Dimitrios Giannakis, Peter Schwander, and Abbas Ourmazd. The symmetries of image formation by scattering. I. Theoretical framework. Optics Express, Vol. 20, Issue 12, pp. 12799-12826 (2012) http://dx.doi.org/10.1364/OE.20.012799
Schwander P, Giannakis D, Yoon CH, Ourmazd A. The symmetries of image formation by scattering. II. Applications. Opt Express. 2012 Jun 4;20(12):12827-49. doi: 10.1364/OE.20.012827.
I: We perceive the world through images formed by scattering. The ability to interpret scattering data mathematically has opened to our scrutiny the constituents of matter, the building blocks of life, and the remotest corners of the universe. Here, we present an approach to image formation based on the symmetry properties of operations in three-dimensional space. Augmented with graph-theoretic means, this approach can recover the three-dimensional structure of objects from random snapshots of unknown orientation at four orders of magnitude higher complexity than previously demonstrated. This is critical for the burgeoning field of structure recovery by X-ray Free Electron Lasers, as well as the more established electron microscopic techniques, including cryo-electron microscopy of biological systems. In a subsequent paper, we demonstrate the recovery of structure and dynamics from experimental, ultralow-signal random sightings of systems with X-rays, electrons, and photons, with no orientational or timing information.
II: We show that the symmetries of image formation by scattering enable graph-theoretic manifold-embedding techniques to extract structural and timing information from simulated and experimental snapshots at extremely low signal. The approach constitutes a physically-based, computationally efficient, and noise-robust route to analyzing the large and varied datasets generated by existing and emerging methods for studying structure and dynamics by scattering. We demonstrate three-dimensional structure recovery from X-ray diffraction and cryo-electron microscope image snapshots of unknown orientation, the latter at 12 times lower dose than currently in use. We also show that ultra-low-signal, random sightings of dynamically evolving systems can be sequenced into high quality movies to reveal their evolution. Our approach offers a route to recovering timing information in time-resolved experiments, and extracting 3D movies from two-dimensional random sightings of dynamic systems.
Initial volume, electron scattering, graph-theory, graph Laplacian