Beta wavelet

Beta distribution

The beta distribution is a continuous probability distribution defined over the interval 0 ≤ t ≤ 1 . It is characterised by a couple of parameters, namely α and β according to:

P ( t ) = 1 B ( α , β ) t α − 1 ⋅ ( 1 − t ) β − 1 , 1 ≤ α , β ≤ + ∞ .

The normalising factor is B ( α , β ) = Γ ( α ) ⋅ Γ ( β ) Γ ( α + β ) ,

where Γ ( ⋅ ) is the generalised factorial function of Euler and B ( ⋅ , ⋅ ) is the Beta function [4].

Gnedenko-Kolmogorov central limit theorem revisited

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

p i ( t ) ≥ 0 , ( ∀ t ) and ∫ − ∞ + ∞ p i ( t ) d t = 1 .

Suppose that all variables are independent.

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

m i = ∫ − ∞ + ∞ τ ⋅ p i ( τ ) d τ , σ i 2 = ∫ − ∞ + ∞ ( τ − m i ) 2 ⋅ p i ( τ ) d τ .

The mean and variance of t are therefore m = ∑ i = 1 N m i and σ 2 = ∑ i = 1 N σ i 2 .

The density p ( t ) of the random variable corresponding to the sum t = ∑ i = 1 N t i is given by the

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

Let { p i ( t ) } be distributions such that S u p p { ( p i ( t ) ) } = ( a i , b i ) ( ∀ i ) .

Let a = ∑ i = 1 N a i < + ∞ , and b = ∑ i = 1 N b i < + ∞ .

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

The random variable t holds, as N → ∞ , p ( t ) ≈ { k ⋅ t α ( 1 − t ) β , o t h e r w i s e

where α = m ( m − m 2 − σ 2 ) σ 2 , and β = ( 1 − m ) ( α + 1 ) m .

Beta wavelets

Since P ( ⋅ | α , β ) is unimodal, the wavelet generated by

ψ b e t a ( t | α , β ) = ( − 1 ) d P ( t | α , β ) d t has only one-cycle (a negative half-cycle and a positive half-cycle).

The main features of beta wavelets of parameters α and β are:

S u p p ( ψ ) = [ − α β α + β + 1 , β α α + β + 1 ] = [ a , b ] .

l e n g t h S u p p ( ψ ) = T ( α , β ) = ( α + β ) α + β + 1 α β .

The parameter R = b / | a | = β / α 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 z e r o c r o s s from the first to the second half cycle is given by

t z e r o c r o s s = ( α − β ) ( α + β − 2 ) α + β + 1 α β .

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

ϕ b e t a ( t | α , β ) = 1 B ( α , β ) T α + β − 1 ⋅ ( t − a ) α − 1 ⋅ ( b − t ) β − 1 , a ≤ t ≤ b .

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

ψ b e t a ( t | α , β ) = − 1 B ( α , β ) T α + β − 1 ⋅ [ α − 1 t − a − β − 1 b − t ] ⋅ ( t − a ) α − 1 ⋅ ( b − t ) β − 1

Beta wavelet spectrum

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

Let ψ b e t a ( t | α , β ) ↔ Ψ B E T A ( ω | α , β ) denote the Fourier transform pair associated with the wavelet.

This spectrum is also denoted by Ψ B E T A ( ω ) for short. It can be proved by applying properties of the Fourier transform that

Ψ B E T A ( ω ) = − j ω ⋅ M ( α , α + β , − j ω ( α + β ) α + β + 1 α β ) ⋅ e x p { ( j ω α ( α + β + 1 ) β ) }

where M ( α , α + β , j ν ) = Γ ( α + β ) Γ ( α ) ⋅ Γ ( β ) ⋅ ∫ 0 1 e j ν t t α − 1 ( 1 − t ) β − 1 d t .

Only symmetrical ( α = β ) cases have zeroes in the spectrum. A few asymmetric ( α ≠ β ) beta wavelets are shown in Fig. Inquisitively, they are parameter-symmetrical in the sense that they hold | Ψ B E T A ( ω | α , β ) | = | Ψ B E T A ( ω | β , α ) | .

Higher derivatives may also generate further beta wavelets. Higher order beta wavelets are defined by ψ b e t a ( t | α , β ) = ( − 1 ) N d N P ( t | α , β ) d t N .

This is henceforth referred to as an N -order beta wavelet. They exist for order N ≤ M i n ( α , β ) − 1 . After some algebraic handling, their closed-form expression can be found:

Ψ b e t a ( t | α , β ) = ( − 1 ) N B ( α , β ) ⋅ T α + β − 1 ∑ n = 0 N s g n ( 2 n − N ) ⋅ Γ ( α ) Γ ( α − ( N − n ) ) ( t − a ) α − 1 − ( N − n ) ⋅ Γ ( β ) Γ ( β − n ) ( b − t ) β − 1 − n .

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 [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.