New Atomicity-Exploiting Algorithms for Super-Resolution X-Ray Crystallography

The X-ray crystallography problem is to reconstruct a crystalline structure from the Fourier magnitude of its diffracted scattering data. This has three major difficulties: (1) Only Fourier magnitude (not phase) data are known; (2) There is no support constraint (since the crystal is periodic); and (3) only low-wavenumber scattering data are available. But it also has two major advantages: (1) the crystal is sparse (atomicity) since it consists of isolated atoms; and (2) the crystal structure often has even symmetry. We exploit atomicity to show that the crystal can be reconstructed easily from only low wavenumber Fourier data. We also propose new algorithms for reconstruction of crystals with even or no symmetry from low-wavenumber Fourier magnitude data using two or one isomorphic replacements (4 algorithms). Small numerical examples illustrate the algorithms. Keywords— X-ray crystallography, sparse images. Phone: 734-763-9810. Fax: 734-763-1503. Email: [email protected]. EDICS: 2-REST.