A posteriori determination of relative projection directions of arbitrarily oriented macromolecules

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Abstract
For the generation of three-dimensional structures from two-dimensional projections a fundamental requirement is knowledge of the individual projection directions. In electron microscopy of single macromolecules the molecular projection directions are not known and must be determined a posteriori from the projection images themselves. The algorithm presented achieves such a determination by using the central-section theorem, geometrical techniques, and quaternion mathematics. The quality of the solution is tested in relation to image noise, angular error in the input data, number of intercompared projections used to generate common-axis data, and number of iterations. Correct determination of mutual alignments is achieved despite significant errors in the input data, indicating that the method should be applicable to electron microscopy problems.