# A fast algorithm for maximum likelihood 3-D signal reconstruction from 2-D projections of unknown orientation and applications to the electron microscopy of viruses

#### Abstract

In a cryo electron microscopy experiment, the data is noisy 2-D projection images of the 3-D electron scattering intensity where the orientation of the projections is unknown. Often the images show randomly selected particles from a mixture of different types of particles or different maturation stages of the same particle. Due to low SNR and the 2-D nature of the data, it is challenging to label the type or stage shown in an individual image. A statistical model and maximum likelihood estimator that computes simultaneous 3-D reconstruction and labels using expectation maximization algorithm exists but requires impractical amounts of computation due to the numerical evaluation of 3- or 5-dimensional integrations of a square matrix of dimension equal to the number of Fourier series coefficients used to describe the 3-D reconstruction. By exploiting the geometry of rotation in 3-D, i.e., the group SO(3), the estimation problem can be transformed so that the inner-most numerical integral has a scalar rather than a matrix integrand. This leads to dramatic reduction in computation, especially as the problem size increases. Numerical examples of the 3-D reconstructions are provided based on various experimental virus images with known structures including the first ever simultaneous 3-D reconstructions from mixtures of particles of different types or maturation stages.^

#### Degree

Ph.D.

#### Advisors

Peter C. Doerschuk, Purdue University.

#### Subject Area

Engineering, Biomedical|Engineering, Electronics and Electrical

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