Beta wavelet

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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 wavelets whose shape is fine-tuned by two parameters and . 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].

Beta distribution[edit]

The beta distribution is a continuous probability distribution defined over the interval . It is characterised by a couple of parameters, namely and according to:

.

The normalising factor is ,

where is the generalised factorial function of Euler and is the Beta function [4].

Gnedenko-Kolmogorov central limit theorem revisited[edit]

Let be a probability density of the random variable , i.e.

, and .

Suppose that all variables are independent.

The mean and the variance of a given random variable are, respectively

.

The mean and variance of are therefore and .

The density of the random variable corresponding to the sum is given by the

Central Limit Theorem for distributions of compact support (Gnedenko and Kolmogorov) [2].

Let be distributions such that .

Let , and .

Without loss of generality assume that and .

The random variable holds, as ,

where and

Beta wavelets[edit]

Since is unimodal, the wavelet generated by

has only one-cycle (a negative half-cycle and a positive half-cycle).

The main features of beta wavelets of parameters and are:

The parameter 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 from the first to the second half cycle is given by

The (unimodal) scale function associated with the wavelets is given by

.

A closed-form expression for first-order beta wavelets can easily be derived. Within their support,

Figure. Unicyclic beta scale function and wavelet for different parameters: a) , b) , c) , .

Beta wavelet spectrum[edit]

The beta wavelet spectrum can be derived in terms of the Kummer hypergeometric function [5].

Let denote the Fourier transform pair associated with the wavelet.

This spectrum is also denoted by for short. It can be proved by applying properties of the Fourier transform that

where .

Only symmetrical cases have zeroes in the spectrum. A few asymmetric beta wavelets are shown in Fig. Inquisitively, they are parameter-symmetrical in the sense that they hold

Higher derivatives may also generate further beta wavelets. Higher order beta wavelets are defined by

This is henceforth referred to as an -order beta wavelet. They exist for order . After some algebraic handling, their closed-form expression can be found:

Figure. Magnitude of the spectrum of beta wavelets, for Symmetric beta wavelet , ,
Figure. Magnitude of the spectrum of beta wavelets, for: Asymmetric beta wavelet , , , .

Application[edit]

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.

References[edit]

  • [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.
  • http://www.iecom.org.br/
  • http://www2.ee.ufpe.br/codec/WEBLET.html
  • http://www2.ee.ufpe.br/codec/beta.html
  • [2] B.V. Gnedenko and A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Reading, Ma: Addison-Wesley, 1954.
  • [3] W.B. Davenport, Probability and Random Processes, McGraw-Hill /Kogakusha, Tokyo, 1970.
  • [4] P.J. Davies, Gamma Function and Related Functions, in: M. Abramowitz; I. Segun (Eds.), Handbook of Mathematical Functions, New York: Dover, 1968.
  • [5] L.J. Slater, Confluent Hypergeometric Function, in: M. Abramowitz; I. Segun (Eds.), Handbook of Mathematical Functions, New York: Dover, 1968.
  • [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
  • [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.
  • [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.
  • [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