Hermitian hat wavelet

The Hermitian hat wavelet is a low-oscillation, complex-valued wavelet. The real and imaginary parts of this wavelet are defined to be the second and first derivatives of a Gaussian respectively:

${\displaystyle \Psi (t)={\frac {2}{\sqrt {5}}}\pi ^{-{\frac {1}{4}}}(1-t^{2}+it)e^{-{\frac {1}{2}}t^{2}}.}$

The Fourier transform of this wavelet is:

${\displaystyle {\hat {\Psi }}(\omega )={\frac {2}{\sqrt {5}}}\pi ^{-{\frac {1}{4}}}\omega (1+\omega )e^{-{\frac {1}{2}}\omega ^{2}}.}$

The Hermitian hat wavelet satisfies the admissibility criterion. The prefactor $C_{{\Psi }}$ in the resolution of the identity of the continuous wavelet transform is:

${\displaystyle C_{\Psi }={\frac {16}{5}}{\sqrt {\pi }}.}$

This wavelet was formulated by Szu in 1997 for the numerical estimation of function derivatives in the presence of noise. The technique used to extract these derivative values exploits only the argument (phase) of the wavelet and, consequently, the relative weights of the real and imaginary parts are unimportant.