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

One of the challenges in single particle reconstruction in cryo-electron microscopy is to find a three-dimensional model of a molecule using its two-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. Our suggested method utilizes self common lines which induce identical lines within the Fourier transform of each projection-image. We show that the location of self common lines admits quite a few favorable geometrical constraints, thus allowing to detect them even in a noisy setting. In addition, for molecules with higher order rotational symmetry, our proposed method exploits the fact that there exist numerous common lines between any two Fourier transformed projection-images of such molecules, thus allowing to determine their relative orientation even under high levels of noise. The efficacy of our proposed method is demonstrated using numerical experiments conducted on simulated and experimental data.