Regularization perspectives on support vector machines

Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform (WT). This transform is proposed in order to rectify the limitations of the WT and the fractional Fourier transform . The FRWT not only inherits the advantages of multiresolution analysis of the WT, but also has the capability of signal representations in the fractional domain which is similar to the FRFT. Compared with the existing FRWT, the FRWT can offer signal representations in the time-fractional-frequency plane.

Definition

The FRWT of a signal or a function $f(t)\in L^2(\mathbb{ℝ})$ is defined as

$W_f^{\alpha }(a,b)={\mathcal{𝒲}}^{\alpha }[f(t)](a,b)=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}f(t){\psi }_{\alpha ,a,b}^{\ast }(t)dt.$

where ${\psi }_{\alpha ,a,b}(t)$ is a continuous affine transformation and chirp modulation of the mother wavelet $\psi (t)$, i.e.,

${\psi }_{\alpha ,a,b}(t)=\frac{1}{\sqrt{a}}\psi \left(\frac{t-b}{a}\right)e^{-j\frac{t^2-b^2}{2}\cot \alpha }$

in which $a\in {\mathbb{ℝ}}^{+}$, $b\in \mathbb{ℝ}$ are scaling and translation parameters, respectively. The inverse FRWT is given by

$f(t)=\frac{1}{2\pi C_{\psi }}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\underset{0}{\overset{+\mathrm{\infty }}{\int }}{W}_f^{\alpha }(a,b){\psi }_{\alpha ,a,b}(t)\frac{da}{a^2}db$

where $C_{\psi }$ is a constant that depends on the wavelet used. The success of the reconstruction depends on this constant called, the admissibility constant, to satisfy the following admissibility condition:

$C_{\psi }=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\frac{|\mathrm{\Psi }(\mathrm{\Omega }){|}^2}{\left|\mathrm{\Omega }\right|}d\mathrm{\Omega }<\mathrm{\infty }$

where $\mathrm{\Psi }(\mathrm{\Omega })$ denotes the FT of $\psi (t)$. The admissibility condition implies that $\mathrm{\Psi }(0)=0$, which is $\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\psi (t)dt=0$. Consequently continuous fractional wavelets must oscillate and behave as bandpass filters in the fractional Fourier domain. Whenever $\alpha =\pi /2$, the FRWT reduces to the classical WT.

References

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