Beta wavelet

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

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

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

where $\backslash Gamma\; (\backslash cdot\; )$ is the generalised factorial function of Euler and $B(\backslash cdot\; ,\backslash cdot\; )$ is the Beta function.{{Cite book|title=Handbook of Mathematical Functions|editor-last=Abramowitz|editor-first=Milton|publisher=Dover|year=1968|isbn=0-486-61272-4|location=New York|pages=253–294|editor-last2=Stegun|editor-first2=Irene|last=Davis|first=Philip J.|chapter=Gamma Function and Related Functions|chapter-url=http://people.math.sfu.ca/~cbm/aands/page_253.htm|editor-link1=Milton Abramowitz|editor-link2=Irene Stegun|author-link=Philip J. Davis}}

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

$p\_\{i\}(t)\backslash ge\; 0$, $(\backslash forall\; t)$ and $\backslash int\_\{-\backslash infty\; \}^\{+\backslash infty\; \}p\_\{i\}(t)dt=1$.

Suppose that all variables are independent.

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

$m\_\{i\}=\backslash int\_\{-\backslash infty\; \}^\{+\backslash infty\; \}\backslash tau\; \backslash cdot\; p\_\{i\}(\backslash tau\; )d\backslash tau\; ,$ $\backslash sigma\; \_\{i\}^\{2\}=\backslash int\_\{-\backslash infty\; \}^\{+\backslash infty\; \}(\backslash tau\; -m\_\{i\})^\{2\}\backslash cdot\; p\_\{i\}(\backslash tau\; )d\backslash tau$.

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

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

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

Let $\backslash \{p\_\{i\}(t)\backslash \}$ be distributions such that $Supp\backslash \{(p\_\{i\}(t))\backslash \}=(a\_\{i\},b\_\{i\})(\backslash forall\; i)$.

Let , and .

Without loss of generality assume that $a=0$ and $b=1$.

The random variable $t$ holds, as $N\backslash rightarrow\; \backslash infty$,

$p(t)\backslash approx$ $\backslash begin\{cases\}\; \{k\; \backslash cdot\; t^\{\backslash alpha\; \}(1-t)^\{\backslash beta\}\},\; \backslash \backslash otherwise\; \backslash end\{cases\}$

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

Since $P(\backslash cdot\; |\backslash alpha\; ,\backslash beta\; )$ is unimodal, the wavelet generated by

$\backslash psi\; \_\{beta\}(t|\backslash alpha\; ,\backslash beta\; )=(-1)\backslash frac\{dP(t|\backslash alpha\; ,\backslash beta\; )\}\{dt\}$

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

The main features of beta wavelets of parameters $\backslash alpha$ and $\backslash beta$ are:

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

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

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

$t\_\{zerocross\}=\backslash frac\{(\backslash alpha\; -\backslash beta\; )\}\{(\backslash alpha\; +\backslash beta\; -2)\}\backslash sqrt\{\backslash frac\{\backslash alpha\; +\backslash beta\; +1\}\{\backslash alpha\; \backslash beta\; \}\}.$

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

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

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

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

Image:Beta scale and wavelet.jpg

The beta wavelet spectrum can be derived in terms of the Kummer hypergeometric function.{{Cite book|title=Handbook of Mathematical Functions|last=Slater|first=Lucy Joan|publisher=Dover|year=1968|isbn=0-486-61272-4|location=New York|pages=503–536|chapter-url=http://people.math.sfu.ca/~cbm/aands/page_503.htm|editor-last=Abramowitz|editor-first=Milton|editor-last2=Stegun|editor-first2=Irene|chapter=Confluent Hypergeometric Function|editor-link1=Milton Abramowitz|editor-link2=Irene Stegun|author-link=Lucy Joan Slater}}

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

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

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

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

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

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

$\backslash psi\; \_\{beta\}(t|\backslash alpha\; ,\backslash beta\; )=(-1)^\{N\}\backslash frac\{d^\{N\}P(t|\backslash alpha\; ,\backslash beta\; )\}\{dt^\{N\}\}.$

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

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

Image:Fig1a.jpg

Image:Fig1b.jpg

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{{Cite journal|title=Beta wavelets. Synthesis and application to lossy image compression|journal=Advances in Engineering Software|url=https://www.researchgate.net/publication/220291306|last1=Ben Amar|first1=Chokri|volume=36|pages=459–474|last2=Zaied|first2=Mourad|publisher=Elsevier|issue=7|year=2005|last3=Alimi|first3=Adel M.|doi= 10.1016/j.advengsoft.2005.01.013}} and its derivative are utilized in several real-time engineering applications such as image compression, bio-medical signal compression,{{Cite journal|title=Electrocardiogram Signal compression Using Beta Wavelets|journal=Journal of Mathematical Modelling and Algorithms|url=https://www.researchgate.net/publication/257581974|last1=Kumar|first1=Ranjeet|volume=11|last2=Kumar|first2=Anil|publisher=Springer Verlag|issue=3|doi=10.1007/s10852-012-9181-9|year=2012|last3=Pandey|first3=Rajesh K.|pages=235–248|s2cid=4667379}}{{Cite journal|title=Beta wavelet based ECG signal compression using lossless encoding with modified thresholding|journal=Computers & Electrical Engineering|url=https://www.researchgate.net/publication/256918193|last1=Kumar|first1=Ranjeet|volume=39|pages=130–140|last2=Kumar|first2=Anil|publisher=Elsevier|issue=1|doi=10.1016/j.compeleceng.2012.04.008|year=2013|last3=Pandey|first3=Rajesh K.}} image recognition [9]{{Cite book|chapter=Training of the Beta wavelet networks by the frames theory: Application to face recognition|title=2008 First Workshops on Image Processing Theory, Tools and Applications|chapter-url=https://ieeexplore.ieee.org/document/4743756|last1=Zaied|first1=Mourad|last2=Jemai|first2=Olfa|publisher=IEEE|doi=10.1109/IPTA.2008.4743756|year=2008|issn=2154-5111|eissn=2154-512X|last3=Ben Amar|first3=Chokri|pages=1–6|isbn=978-1-4244-3321-6|s2cid=12230926}} etc.

{{Reflist}}

• 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

Category:Continuous wavelets