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Inference from Incomplete Data in Coherent Diffraction Imaging

  • Author(s): Salha, Sara
  • Advisor(s): Miao, Jianwei
  • et al.

Progress in nanotechnology and biotechnology are propelled by our ability to manipulate and resolve the structure of matter on fine scales. As imaging at higher resolution is limited by the probing light source and the numerical aperture, lensless imaging offers an advantage over lensed microscopy. Dispensing with lenses allows one to overcome certain intrinsic aberrations and to bypass fabrication costs, in the optical and the X-ray regimes. The long penetration depth of X-rays renders coherent X-ray diffraction imaging (CXDI) the method of choice for high resolution structure determination with broad applications from materials science to biology; moreover, the same methodology is extensible to electrons, optical photons, or even gamma rays or neutrons. Since coherent diffraction imaging (CDI) bypasses the need for focusing optics, it relies upon computer algorithms to reconstruct the structure of the scattering object. Currently, one of the main obstacles to nanometer resolution of biological imaging is noisy, incomplete data due to radiation damage. With the rapid development of new light source facilities and the advancement in image reconstruction techniques, determining the structure of individual virons or cells at high resolution is becoming more feasible. In particular, the femtosecond pulse of a free electron laser (FEL) is shorter than the coulomb explosion of the specimen, and thus, it is possible to collect diffraction data prior to radiation damage. However, to fully exploit the computational aspect of lensless imaging, prior knowledge about the object should be incorporated into the image reconstruction process and yet so far such methods are generally lacking. In this thesis, we develop tools that incorporate prior knowledge and reduce the amount of necessary data to recover the structure. We begin by a brief overview of lensless imaging and its place in the natural sciences. we then review the process of image formation in coherent X-ray scattering, the corresponding phase problem and the current state of image recovery. The contributions to this field are two fold. We first demonstrate that three dimensional information can be extracted from a two dimensional diffraction pattern collected at a high numerical aperture. Second, we present a framework for image discovery through Bayesian inference, where we introduce four general constraints: symmetry, sparsity and bounded local and total variation. Using simulated noisy, incomplete data, we recover the solution in situations where traditional algorithms fail. We anticipate that these results will encourage the broader application of Bayesian learning into the phase retrieval problem from noisy, incomplete diffraction data and further enhance the possibility of single shot three dimensional structure determination.

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