Exact Reduced-Complexity Maximum Likelihood Reconstruction of Multiple 3-D Objects From Unlabeled Unoriented 2-D Projections and Electron Microscopy of Viruses

In cryo-electron microscopy, the data is comprised of noisy 2D projection images of the 3D electron scattering intensity of the object where the orientation of the projections is unknown. Often, the images show randomly selected objects from a mixture of different types of objects. Objects of different type may be unrelated, e.g., different species of virus, or related, e.g., different conformations of the same species of virus. Due to the low SNR and the 2D nature of the data, it is challenging to determine the type of the object shown in an individual image. A statistical model and maximum likelihood estimator that computes simultaneous 3D reconstruction and labels using an expectation maximization algorithm exists but requires extensive computation due to the numerical evaluation of 3D or 5D integrations of a square matrix of dimension equal to the number of degrees of freedom in the 3D reconstruction. By exploiting the geometry of rotations in 3D, the estimation problem can be transformed so that the inner-most numerical integral has a scalar rather than a matrix integrand. This leads to a dramatic reduction in computation, especially as the number of degrees of freedom in the 3D reconstruction increases. Numerical examples of the 3D reconstructions are provided based on synthetic and experimental images where the objects are small spherical viruses.