Beta wavelet

The beta distribution is a continuous probability distribution defined over the interval ${\displaystyle 0\leq t\leq 1}$. It is characterised by a couple of parameters, namely ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ according to:

${\displaystyle P(t)={\frac {1}{B(\alpha ,\beta )}}t^{\alpha -1}\cdot (1-t)^{\beta -1},\quad 1\leq \alpha ,\beta \leq +\infty }$.

The normalising factor is ${\displaystyle B(\alpha ,\beta )={\frac {\Gamma (\alpha )\cdot \Gamma (\beta )}{\Gamma (\alpha +\beta )}}}$,

where ${\displaystyle \Gamma (\cdot )}$ is the generalised factorial function of Euler and ${\displaystyle B(\cdot ,\cdot )}$ is the Beta function.[3]

Let ${\displaystyle p_{i}(t)}$ be a probability density of the random variable ${\displaystyle t_{i}}$, ${\displaystyle i=1,2,3..N}$ i.e.

${\displaystyle p_{i}(t)\geq 0}$, ${\displaystyle (\forall t)}$ and ${\displaystyle \int _{-\infty }^{+\infty }p_{i}(t)dt=1}$.

Suppose that all variables are independent.

The mean and the variance of a given random variable ${\displaystyle t_{i}}$ are, respectively

${\displaystyle m_{i}=\int _{-\infty }^{+\infty }\tau \cdot p_{i}(\tau )d\tau ,}$ ${\displaystyle \sigma _{i}^{2}=\int _{-\infty }^{+\infty }(\tau -m_{i})^{2}\cdot p_{i}(\tau )d\tau }$.

The mean and variance of ${\displaystyle t}$ are therefore ${\displaystyle m=\sum _{i=1}^{N}m_{i}}$ and ${\displaystyle \sigma ^{2}=\sum _{i=1}^{N}\sigma _{i}^{2}}$.

The density ${\displaystyle p(t)}$ of the random variable corresponding to the sum ${\displaystyle t=\sum _{i=1}^{N}t_{i}}$ is given by the

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

Let ${\displaystyle \{p_{i}(t)\}}$ be distributions such that ${\displaystyle Supp\{(p_{i}(t))\}=(a_{i},b_{i})(\forall i)}$.

Let ${\displaystyle a=\sum _{i=1}^{N}a_{i}<+\infty }$, and ${\displaystyle b=\sum _{i=1}^{N}b_{i}<+\infty }$.

Without loss of generality assume that ${\displaystyle a=0}$ and ${\displaystyle b=1}$.

The random variable ${\displaystyle t}$ holds, as ${\displaystyle N\rightarrow \infty }$, ${\displaystyle p(t)\approx }$ ${\displaystyle {\begin{cases}{k\cdot t^{\alpha }(1-t)^{\beta }},\\otherwise\end{cases}}}$

where ${\displaystyle \alpha ={\frac {m(m-m^{2}-\sigma ^{2})}{\sigma ^{2}}},}$ and ${\displaystyle \beta ={\frac {(1-m)(\alpha +1)}{m}}.}$

Since ${\displaystyle P(\cdot |\alpha ,\beta )}$ is unimodal, the wavelet generated by

${\displaystyle \psi _{beta}(t|\alpha ,\beta )=(-1){\frac {dP(t|\alpha ,\beta )}{dt}}}$ has only one-cycle (a negative half-cycle and a positive half-cycle).

The main features of beta wavelets of parameters ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are:

${\displaystyle Supp(\psi )=[-{\sqrt {\frac {\alpha }{\beta }}}{\sqrt {\alpha +\beta +1}},{\sqrt {\frac {\beta }{\alpha }}}{\sqrt {\alpha +\beta +1}}]=[a,b].}$

${\displaystyle lengthSupp(\psi )=T(\alpha ,\beta )=(\alpha +\beta ){\sqrt {\frac {\alpha +\beta +1}{\alpha \beta }}}.}$

The parameter ${\displaystyle R=b/|a|=\beta /\alpha }$ 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 ${\displaystyle t_{zerocross}}$ from the first to the second half cycle is given by

${\displaystyle t_{zerocross}={\frac {(\alpha -\beta )}{(\alpha +\beta -2)}}{\sqrt {\frac {\alpha +\beta +1}{\alpha \beta }}}.}$

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

${\displaystyle \phi _{beta}(t|\alpha ,\beta )={\frac {1}{B(\alpha ,\beta )T^{\alpha +\beta -1}}}\cdot (t-a)^{\alpha -1}\cdot (b-t)^{\beta -1},}$ ${\displaystyle a\leq t\leq b}$.

