A Semidiscrete Version of the Citti-Petitot-Sarti Model as a Plausible Model for Anthropomorphic Image Reconstruction and Pattern Recognition pp 35-42 | Cite as

# Lifts

## Abstract

Let \({\mathbb {G}}= {\mathbb {K}}\ltimes {\mathbb {H}}\) be a semi-direct product, as considered in Sect. 2.2. Here, we are interested in operators \(L:L^2({\mathbb {H}})\rightarrow L^2({\mathbb {G}})\), which we call *lift operators* for obvious reasons. Observe that, via the isomorphism \(\sigma ^*:B_2({\mathbb {H}})\rightarrow L^2({\mathbb {H}}^\flat )\) any lift \(L:L^2({\mathbb {H}}^\flat )\rightarrow L^2({\mathbb {G}}^\flat )\) induces a lift \(L':B_2({\mathbb {H}})\rightarrow B_2({\mathbb {G}})\) of Besicovitch almost periodic functions. We are mainly interested in identifying the action of the quasi-regular representation on \(f\in L^2({\mathbb {H}})\) by analyzing the Fourier transform of the lift \(Lf\in L^2({\mathbb {G}})\). Thus, the first, and more natural, requirement on the lift operation is to intertwine the quasi-regular representation acting on \(L^2({\mathbb {H}})\) with the left regular representations on \(L^2({\mathbb {G}})\). We call these type of lifts *left-invariant*. We show that, under some mild regularity assumptions on *L*, left-invariant lifts coincide with wavelet transforms, as defined in Sect. 2.2.3. These kind of lifts have been extensively studied in, e.g., [33], and related works. Unfortunately, left-invariant lifts have a huge drawback for our purposes: they never have an invertible non-commutative Fourier transform \(\widehat{Lf}(T^\lambda )\). The second part of this chapter is then devoted to the generalization of the concept of *cyclic lift*, introduced in [73] exactly to overcome the above problem. In this general context, we will present a cyclic lift as a combination of an *almost-left-invariant lift* and a centering operation, as defined in Definition 2.2. As a consequence, we obtain a precise characterization of the invertibility of \(\widehat{Lf}(T^\lambda )\) for these lifts.