Geometric multiscale analysis via nonstationary subdivision schemes

Abstract

Pyramid transforms are constructive methods for analyzing sequences in a multiscale fashion. Traditionally, these transforms rely on stationary upsampling and downsampling operations. In this paper, we propose employing nonstationary subdivision schemes as upsampling operators that vary with the refinement level. These schemes offer greater flexibility, enabling the development of more expressive multiscale transforms, including geometric multiscale analysis. We establish the fundamental properties of the resulting nonstationary transforms, namely the decay of their detail coefficients and the stability of both the decomposition and the reconstruction, and we demonstrate their effectiveness in capturing and analyzing geometric features. In particular, we apply the framework to assess the consistency of planar samples with a circular shape and to detect localized geometric anomalies, including a study of circles generated by a neural network. The accompanying code is publicly available.