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

${\displaystyle \psi _{beta}(t|\alpha ,\beta )={\frac {-1}{B(\alpha ,\beta )T^{\alpha +\beta -1}}}\cdot [{\frac {\alpha -1}{t-a}}-{\frac {\beta -1}{b-t}}]\cdot (t-a)^{\alpha -1}\cdot (b-t)^{\beta -1}}$

Figure. Unicyclic beta scale function and wavelet for different parameters: a) ${\displaystyle \alpha =4}$, ${\displaystyle \beta =3}$ b) ${\displaystyle \alpha =3}$, ${\displaystyle \beta =7}$ c) ${\displaystyle \alpha =5}$, ${\displaystyle \beta =17}$.

The beta wavelet spectrum can be derived in terms of the Kummer hypergeometric function.[4]

Let ${\displaystyle \psi _{beta}(t|\alpha ,\beta )\leftrightarrow \Psi _{BETA}(\omega |\alpha ,\beta )}$ denote the Fourier transform pair associated with the wavelet.

This spectrum is also denoted by ${\displaystyle \Psi _{BETA}(\omega )}$ for short. It can be proved by applying properties of the Fourier transform that

${\displaystyle \Psi _{BETA}(\omega )=-j\omega \cdot M(\alpha ,\alpha +\beta ,-j\omega (\alpha +\beta ){\sqrt {\frac {\alpha +\beta +1}{\alpha \beta }}})\cdot exp\{(j\omega {\sqrt {\frac {\alpha (\alpha +\beta +1)}{\beta }}})\}}$

where ${\displaystyle M(\alpha ,\alpha +\beta ,j\nu )={\frac {\Gamma (\alpha +\beta )}{\Gamma (\alpha )\cdot \Gamma (\beta )}}\cdot \int _{0}^{1}e^{j\nu t}t^{\alpha -1}(1-t)^{\beta -1}dt}$.

Only symmetrical ${\displaystyle (\alpha =\beta )}$ cases have zeroes in the spectrum. A few asymmetric ${\displaystyle (\alpha \neq \beta )}$ beta wavelets are shown in Fig. Inquisitively, they are parameter-symmetrical in the sense that they hold ${\displaystyle |\Psi _{BETA}(\omega |\alpha ,\beta )|=|\Psi _{BETA}(\omega |\beta ,\alpha )|.}$

Higher derivatives may also generate further beta wavelets. Higher order beta wavelets are defined by ${\displaystyle \psi _{beta}(t|\alpha ,\beta )=(-1)^{N}{\frac {d^{N}P(t|\alpha ,\beta )}{dt^{N}}}.}$

This is henceforth referred to as an ${\displaystyle N}$-order beta wavelet. They exist for order ${\displaystyle N\leq Min(\alpha ,\beta )-1}$. After some algebraic handling, their closed-form expression can be found:

${\displaystyle \Psi _{beta}(t|\alpha ,\beta )={\frac {(-1)^{N}}{B(\alpha ,\beta )\cdot T^{\alpha +\beta -1}}}\sum _{n=0}^{N}sgn(2n-N)\cdot {\frac {\Gamma (\alpha )}{\Gamma (\alpha -(N-n))}}(t-a)^{\alpha -1-(N-n)}\cdot {\frac {\Gamma (\beta )}{\Gamma (\beta -n)}}(b-t)^{\beta -1-n}.}$

Figure. Magnitude of the spectrum ${\displaystyle \Psi _{BETA}(\omega )}$ of beta wavelets, ${\displaystyle |\Psi _{BETA}(\omega \alpha ,\beta )|}$ ${\displaystyle \times \omega }$ for Symmetric beta wavelet ${\displaystyle \alpha =\beta =3}$, ${\displaystyle \alpha =\beta =4}$, ${\displaystyle \alpha =\beta =5}$

Figure. Magnitude of the spectrum ${\displaystyle \Psi _{BETA}(\omega )}$ of beta wavelets, ${\displaystyle |\Psi _{BETA}(\omega \alpha ,\beta )|}$ ${\displaystyle \times \omega }$ for: Asymmetric beta wavelet ${\displaystyle \alpha =3}$, ${\displaystyle \beta =4}$, ${\displaystyle \alpha =3}$, ${\displaystyle \beta =5}$.

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][5] and its derivative are utilized in several real-time engineering applications such as image compression,[5] bio-medical signal compression,[6][7] image recognition [9][8] etc.

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

• 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