  We demonstrate that the Plancherel transform for Type-I groups provides one
with a natural, unified perspective for the generalized continuous wavelet
transform, on the one hand, and for a class of Wigner functions, on the other.
The wavelet transform of a signal is an $L^2$-function on an appropriately
chosen group, while the Wigner function is defined on a coadjoint orbit of the
group and serves as an alternative characterization of the signal, which is
often used in practical applications. The Plancherel transform maps
$L^2$-functions on a group unitarily to fields of Hilbert-Schmidt operators,
indexed by unitary irreducible representations of the group. The wavelet
transform can essentiallly be looked upon as restricted inverse Plancherel
transform, while Wigner functions are modified Fourier transforms of inverse
Plancherel transforms, usually restricted to a subset of the unitary dual of
the group. Some known results both on Wigner functions and wavelet transforms,
appearing in the literature from very different perspectives, are naturally
unified within our approach. Explicit computations on a number of groups
illustrate the theory.
