Continuous wavelets of compact support alpha 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
\alpha
\beta
The beta distribution is a continuous probability distribution defined over the interval
0\leqt\leq1
\alpha
\beta
| P(t)= | 1 |
| B(\alpha,\beta) |
t\alpha ⋅ (1-t)\beta, 1\leq\alpha,\beta\leq+infty
The normalising factor is
B(\alpha,\beta)=
| \Gamma(\alpha) ⋅ \Gamma(\beta) | |
| \Gamma(\alpha+\beta) |
where
\Gamma( ⋅ )
B( ⋅ , ⋅ )
Let
pi(t)
ti
i=1,2,3..N
pi(t)\ge0
(\forallt)
| +infty | |
| \int | |
| -infty |
pi(t)dt=1
Suppose that all variables are independent.
The mean and the variance of a given random variable
ti
mi
| +infty | |
| =\int | |
| -infty |
\tau ⋅ pi(\tau)d\tau,
\sigma
| 2 | |
| i |
| +infty | |
| =\int | |
| -infty |
(\tau-mi)2 ⋅ pi(\tau)d\tau
The mean and variance of
t
| N | |
| m=\sum | |
| i=1 |
mi
\sigma2
| N | |
| =\sum | |
| i=1 |
\sigma
| 2 | |
| i |
The density
p(t)
| N | |
| t=\sum | |
| i=1 |
ti
Central Limit Theorem for distributions of compact support (Gnedenko and Kolmogorov).[2]
Let
\{pi(t)\}
Supp\{(pi(t))\}=(ai,bi)(\foralli)
Let
| N | |
| a=\sum | |
| i=1 |
ai<+infty
| N | |
| b=\sum | |
| i=1 |
bi<+infty
Without loss of generality assume that
a=0
b=1
t
N → infty
p(t) ≈
\begin{cases}{k ⋅ t\alpha(1-t)\beta
where
\alpha=
| m(m-m2-\sigma2) | |
| \sigma2 |
,
\beta=
| (1-m)(\alpha+1) | |
| m |
.
Since
P( ⋅ |\alpha,\beta)
\psibeta(t|\alpha,\beta)=(-1)
| dP(t|\alpha,\beta) | |
| dt |
The main features of beta wavelets of parameters
\alpha
\beta
Supp(\psi)=[-\sqrt{
| \alpha | |
| \beta |
lengthSupp(\psi)=T(\alpha,\beta)=(\alpha+\beta)\sqrt{
| \alpha+\beta+1 | |
| \alpha\beta |
The parameter
R=b/|a|=\beta/\alpha
tzerocross
tzerocross=
| (\alpha-\beta) | \sqrt{ | |
| (\alpha+\beta-2) |
| \alpha+\beta+1 | |
| \alpha\beta |
The (unimodal) scale function associated with the wavelets is given by
\phibeta(t|\alpha,\beta)=
| 1 | |
| B(\alpha,\beta)T\alpha |
⋅ (t-a)\alpha ⋅ (b-t)\beta,
a\leqt\leqb
A closed-form expression for first-order beta wavelets can easily be derived. Within their support,
\psibeta(t|\alpha,\beta)=
| -1 | |
| B(\alpha,\beta)T\alpha |
⋅ [
| \alpha-1 | - | |
| t-a |
| \beta-1 | |
| b-t |
] ⋅ (t-a)\alpha ⋅ (b-t)\beta
The beta wavelet spectrum can be derived in terms of the Kummer hypergeometric function.[4]
Let
\psibeta(t|\alpha,\beta)\leftrightarrow\PsiBETA(\omega|\alpha,\beta)
This spectrum is also denoted by
\PsiBETA(\omega)
\PsiBETA(\omega)=-j\omega ⋅ M(\alpha,\alpha+\beta,-j\omega(\alpha+\beta)\sqrt{
| \alpha+\beta+1 | |
| \alpha\beta |
where
M(\alpha,\alpha+\beta,j\nu)=
| \Gamma(\alpha+\beta) | |
| \Gamma(\alpha) ⋅ \Gamma(\beta) |
⋅
| 1 | |
| \int | |
| 0 |
ej\nut\alpha(1-t)\betadt
Only symmetrical
(\alpha=\beta)
(\alpha ≠ \beta)
|\PsiBETA(\omega|\alpha,\beta)|=|\PsiBETA(\omega|\beta,\alpha)|.
Higher derivatives may also generate further beta wavelets. Higher order beta wavelets are defined by
\psibeta(t|\alpha,\beta)=(-1)N
| dNP(t|\alpha,\beta) | |
| dtN |
.
This is henceforth referred to as an
N
N\leqMin(\alpha,\beta)-1
\Psibeta(t|\alpha,\beta)=
| (-1)N | |
| B(\alpha,\beta) ⋅ T\alpha |
| N | |
| \sum | |
| n=0 |
sgn(2n-N) ⋅
| \Gamma(\alpha) | |
| \Gamma(\alpha-(N-n)) |
(t-a)\alpha ⋅
| \Gamma(\beta) | |
| \Gamma(\beta-n) |
(b-t)\beta.
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.