Haar functions and Haar system
For every pair ''n'', ''k'' of integers in , the Haar function ''ψ''''n'',''k'' is defined on the real line by the formula : This function is supported on the right-open interval , ''i.e.'', it vanishes outside that interval. It has integral 0 and norm 1 in the Hilbert space ''L''2(), : The Haar functions are pairwise orthogonal, : where represents theHaar wavelet properties
The Haar wavelet has several notable properties: #Any continuous real function with compact support can be approximated uniformly 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 on , 1can be approximated uniformly on , 1by linear combinations of the constant function 1, and their shifted functions. # Orthogonality in the form Here, represents theHaar system on the unit interval and related systems
In this section, the discussion is restricted to theThe Faber–Schauder system
The Faber–Schauder systemFaber, Georg (1910), "Über die Orthogonalfunktionen des Herrn Haar", ''Deutsche Math.-Ver'' (in German) 19: 104–112. ; http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X ; http://resolver.sub.uni-goettingen.de/purl?GDZPPN002122553 is the family of continuous functions on , 1consisting of the constant function 1, and of multiples of indefinite integrals of the functions in the Haar system on , 1 chosen to have norm 1 in theThe Franklin system
The Franklin system is obtained from the Faber–Schauder system by the Gram–Schmidt orthonormalization procedure. Since the Franklin system has the same linear span as that of the Faber–Schauder system, this span is dense in ''C''( , 1, hence in ''L''2( , 1. The Franklin system is therefore an orthonormal basis for ''L''2( , 1, consisting of continuous piecewise linear functions. P. Franklin proved in 1928 that this system is a Schauder basis for ''C''( , 1. The Franklin system is also an unconditional Schauder basis for the space ''L''''p''( , 1 when .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. Bočkarev's construction of a Schauder basis in ''A''(''D'') goes as follows: let ''f'' be a complex valued Lipschitz function on , 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, : Bočkarev's basis for ''A''(''D'') is formed by the images under ''T'' of the functions in the Franklin system on , π Bočkarev's equivalent description for the mapping ''T'' starts by extending ''f'' to an even Lipschitz function ''g''1 on minus;π, π identified with a Lipschitz function on theHaar 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. 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. : :where and is the Kronecker product. The Kronecker product of , where is an m×n matrix and is a p×q matrix, is expressed as : An un-normalized 8-point Haar matrix 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 , one can observe that, unlike the Fourier transform, has only real elements (i.e., 1, -1 or 0) and is non-symmetric. Take the 8-point Haar matrix as an example. The first row of measures the average value, and the second row of 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.Haar transform
The Haar transform is the simplest of theIntroduction
The Haar transform is one of the oldest transform functions, proposed in 1910 by the Hungarian mathematicianProperty
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 thanHaar transform and Inverse Haar transform
The Haar transform ''y''''n'' of an n-input function ''x''''n'' is : The Haar transform matrix is real and orthogonal. Thus, the inverse Haar transform can be derived by the following equations. : : where is the identity matrix. For example, when n = 4 : Thus, the inverse Haar transform is :Example
The Haar transform coefficients of a n=4-point signal can be found as : The input signal can then be perfectly reconstructed by the inverse Haar transform :See also
*Notes
References
* * Charles K. Chui, ''An Introduction to Wavelets'', (1992), Academic Press, San Diego, * English Translation of Haar's seminal articleExternal links
*Haar transform