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 thesecond and first derivatives of a Gaussian respectively:

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

The Fourier transform of this wavelet is:

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

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

$C\_\{Psi\}=frac\{16\}\{5\}sqrt\{pi\}.$

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