Pyramid Transform of Manifold Data via Subdivision Operators

Abstract: Multiscale transforms have become a key ingredient in many data processing tasks. With technological development, we observe a growing demand for methods to cope with non-linear data structures such as manifold values. In this paper, we propose a multiscale approach for analyzing manifold-valued data using a pyramid transform. The transform uses a unique class of downsampling operators that enable a non-interpolating subdivision schemes as upsampling operators. We describe this construction in detail and present its analytical properties, including stability and coefficient decay. Next, we numerically demonstrate the results and show the application of our method to denoising and anomaly detection.