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| [[Continuous wavelets]] of [[compact support]] can be built [1], which are related to the [[beta distribution]]. The process is derived from probability distributions using blur derivative. These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a ''soft variety'' of [[Haar wavelet]]s whose shape is fine-tuned by two parameters <math>\alpha</math> and <math>\beta</math>. Closed-form expressions for beta wavelets and scale functions as well as their spectra are derived. Their importance is due to the [[Central Limit Theorem]] by Gnedenko and Kolmogorov applied for compactly supported signals [2].
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| == Beta distribution ==
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| The [[beta distribution]] is a continuous probability distribution defined over the interval <math>0\leq t\leq 1</math>. It is characterised by a couple of parameters, namely <math>\alpha</math> and <math>\beta</math> according to:
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| <math>P(t)=\frac{1}{B(\alpha ,\beta )}t^{\alpha -1}\cdot (1-t)^{\beta -1},\quad 1\leq \alpha ,\beta \leq +\infty </math>.
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| The normalising factor is <math>B(\alpha ,\beta )=\frac{\Gamma (\alpha )\cdot \Gamma (\beta )}{\Gamma (\alpha +\beta )}</math>,
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| where <math> \Gamma (\cdot )</math> is the generalised factorial function of Euler and <math>B(\cdot ,\cdot )</math> is the Beta function [4].
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| == Gnedenko-Kolmogorov central limit theorem revisited ==
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| Let <math>p_{i}(t)</math> be a probability density of the random variable <math>t_{i}</math>, <math>i=1,2,3..N</math> i.e.
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| <math>p_{i}(t)\ge 0</math>, <math>(\forall t)</math> and <math>\int_{-\infty }^{+\infty }p_{i}(t)dt=1</math>.
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| Suppose that all variables are independent.
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| The mean and the variance of a given random variable <math>t_{i}</math> are, respectively
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| <math>m_{i}=\int_{-\infty }^{+\infty }\tau \cdot p_{i}(\tau )d\tau ,</math> <math>\sigma _{i}^{2}=\int_{-\infty }^{+\infty }(\tau -m_{i})^{2}\cdot p_{i}(\tau )d\tau </math>.
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| The mean and variance of <math>t</math> are therefore <math>m=\sum_{i=1}^{N}m_{i}</math> and <math>\sigma^2 =\sum_{i=1}^{N}\sigma _{i}^{2}</math>.
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| The density <math>p(t)</math> of the random variable corresponding to the sum <math>t=\sum_{i=1}^{N}t_{i}</math> is given by the
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| '''Central Limit Theorem for distributions of compact support (Gnedenko and Kolmogorov) [2].'''
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| Let <math>\{p_{i}(t)\}</math> be distributions such that <math>Supp\{(p_{i}(t))\}=(a_{i},b_{i})(\forall i)</math>.
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| Let <math>a=\sum_{i=1}^{N}a_{i}<+\infty </math>, and <math>b=\sum_{i=1}^{N}b_{i}<+\infty</math>.
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| Without loss of generality assume that <math>a=0</math> and <math>b=1</math>.
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| The random variable <math>t</math> holds, as <math>N\rightarrow \infty </math>,
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| <math>p(t)\approx </math> <math>\begin{cases} {k \cdot t^{\alpha }(1-t)^{\beta}}, \\otherwise \end{cases}</math>
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| where <math>\alpha =\frac{m(m-m^{2}-\sigma ^{2})}{\sigma ^{2}},</math> and <math>\beta =\frac{(1-m)(\alpha +1)}{m}.</math>
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| == Beta wavelets ==
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| Since <math>P(\cdot |\alpha ,\beta )</math> is unimodal, the wavelet generated by
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| <math>\psi _{beta}(t|\alpha ,\beta )=(-1)\frac{dP(t|\alpha ,\beta )}{dt}</math>
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| has only one-cycle (a negative half-cycle and a positive half-cycle).
