3-D image reconstruction for bio nanomachines with helical symmetry: Image formation theory

Cryo electron microscopy leads to 3-D image reconstruction problems. The image is one projection of each of many different instances of the object. Symmetry of the object is important because then the image shows the motif in several projections with known relative orientations even though the projections of the different motifs overlap. This paper derives image formation models for helical objects based on arbitrary motifs and where the object may, in fact, not be helical but just have a rotational and translational shift between successive motifs. This image formation model potentially has computational advantages when the number of motif repetitions per helical period is large and also allows the description of imperfect helices by stochastic variation in the shifts, which may be useful in stochastic reconstruction problems for image data sets that cannot currently be successfully processed.