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One of the challenges in single particle reconstruction in cryo-electronmicroscopy is to find a three-dimensional model of a molecule using itstwo-dimensional noisy projection-images. In this paper, we propose a robust"angular reconstitution" algorithm for molecules with $n$-fold cyclic symmetry,that estimates the orientation parameters of the projections-images. Oursuggested method utilizes self common lines which induce identical lines withinthe Fourier transform of each projection-image. We show that the location ofself common lines admits quite a few favorable geometrical constraints, thusallowing to detect them even in a noisy setting. In addition, for moleculeswith higher order rotational symmetry, our proposed method exploits the factthat there exist numerous common lines between any two Fourier transformedprojection-images of such molecules, thus allowing to determine their relativeorientation even under high levels of noise. The efficacy of our proposedmethod is demonstrated using numerical experiments conducted on simulated andexperimental data.

A common lines approach for ab-initio modeling of cyclically-symmetric molecules