Hermitian wavelet

Hermitian wavelets are a family of discrete and continuous wavelets, used in the continuous and discrete Hermite wavelet transform. The ${\displaystyle n^{\textrm {th}}}$ Hermitian wavelet is defined as the ${\displaystyle n^{\textrm {th}}}$ derivative of a Gaussian distribution for each positive ${\displaystyle n}$: ^[1]

$${\displaystyle \Psi _{n}(x)=(2n)^{-{\frac {n}{2}}}c_{n}\operatorname {He} _{n}\left(x\right)e^{-{\frac {1}{2}}x^{2}},}$$

where in this case we consider the (probabilist) Hermite polynomial ${\displaystyle \operatorname {He} _{n}(x)}$. The normalization coefficient ${\displaystyle c_{n}}$ is given by,

$${\displaystyle c_{n}=\left(n^{{\frac {1}{2}}-n}\Gamma \left(n+{\frac {1}{2}}\right)\right)^{-{\frac {1}{2}}}=\left(n^{{\frac {1}{2}}-n}{\sqrt {\pi }}2^{-n}(2n-1)!!\right)^{-{\frac {1}{2}}}\quad n\in \mathbb {N} .}$$

The function ${\displaystyle \Psi \in L_{\rho ,\mu }(-\infty ,\infty )}$ is said to be an admissible Hermite wavelet if it satisfies the admissibility relation: ^[2]

$${\displaystyle C_{\Psi }=\sum _{n=0}^{\infty }{\frac {\|{\hat {\Psi }}(n)\|^{2}}{\|n\|}}<\infty }$$

where ${\displaystyle {\hat {\Psi }}(n)}$ is the Hermite transform of ${\displaystyle \Psi }$.

The perfector ${\displaystyle C_{\Psi }}$ in the resolution of the identity of the continuous wavelet transform for this wavelet is given by the formula,^{[ further explanation needed ]}

$${\displaystyle C_{\Psi }={\frac {4\pi n}{2n-1}}}$$

In computer vision and image processing, Gaussian derivative operators of different orders are frequently used as a basis for expressing various types of visual operations; see scale space and N-jet. ^[3]

Examples

The first three derivatives of the Gaussian function with ${\displaystyle \mu =0,\;\sigma =1}$:

$${\displaystyle f(t)=\pi ^{-1/4}e^{(-t^{2}/2)},}$$

are:

$${\displaystyle {\begin{aligned}f'(t)&=-\pi ^{-1/4}te^{(-t^{2}/2)},\\f''(t)&=\pi ^{-1/4}(t^{2}-1)e^{(-t^{2}/2)},\\f^{(3)}(t)&=\pi ^{-1/4}(3t-t^{3})e^{(-t^{2}/2)},\end{aligned}}}$$

and their ${\displaystyle L^{2}}$ norms ${\displaystyle ||f'||={\sqrt {2}}/2,||f''||={\sqrt {3}}/2,||f^{(3)}||={\sqrt {30}}/4}$. Normalizing the derivatives yields three Hermitian wavelets:

$${\displaystyle {\begin{aligned}\Psi _{1}(t)&={\sqrt {2}}\pi ^{-1/4}te^{(-t^{2}/2)},\\\Psi _{2}(t)&={\frac {2}{3}}{\sqrt {3}}\pi ^{-1/4}(1-t^{2})e^{(-t^{2}/2)},\\\Psi _{3}(t)&={\frac {2}{15}}{\sqrt {30}}\pi ^{-1/4}(t^{3}-3t)e^{(-t^{2}/2)}.\end{aligned}}}$$

See also

• Wavelet
• The Ricker wavelet is the ${\displaystyle n=2}$ Hermitian wavelet

References

1. Brackx, F.; De Schepper, H.; De Schepper, N.; Sommen, F. (2008-02-01). "Hermitian Clifford-Hermite wavelets: an alternative approach". Bulletin of the Belgian Mathematical Society, Simon Stevin. 15 (1). doi: 10.36045/bbms/1203692449 . ISSN   1370-1444.
2. "Continuous and Discrete Wavelet Transforms Associated with Hermite Transform". International Journal of Analysis and Applications. 2020. doi: 10.28924/2291-8639-18-2020-531 .
3. Wah, Benjamin W., ed. (2007-03-15). Wiley Encyclopedia of Computer Science and Engineering (1 ed.). Wiley. doi:10.1002/9780470050118.ecse609. ISBN   978-0-471-38393-2.

External links

• Hermitian Clifford–Hermite Wavelets (Department of Mathematical Analysis, Faculty of Engineering, Ghent University)