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| The main features of beta wavelets of parameters <math>\alpha</math> and <math>\beta</math> are:
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| <math>Supp(\psi )=[ -\sqrt{\frac{\alpha}{\beta}}\sqrt{\alpha + \beta +1},\sqrt{ \frac{\beta }{\alpha }} \sqrt{\alpha +\beta +1}]=[a,b].</math>
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| <math>lengthSupp(\psi )=T(\alpha ,\beta )=(\alpha +\beta )\sqrt{\frac{\alpha +\beta +1}{\alpha \beta }}.</math>
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| The parameter <math>R=b/|a| =\beta / \alpha</math> is referred to as “cyclic balance”, and is defined as the ratio between the lengths of the causal and non-causal piece of the wavelet. The instant of transition <math>t_{zerocross}</math> from the first to the second half cycle is given by
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| <math>t_{zerocross}=\frac{(\alpha -\beta )}{(\alpha +\beta -2)}\sqrt{\frac{\alpha +\beta +1}{\alpha \beta }}.</math>
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| The (unimodal) scale function associated with the wavelets is given by
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| <math>\phi _{beta}(t|\alpha ,\beta )=\frac{1}{B(\alpha ,\beta )T^{\alpha +\beta -1}}\cdot (t-a)^{\alpha -1}\cdot (b-t)^{\beta -1},</math> <math>a\leq t\leq b </math>.
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| A closed-form expression for first-order beta wavelets can easily be derived. Within their support,
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| <math>\psi_{beta}(t|\alpha ,\beta ) =\frac{-1}{B(\alpha ,\beta )T^{\alpha +\beta -1}} \cdot [\frac{\alpha -1}{t-a}-\frac{\beta -1}{b-t}] \cdot(t-a)^{\alpha -1} \cdot(b-t)^{\beta -1}</math>
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| [[Image:Beta scale and wavelet.jpg|frame|right|'''Figure. Unicyclic beta scale function and wavelet for different parameters: a) <math>\alpha =4</math>, <math>\beta =3</math> b) <math>\alpha =3</math>, <math>\beta =7</math> c) <math>\alpha =5</math>, <math>\beta =17</math>.''']]
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| == Beta wavelet spectrum ==
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| The beta wavelet spectrum can be derived in terms of the Kummer hypergeometric function [5].
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| Let <math>\psi _{beta}(t|\alpha ,\beta )\leftrightarrow \Psi _{BETA}(\omega |\alpha ,\beta )</math> denote the Fourier transform pair associated with the wavelet.
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| This spectrum is also denoted by <math>\Psi _{BETA}(\omega)</math> for short. It can be proved by applying properties of the Fourier transform that
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| <math>\Psi _{BETA}(\omega ) =-j\omega \cdot M(\alpha ,\alpha +\beta ,-j\omega (\alpha +\beta )\sqrt{\frac{\alpha +\beta +1}{\alpha \beta}})\cdot exp\{(j\omega \sqrt{\frac{\alpha (\alpha +\beta +1)}{\beta }})\}</math>
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| where <math>M(\alpha ,\alpha +\beta ,j\nu )=\frac{\Gamma (\alpha +\beta )}{\Gamma (\alpha )\cdot \Gamma (\beta )}\cdot \int_{0}^{1}e^{j\nu t}t^{\alpha -1}(1-t)^{\beta -1}dt</math>.
