Rick P. Millane

Phase retrieval in crystallography and optics

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.

Fluorescence optical diffusion tomography

A nonlinear, Bayesian optimization scheme is presented for reconstructing fluorescent yield and lifetime, the absorption coefficient, and the diffusion coefficient in turbid media, such as biological tissue. The method utilizes measurements at both the excitation and the emission wavelengths to reconstruct all unknown parameters. The effectiveness of the reconstruction algorithm is demonstrated by simulation and by application to experimental data from a tissue phantom containing the fluorescent agent Indocyanine Green.

The crystal structure of gellan

Abstract Gellan is a nonsulfated, anionic, extracellular polytetrasaccharide secreted by the bacterium Auromonas elodea . It is potentially useful in the food industry because of its gel-forming properties. The molecular basis of these properties had been investigated by X-ray diffraction analysis of oriented fibers, but an exhaustive study by Upstill et al. in 1986 produced no molecular model with a remotely acceptable fit to the observed X-ray intensities. We describe here a successful re-examination of the crystal structure of gellan; the gellan chains have backbone conformations different from those previously considered. Two left-handed, 3-fold helical chains are organized in parallel fashion in an intertwined duplex in which each chain is translated half a pitch ( p = 5.64 nm) with respect to the other. The duplex is stabilized by interchain hydrogen bonds at each carboxylate group. There are two molecules in each trigonal unit cell ( a = 1.56 nm and c = 2.82 nm).

The molecular structure of kappa-carrageenan and comparison with iota-carrageenan

Abstract The ordered conformation of kappa-carrageenan molecules in condensed but well-hydrated systems has been investigated by refining stereochemically plausible models to fit the continuous X-ray diffraction data obtained from oriented fibers. In the best model, the molecules are coaxial duplexes comprising right-handed, 3-fold helical chains of pitch 25.0 A. As with iota-carrageenan, the chains are parallel but their juxtaposition in kappa-carrageenan is significantly different since they are offset from the half-staggered arrangement by a 28° rotation and a 1.0-A translation. Alternative models (single helices, coaxial duplexes containing 6-fold chains, noncoaxial dimers, and mixtures of single and double helices) are quite incompatible with the diffraction data. Some antiparallel, coaxial duplex models approach the best model either in stereochemical plausibility or fit with the diffraction data, but none is as convincing overall as the best (parallel-stranded) model.

Optical diffusion tomography by iterative-coordinate-descent optimization in a Bayesian framework

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 iterative-coordinate-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.

Nonlinear multigrid algorithms for Bayesian optical diffusion tomography

Optical diffusion tomography is a technique for imaging a highly scattering medium using measurements of transmitted modulated light. Reconstruction of the spatial distribution of the optical properties of the medium from such data is a difficult nonlinear inverse problem. Bayesian approaches are effective, but are computationally expensive, especially for three-dimensional (3-D) imaging. This paper presents a general nonlinear multigrid optimization technique suitable for reducing the computational burden in a range of nonquadratic optimization problems. This multigrid method is applied to compute the maximum a posteriori (MAP) estimate of the reconstructed image in the optical diffusion tomography problem. The proposed multigrid approach both dramatically reduces the required computation and improves the reconstructed image quality.

Structure of the α-form of poly[d(A)] · poly[d(T)] and related polynucleotide duplexes☆

Abstract The α-form of poly[d(A)] · poly[d(T)], observed in fibers at high (> 80%) relative humidity, is a 10-fold double-helical structure of pitch 3.2 nm. This new X-ray analysis shows that the two strands of the double helix are of the same kind conformationally and both B -like in containing C-2′- endo -puckered deoxyribose rings. Nevertheless, the two strands are different enough for the overall morphology of the duplex to resemble that of the heteromerous model for the drier (β) form of poly[d(A)] · poly[d(T)]in which one strand has C-2′- endo rings and the other C-3′- endo . Since the orientations of the bases in poly[d(A)] · poly[d(T)]are persistently different from those of classical B -DNA it is likely that there will be local bending (about 10 °) at the junctions between general sequence tracts and the oligo [d(A)] · oligo[d(T)]tracts that occur in some native DNAs. The conclusions about the structure of α-poly[d(A)] · poly[d(T)]are reinforced by independent analyses of similar X-ray diffraction patterns from poly[d(A)] · poly[d(U)]and poly[d(A-I)] · poly[d(C-T)].

Reconstruction of an object from its symmetry-averaged diffraction pattern.

The problem of reconstructing an object from diffraction data that has been incoherently averaged over a discrete group of symmetries is considered. A necessary condition for such data to uniquely specify the object is derived in terms of the object support and the symmetry group. An algorithm is introduced for reconstructing objects from symmetry-averaged data and its use with simulations is demonstrated. The results demonstrate the feasibility of structure determination using a recent proposal for aligning molecules by means of their anisotropic dielectric interaction with an intense light field

Modified distorted Born iterative method with an approximate Fréchet derivative for optical diffusion tomography

In frequency-domain optical diffusion imaging, the magnitude and the phase of modulated light propagated through a highly scattering medium are used to reconstruct an image of the scattering and absorption coefficients in the medium. Although current reconstruction algorithms have been applied with some success, there are opportunities for improving both the accuracy of the reconstructions and the speed of convergence. In particular, conventional integral equation approaches such as the Born iterative method and the distorted Born iterative method can suffer from slow convergence, especially for large spatial variations in the constitutive parameters. We show that slow convergence of conventional algorithms is due to the linearized integral equations’ not being the correct Frechet derivative with respect to the absorption and scattering coefficients. The correct Frechet derivative operator is derived here. However, the Frechet derivative suffers from numerical instability because it involves gradients of both the Green’s function and the optical flux near singularities, a result of the use of near-field imaging data. To ameliorate these effects we derive an approximation to the Frechet derivative and implement it in an inversion algorithm. Simulation results show that this inversion algorithm outperforms conventional iterative methods.

Prior estimate-based compressed sensing in parallel MRI

Two improved compressed sensing (CS)‐based image reconstruction methods for MRI are proposed: prior estimate‐based compressed sensing (PECS) and sensitivity encoding‐based compressed sensing (SENSECS). PECS allows prior knowledge of the underlying image to be intrinsically incorporated in the image recovery process, extending the use of data sorting as first proposed by Adluru and DiBella (Int J Biomed Imaging 2008: 341648). It does so by rearranging the elements in the underlying image based on the magnitude information gathered from a prior image estimate, so that the underlying image can be recovered in a new form that exhibits a higher level of sparsity. SENSECS is an application of PECS in parallel imaging. In SENSECS, image reconstruction is carried out in two stages: SENSE and PECS, with the SENSE reconstruction being used as a image prior estimate in the following PECS reconstruction. SENSECS bypasses the conflict of sampling pattern design in directly applying CS recovery in multicoil data sets and exploits the complementary characteristics of SENSE‐type and CS‐type reconstructions, hence achieving better image reconstructions than using SENSE or CS alone. The characteristics of PECS and SENSECS are investigated using experimental data. Magn Reson Med, 2010.