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:
The Fourier transform of this wavelet is:
The Hermitian hat wavelet satisfies the admissibility criterion. The prefactor in the resolution of the identity of the continuous wavelet transform is:
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.
Famous quotes containing the words hat and/or wavelet:
“Lord Hamlet, with his doublet all unbraced,
No hat upon his head, his stockings fouled,
Ungartered, and down-gyved to his ankle,
Pale as his shirt, his knees knocking each other,
And with a look so piteous in purport
As if he had been loosed out of hell
To speak of horrors.”
—William Shakespeare (1564–1616)
“These facts have always suggested to man the sublime creed that the world is not the product of manifold power, but of one will, of one mind; and that one mind is everywhere active, in each ray of the star, in each wavelet of the pool; and whatever opposes that will is everywhere balked and baffled, because things are made so, and not otherwise.”
—Ralph Waldo Emerson (1803–1882)