The problem of computing 3-D density maps of macromolecules from a set of 2-D projection images taken by an electron microscope is becoming increasingly important in biological research. The problem is challenging in a number of ways:

1) Limited by radiation damage, the 2-D images are obtained by taking a single shot of many identical but randomly oriented molecules embedded in a thin layer of vitreous ice (a process known as cryo-electron microscopy). As a result, the relative orientations of the 2-D images are unknown, and must be determined as part of the solution to the 3-D image reconstruction problem.

2) To avoid radiation damage, the electron dose of the microscope has to be kept low. This limited exposure of molecule to electron beams results in a typical low signal-to-noise ratio in the 2-D projection images. These 2-D images must also be corrected by a contrast transfer function to account for defocus and aberration artifacts introduced by the electron

microscope.

3) To obtain a high resolution reconstruction, a large number of 2-D projection images must be collected to ensure a uniform and detailed coverage of projection directions. In this talk, we will give an overview on the mathematical

formulation of the image reconstruction problem, and provide a survey on numerical techniques currently employed

at various stages of the reconstruction process.