Pseudo-reversing and its application for multiscaling of manifold-valued data

The well-known Wiener's lemma is a valuable statement in harmonic analysis; in the Banach space of functions with absolutely convergent Fourier series, the lemma proposes a sufficient condition for the existence of a pointwise multiplicative inverse. We call the functions that admit an inverse as \emph{reversible}. In this paper, we introduce a simple and efficient method for approximating the inverse of functions, which are not necessarily reversible, with elements from the space. We term this process \emph{pseudo-reversing}. In addition, we define a condition number to measure the reversibility of functions and study the reversibility under pseudo-reversing. Then, we exploit pseudo-reversing to construct a multiscale pyramid transform based on a refinement operator and its pseudo-reverse for analyzing real and manifold-valued data. Finally, we present the properties of the resulting multiscale methods and numerically illustrate different aspects of pseudo-reversing, including the applications of its resulting multiscale transform to data compression and contrast enhancement of manifold-valued sequence.