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The Symmetries of Image Formation by Scattering. I. Theoretical Framework The symmetries of image formation by scattering. I. Theoretical framework Dimitrios Giannakis,1 Peter Schwander,2 and Abbas Ourmazd2,∗ 1Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, New York 10012, USA 2Department of Physics, University of Wisconsin Milwaukee, 1900 E Kenwood Blvd, Milwaukee, Wisconsin 53211, USA ∗[email protected] Abstract: We perceive the world through images formed by scatte- ring. 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. © 2012 Optical Society of America OCIS codes: (290.5825) Scattering theory; (290.5840) Scattering, molecules; (290.3200) In- verse scattering; (140.2600) Free-electron lasers (FELs); (180.6900) Three-dimensional mi- croscopy. References and links 1. W. A. Freiwald and D. Y. 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