Hermitian wavelet

Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. The n th {displaystyle n^{ extrm {th}}} Hermitian wavelet is defined as the n th {displaystyle n^{ extrm {th}}} derivative of a Gaussian distribution: Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. The n th {displaystyle n^{ extrm {th}}} Hermitian wavelet is defined as the n th {displaystyle n^{ extrm {th}}} derivative of a Gaussian distribution: Ψ n ( t ) = ( 2 n ) − n 2 c n H e n ( t ) e − 1 2 n t 2 {displaystyle Psi _{n}(t)=(2n)^{-{frac {n}{2}}}c_{n}He_{n}left(t ight)e^{-{frac {1}{2n}}t^{2}}} where H e n ( x ) {displaystyle He_{n}left({x} ight)} denotes the n th {displaystyle n^{ extrm {th}}} Hermite polynomial. The normalisation coefficient c n {displaystyle c_{n}} is given by: c n = ( n 1 2 − n Γ ( n + 1 2 ) ) − 1 2 = ( n 1 2 − n π 2 − n ( 2 n − 1 ) ! ! ) − 1 2 n ∈ Z . {displaystyle c_{n}=left(n^{{frac {1}{2}}-n}Gamma (n+{frac {1}{2}}) ight)^{-{frac {1}{2}}}=left(n^{{frac {1}{2}}-n}{sqrt {pi }}2^{-n}(2n-1)!! ight)^{-{frac {1}{2}}}quad nin mathbb {Z} .} The prefactor C Ψ {displaystyle C_{Psi }} in the resolution of the identity of the continuous wavelet transform for this wavelet is given by: C Ψ = 4 π n 2 n − 1 {displaystyle C_{Psi }={frac {4pi n}{2n-1}}} i.e. Hermitian wavelets are admissible for all positive n {displaystyle n} . 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.