REPRESENTATION THEORETIC PATTERNS IN THREE DIMENSIONAL CRYO-ELECTRON MICROSCOPY III- PRESENCE OF POINT SYMMETRIES

Abstract. In this paper we present a novel algorithm, referred to as the non-linear intrinsic reconstitution algorithm, for three-dimensional structure deter-mination of large biological molecules from cryo-electron microscopy projection images (cryo-EM for short), focusing our attention on the situation where the molecule admits non-trivial symmetries. Our algorithm constitutes a far reach-ing generalization of the intrinsic reconstitution algorithm presented in [13], which does not account for molecules with non-trivial symmetry. The formal justi cation of the algorithmic procedure is based on studying the spectrum of various integral operators, related to parallel transportation on the two-dimensional sphere, called invariant transport operators. In this regard, the main technical result of this paper is a complete description of the spectrum of the invariant transport operators, generalizing earlier results presented in [11] and [12]. Along the way, we continue to develop the mathematical foundations of three-dimensional cryo-EM, further elucidating the central role played by representation theoretic principles in this scienti\u85c discipline