Phase-contrast imaging produces cryo-electron microsope (cryo-EM) images which appear visually to have much higher signal to noise ratio (S/N) than corresponding images obtained by conventional means . The importance of this advance should not be underestimated. It has been known for a long time that noise-free 2D projections almost trivially enable 3D reconstruction, while finite S/N implies that at least some algorithms for 3D reconstruction become NP-complete , explaining the predominance of heuristic methods.
Understanding the noise-free reconstruction problem is roughly equivalent to understanding the Penrose tribar . (Wikipedia has entries on the tribar in several languages.) In addition to a thorough discussion of the underlying mathematics, Ye and Lim  reference at least a slice of the considerable literature on the cryo-EM reconstruction problem.
Among widely used algorithms, as implemented in RELION  for example, many require a “close” starting model, implementing, as they do, structure refinement rather than ab initio reconstruction. A new entrant, PRIME , seems to side-step this problem. The recent review by Elmlund and Elmlund  compares most of the relevant algorithms in use.
 Danev and Baumeister, “Cryo-EM single particle analysis with the Volta phase plate”, eLife 2016;5:e13046. doi: 10.7554/eLife.13046.
 Mielikäinen, Ravantti and Ukkonen, “The Computational Complexity of Orientation Search Problems in cryo-Electron Microscopy”, arXiv:cs/0406043.
 Ye and Lim, “Cohomology of cryo-electron microscopy”, arXiv:1604.01319.
 Scheres SHW, “RELION: implementation of a Bayesian approach to cryo-EM structure determination”. J. Struct. Biol. 2012, 180:519–30. doi: 10.1016/j.jsb.2012.09.006.
 Elmlund H, Elmlund D, Bengio S, “PRIME: probabilistic initial 3D model generation for single-particle cryo–electron microscopy.” Structure 2013, 21:1299–306. doi: 10.1016/j.str.2013.07.002.
 Elmlund and Elmlund, “Cryogenic Electron Microscopy and Single-Particle Analysis”, Annu. Rev. Biochem. 2015. 84:20.1–20.19. doi: 10.1146/annurev-biochem-060614-034226.