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Abstract The Fourier shell correlation (FSC) is a measure of the similarity between two signals computed over corresponding shells in the frequency domain and has broad applications in microscopy. In structural biology, the FSC is ubiquitous in methods for validation, resolution determination, and signal enhancement. Computing the FSC usually requires two independent measurements of the same underlying signal, which can be limiting for some applications. Here, we analyze and extend on an approach to estimate the FSC from a single measurement. In particular, we derive the necessary conditions required to estimate the FSC from downsampled versions of a single noisy measurement. These conditions reveal additional corrections which we implement to increase the applicability of the method. We then illustrate two applications of our approach, first as an estimate of the global resolution from a single 3-D structure and second as a data-driven method for denoising tomographic reconstructions in electron cryo-tomography. These results provide general guidelines for computing the FSC from a single measurement and suggest new applications of the FSC in microscopy. 

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.