Page 282 - Introduction to Information Optics

5.5. Physical Wavelet Transform 26 7
of the electromagnetic field is related to the 3D potential vector by a relation
as f(p) — 2p 0a(p), so that the integral
i r d 3 P i

defines a norm of solutions of the Maxwell equations that are invariant under
the Lorentz transformation, because f(p)/p 0 with p 0 = ± w has a dimension of
the vector potential a(p). The square module of the latter is invariant under the
Lorentz transformation, and the Lorentz-invariant inner products must be
2
defined according to the norm of field. That is the reason for the factor l/(co/c)
on the right-hand side of Eq. (5.16), which is the definition of the electromag-
netic wavelet transform. On the other hand, in the definition of the electromag-
2
netic wavelet, we introduced a factor (co/c) in Eq. (5.13). As a result, the
wavelet transform defined in Eq. (5.16) is invariant under the Lorentz trans-
formation. The inner products defined in Eq. (5.16) form a Helbert space where
the wavelet transform is defined.
The scalar electromagnetic field /(r) may be reconstructed from the inverse
wavelet transform by [16]

f .
3
/(r) = d tdshs, s(r)W f(£,s), (5.17)
JE

where the Euclidean region E is the group of spatial translations and scaling,
3
which act on the real-valued space-time with ceR and seR i=- 0. The scalar
wavelet transform is defined on £, which is a Euclidean region in the Helbert
space.
According to Eq. (5.17), an electromagnetic field is decomposed as a linear
combination of the scaled and shifted electromagnetic wavelets. In this wavelet
decomposition in the space-time domain, the integral on the right-hand side
of Eq. (5.17) is a function of the time translation t' with a component of
c — (c,f')> while the signal /(r), on the left-hand side of Eq. (5.17), is not a
function of the time translation t'. We shall show below that, in fact, t' = 0
because the translations in the time and space domains are not independent
from each other, according to the electromagnetic theory [18].
The wavelet transform with the electromagnetic wavelet is a decomposition
of an electromagnetic field into a linear combination in the scaled and shifted
electromagnetic wavelet family, h^ s(r). However, we must note that the
electromagnetic wavelet is a solution of the wave equation; its space and time
translations cannot be arbitrary, but must be constrained by the light cone in
2 2
the space- time c *'