Continuous wavelets of compact support alpha can be built, 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. 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.

Beta distribution

The beta distribution is a continuous probability distribution according to:

.

,

is the generalised factorial function of Euler is the Beta function.

Gnedenko-Kolmogorov central limit theorem revisited

i.e.

.

Suppose that all variables are independent.

are, respectively

.

.

of the random variable is given by the

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

.

.

Without loss of generality.

Beta wavelets

is unimodal, the wavelet generated by

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

are:

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,

Beta wavelet spectrum

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

denote the Fourier transform pair associated with the wavelet.

for short. It can be proved by applying properties of the Fourier transform that

.

. After some algebraic handling, their closed-form expression can be found:

Application

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 and its derivative are utilized in several real-time engineering applications such as image compression, bio-medical signal compression, image recognition [9] etc.

References

Further reading

• W.B. Davenport, Probability and Random Processes, McGraw-Hill, Kogakusha, Tokyo, 1970.

External links

• https://jcis.sbrt.org.br/jcis/issue/view/27

• http://www.de.ufpe.br/~hmo/WEBLET.html

• http://www.de.ufpe.br/~hmo/beta.html