Mexican hat wavelet
Key formulae: Mexican hat wavelet
• Gaussian filter
F
σ
(t) =
1
2
√
πσ
e
−
t
2
4σ
• The scaled wavelet
ψ
2
(σ, t) =
dF
σ
dσ
=
d
2
F
σ
dt
2
• Transform
e
u
2
(σ, t) =
Z
u(t
0
)ψ
2
(σ, t − t
0
) dt
0
• Compatibility condition
∂
σ
e
u
2
= ∂
2
t
e
u
2
• Inverse transform
u(t) = 4
Z Z
e
u
2
(σ, t
0
) ψ
2
(σ, t − t
0
) dσ dt
0
• Multiresolution
e
U = −σ
e
u
2
• Local reconstruction
u(t) =
Z
∞
0
e
U
dσ
σ
• Parseval (Plancherel)
Z
1
2
u
2
dx =
Z
ds
s
Z
2
e
U
2
dx
The normalization constant c
p
si is absorbed in the definition of ψ
2
; the scale σ corresponds
to a frequency ω = 1/2π
√
σ; logarithmic scale integration for inverse and Parseval are
natural; and the compatibility equation facilitates analytical manipulations.
Starting from the normalized Gaussian filter
F
σ
(t) =
1
2
√
πσ
e
−
t
2
4σ
(1)
1