Phase problems occur in many scientific disciplines, particularly those involving remote sensing using a wave field. Although there has been much interest in phase retrieval in optics and in imaging in general over the past decade, phase retrieval has a much longer history in x-ray crystallography, and a variety of powerful and practical techniques have been developed. The nature of crystallography means that crystallographic phase problems are distinct from those in other imaging contexts, but there are a number of commonalities. Here the principles of phase retrieval in crystallography are outlined and are compared and contrasted with phase retrieval in general imaging. Uniqueness results are discussed, but the emphasis is on phase-retrieval algorithms and areas in which results in one discipline have, and may, contribute to the other.

We present a technique for reconstructing the spatially dependent dynamics of a fluorescent contrast agent in turbid media. The dynamic behavior is described by linear and nonlinear parameters of a compartmental model or some other model with a deterministic functional form. The method extends our previous work in fluorescence optical diffusion tomography by parametrically reconstructing the time-dependent fluorescent yield. The reconstruction uses a Bayesian framework and parametric iterative coordinate descent optimization, which is closely related to Gauss-Seidel methods. We demonstrate the method with a simulation study.

Frequency-domain diffusion imaging uses the magnitude and phase of modulated light propagating through a highly scattering medium to reconstruct an image of the spatially dependent scattering or absorption coefficients in the medium. An inversion algorithm is formulated in a Bayesian framework and an efficient optimization technique is presented for calculating the maximum a posteriori image. In this framework the data are modeled as a complex Gaussian random vector with shot-noise statistics, and the unknown image is modeled as a generalized Gaussian Markov random field. The shot-noise statistics provide correct weighting for the measurement, and the generalized Gaussian Markov random field prior enhances the reconstruction quality and retains edges in the reconstruction. A localized relaxation algorithm, the iterativecoordinate-descent algorithm, is employed as a computationally efficient optimization technique. Numerical results for two-dimensional images show that the Bayesian framework with the new optimization scheme outperforms conventional approaches in both speed and reconstruction quality.

The crystal structure of Valonia cellulose Iβ is determined by X-ray fiber diffraction analysis. A careful reanalysis of two existing X-ray diffraction data sets for Valonia cellulose I leads to a consistent, definitive structure of the β-phase. The results resolve ambiguities in existing X-ray analyses between “parallel up” and “parallel down” packing of the cellulose sheets in native Valonia cellulose. The molecular and sheet structures are essentially identical to those previously determined, but these results define precisely the packing of the sheets. The sheet packing is discussed in terms of the energy and density of the crystal structure. The structure of the β-phase has specific implications for the packing of the α-phase of Valonia cellulose I.

A general class of iterative projection algorithms is described and proposed as a tool for phasing in protein crystallography in order to improve the radius of convergence over that of conventional density-modification algorithms. Their relationship to conventional density modification is described. The common iterative projection algorithms, their convergence properties and their application to protein crystallography are described. These algorithms offer the possibility of protein structure determination starting with only information on the molecular envelope and low-order non-crystallographic symmetry.

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A method is presented for fluorescence optical diffusion tomography in turbid media using multiple-frequency data. The method uses a frequency-domain diffusion equation model to reconstruct the fluorescent yield and lifetime by means of a Bayesian framework and an efficient, nonlinear optimizer. The method is demonstrated by using simulations and laboratory experiments to show that reconstruction quality can be improved in certain problems through the use of more than one frequency. A broadly applicable mutual information performance metric is also presented and used to investigate the advantages of using multiple modulation frequencies compared with using only one.