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Abstract In synchronization problems, the goal is to estimate elements of a group from noisy measurements of their ratios. A popular estimation method for synchronization is the spectral method. It extracts the group elements from eigenvectors of a block matrix formed from the measurements. The eigenvectors must be projected, or ‘rounded’, onto the group. The rounding procedures are constructed ad hoc and increasingly so when applied to synchronization problems over non-compact groups. In this paper, we develop a spectral approach to synchronization over the non-compact group $$\mathrm{SE}(3)$$, the group of rigid motions of $$\mathbb{R}^{3}$$. We based our method on embedding $$\mathrm{SE}(3)$$ into the algebra of dual quaternions, which has deep algebraic connections with the group $$\mathrm{SE}(3)$$. These connections suggest a natural rounding procedure considerably more straightforward than the current state of the art for spectral $$\mathrm{SE}(3)$$ synchronization, which uses a matrix embedding of $$\mathrm{SE}(3)$$. We show by numerical experiments that our approach yields comparable results with the current state of the art in $$\mathrm{SE}(3)$$ synchronization via the spectral method. Thus, our approach reaps the benefits of the dual quaternion embedding of $$\mathrm{SE}(3)$$ while yielding estimators of similar quality. 

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 nonlinear 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 noninterpolating 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.