orthonormalization procedure]].<ref>see Z. Ciesielski, ''Properties of the orthonormal Franklin system''. Studia Math. 23 1963 141–157.</ref> |
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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 ''L''<sup>2</sup>([0, 1]). The Franklin system is therefore an orthonormal basis for ''L''<sup>2</sup>([0, 1]), consisting of continuous piecewise linear functions. P. Franklin proved in 1928 that this system is a Schauder basis for ''C''([0, 1]).<ref>Philip Franklin, ''A set of continuous orthogonal functions'', Math. Ann. 100 (1928), 522-529.</ref> |
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The Franklin system is also an unconditional basis for the space ''L''<sup>''p''</sup>([0, 1]) when {{nowrap|1 < ''p'' < ∞}}.<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> |
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It is a Schauder basis in the [[disk algebra]] ''A''(''D''):<ref name=Bo></ref> |
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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. |
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==Haar matrix== |
==Haar matrix== |
Revision as of 07:42, 12 May 2013
This article provides insufficient context for those unfamiliar with the subject.(October 2009) |
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.[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, 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.[2]
The Haar wavelet's mother wavelet function can be described as
Its scaling function can be described as
Haar functions and Haar system
For every pair n, k of integers in Z, the Haar function ψn, k is defined on the real line R by the formula
This function is supported on the right-open interval In, 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),
The Haar functions are pairwise orthogonal,
where δi,j represents the Kronecker delta. Here is the reason for orthogonality: when the two supporting intervals and are not equal, then they are either disjoint, or else, the smaller of the two supports, say , is contained in the lower or in the upper half of the other interval, on which the function remains constant.
The Haar system on the real line is the set of functions
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:
- Any continuous real function can be approximated by linear combinations of and their shifted functions. This extends to those function spaces where any function therein can be approximated by continuous functions.
- Any continuous real function can be approximated by linear combinations of the constant function, and their shifted functions.
- Orthogonality in the form
- Here δi,j represents the Kronecker delta. The dual function of is itself.
- 4. Wavelet/scaling functions with different scale m have a functional relationship:[3]
- 5. Coefficients of scale m can be calculated by coefficients of scale m+1:
- If
- and
- then
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
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 Lp([0, 1]) when 1 ≤ p < ∞.[4] This basis is unconditional for p > 1.[5]
There is a related Rademacher system consisting of sums of Haar functions,
This is an orthogonal system but it is not complete.[6][7] 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 Lp([0, 1]), 1 ≤ p < ∞, this sequence is equivalent to the unit vector basis in ℓ2.[8] In particular, the closed linear span of the Rademacher sequence in Lp([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 sn, k are defined by the formula
These functions sn, k are continuous, piecewise linear, supported by the interval In, k that also supports ψn, k. The function sn, k is equal to 1 at the midpoint xn, k of the interval In, 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].[9] For every f in C([0, 1]), the partial sum
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
gives a way to compute the expansion of f step by step. Since f is uniformly continuous, the sequence {fn} 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.[10] 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]).[11] The Franklin system is also an unconditional basis for the space Lp([0, 1]) when 1 < p < ∞.[12] It is a Schauder basis in the disk algebra A(D):[12] 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
Using the discrete wavelet transform, one can transform any sequence of even length into a sequence of two-component-vectors . If one right-multiplies each vector with the matrix , one gets the result 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.[13]
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.
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.[14]
The Haar transform is derived from the Haar matrix. An example of a 4x4 Haar transformation matrix is shown below.
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
Notes
- ^ Haar, Alfred (1910). "Zur Theorie der orthogonalen Funktionensysteme". Mathematische Annalen. 69 (3): 331–371. doi:10.1007/BF01456326.
- ^ Lee, B.; Tarng, Y. S. (1999). "Application of the discrete wavelet transform to the monitoring of tool failure in end milling using the spindle motor current". International Journal of Advanced Manufacturing Technology. 15 (4): 238–243. doi:10.1007/s001700050062.
- ^ Vidakovic, Brani (2010). Statistical Modeling by Wavelets (2 ed.). pp. 60, 63. doi:10.1002/9780470317020.
- ^ 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.
- ^ 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.
- ^ "Orthogonal system". Encyclopaedia of Mathematics.
- ^ Shen, Gilbert G. (2001). Wavelets and Other Orthogonal Systems. Boca Raton: Chapman. ISBN 1-58488-227-1.
{{cite book}}
:|first2=
missing|last2=
(help) - ^ 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.
- ^ 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.
- ^ see Z. Ciesielski, Properties of the orthonormal Franklin system. Studia Math. 23 1963 141–157.
- ^ Philip Franklin, A set of continuous orthogonal functions, Math. Ann. 100 (1928), 522-529.
- ^ a b 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.
- ^ Ruch, David K.; Van Fleet, Patrick J. (2009). Wavelet Theory: An Elementary Approach with Applications. John Wiley & Sons. ISBN 978-0-470-38840-2.
- ^ 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 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