Two ways to incorporate scale in the Heisenberg group with an intertwining operator

This paper presents two different representations associated with the Heisenberg group in order to incorporate multiscale resolution.The first representation incorporates scale and phase scale with an appropriate intertwining operator analogous to the linear Schrödinger representation associated with the Heisenberg group of shift and phase shift and with the Zak transform. This is done in a simple way by operating the Heisenberg group over an appropriate manifold so that the actions of scale and phase scale are translated to a local shift and a phase shift. As a result, a new signal decomposition of multiscale resolution is determined with the ability to observe new information on scale resolution that may be useful for image compression and extension.The second representation extends the Schrödinger representation to include scaling. The hybrid multiscale-Heisenberg representation determines a four-parameter linear group. The new group contains the Heisenberg and the affine as subgroups, where the former is also a normal subgroup. This leads to various signal representation, including Heisenberg—Gabor wavelets, multiscale wavelets, and hybrid multiscale-Heisenberg wavelets.