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| Only symmetrical <math>(\alpha =\beta )</math> cases have zeroes in the spectrum. A few asymmetric <math>(\alpha \neq \beta )</math> beta wavelets are shown in Fig. Inquisitively, they are parameter-symmetrical in the sense that they hold <math>|\Psi _{BETA}(\omega |\alpha ,\beta )|=|\Psi _{BETA}(\omega |\beta ,\alpha )|.</math>
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| Higher derivatives may also generate further beta wavelets. Higher order beta wavelets are defined by
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| <math>\psi _{beta}(t|\alpha ,\beta )=(-1)^{N}\frac{d^{N}P(t|\alpha ,\beta )}{dt^{N}}.</math>
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| This is henceforth referred to as an <math>N</math>-order beta wavelet. They exist for order <math>N\leq Min(\alpha ,\beta )-1</math>. After some algebraic handling, their closed-form expression can be found:
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| <math>\Psi _{beta}(t|\alpha ,\beta ) =\frac{(-1)^{N}}{B(\alpha ,\beta ) \cdot T^{\alpha +\beta -1}} \sum_{n=0}^{N}sgn(2n-N)\cdot \frac{\Gamma (\alpha )}{\Gamma (\alpha -(N-n))}(t-a)^{\alpha -1-(N-n)} \cdot \frac{\Gamma (\beta )}{\Gamma (\beta -n)}(b-t)^{\beta -1-n}.</math>
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| [[Image:Fig1a.jpg|40px|frame|right|'''Figure. Magnitude of the spectrum <math>\Psi _{BETA}(\omega )</math> of beta wavelets, <math>|\Psi _{BETA}(\omega \alpha ,\beta )|</math> <math>\times \omega</math> for Symmetric beta wavelet <math>\alpha = \beta = 3</math>, <math>\alpha = \beta = 4</math>, <math>\alpha = \beta = 5</math>''']]
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| [[Image:Fig1b.jpg|40px|frame|right|'''Figure. Magnitude of the spectrum <math>\Psi _{BETA}(\omega )</math> of beta wavelets, <math>|\Psi _{BETA}(\omega \alpha ,\beta )|</math> <math>\times \omega</math> for: Asymmetric beta wavelet <math>\alpha =3</math>, <math>\beta =4</math>, <math>\alpha =3</math>, <math>\beta =5</math>.''']]
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| == Application ==
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| Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of time-frequency representation for continuous-time (analog) signals and so are related to harmonic analysis. Almost all practically useful discrete wavelet transforms use discrete-time filter banks. Similarly, Beta wavelet [1][6] and its derivative are utilized in several real-time engineering applications such as image compression[6],bio-medical signal compression[7][8], image recognition [9] etc.
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| == References ==
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| * [1] H.M. de Oliveira, G.A.A. Araújo, Compactly Supported One-cyclic Wavelets Derived from Beta Distributions, ''Journal of Communication and Information Systems'', vol.20, n.3, pp. 27–33, 2005.
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| * http://www.iecom.org.br/
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| * http://www2.ee.ufpe.br/codec/WEBLET.html
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| * http://www2.ee.ufpe.br/codec/beta.html
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| * [2] B.V. Gnedenko and A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Reading, Ma: Addison-Wesley, 1954.
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| * [3] W.B. Davenport, Probability and Random Processes, McGraw-Hill /Kogakusha, Tokyo, 1970.
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| * [4] P.J. Davies, Gamma Function and Related Functions, in: M. Abramowitz; I. Segun (Eds.), Handbook of Mathematical Functions, New York: Dover, 1968.
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| * [5] L.J. Slater, Confluent Hypergeometric Function, in: M. Abramowitz; I. Segun (Eds.), Handbook of Mathematical Functions, New York: Dover, 1968.
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| * [6] B.C. Amar, M. Zaied, M.A. Alimi, "Beta wavelet synthesis and application to lossy image compression" Adv Eng Softw, 36 (2005), pp. 459–474
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| * [7] Ranjeet Kumar, A. Kumar and Rajesh K. Pandey, “Electrocardiogram Signal compression Using Beta Wavelets” Journal of Mathematical Modeling and Algorithms, Vol. 11, pp. 235-248, 2012.
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| * [8] Ranjeet Kumar , A. Kumar and Rajesh K Pandey “Beta Wavelet Based ECG Signal Compression using Loss-less Encoding with Modified Thresholding” Computers & Electrical Engineering, Vol. 39, Issue. 1, pp.130- 140, 2013.
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| * [9] Zaied, M., Jemai, O. , Ben Amar, C., "Training of the Beta wavelet networks by the frames theory: Application to face recognition", Image Processing Theory, Tools and Applications, 2008. DOI: 10.1109/IPTA.2008.4743756
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| [[Category:Continuous wavelets]]
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| [[Category:Functional analysis]]
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