Pyramid Transform of Manifold Data via Subdivision Operators

TL;DR

This paper introduces a multiscale pyramid transform for manifold-valued data, utilizing subdivision operators for stable analysis, with applications in denoising and anomaly detection.

Contribution

It presents a novel multiscale transform framework for manifold data using non-interpolating subdivision schemes for upsampling.

Findings

The transform is stable and exhibits coefficient decay.

Numerical experiments demonstrate effective denoising.

Applications include anomaly detection on manifold data.

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.