Integral geometry and three-dimensional reconstruction of randomly oriented identical particles from their electron microphotos

A new method for the three-dimensional reconstruction of a structure from projections of randomly oriented particles on a plane is proposed. Reconstruction is performed in two steps. First, we find mutual orientation of particles, i.e., Euler angles, describing the angle of one particle with respect to another. Almost all the paper is devoted to solving this problem. Then we perform the three-dimensional reconstruction of an object from its projections in already-known directions.The stability of the method with respect to experimental errors is shown. Three-dimensional reconstruction of asymmetric biological objects might be one of its applications.