Hermitian wavelet

Hermitian wavelets are a family of continuous wavelets, used in the continuous 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]

where ${\displaystyle \operatorname {He} _{n}(x)}$ denotes the ${\displaystyle n^{\textrm {th}$ probabilist's Hermite polynomial.

The normalization coefficient ${\displaystyle c_{n}$ is given by,

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]}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.

Examples

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

are: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:

See also

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

References

External links

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