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James A Pollock1, Kaye S Morgan2, Linda C P Croton2

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This study rederives phase retrieval algorithms using discrete mathematics for X-ray imaging. This approach enhances spatial resolution, particularly with photon-counting detectors, improving image reconstruction accuracy.

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Area of Science:

  • Optics and Photonics
  • X-ray Imaging
  • Computational Mathematics

Background:

  • Phase retrieval is an ill-posed problem in optics with broad applications in wave imaging.
  • Current algorithms often use continuous mathematics, leading to spatial resolution loss in reconstructions.
  • Discrete Fourier transforms are computationally efficient but can be limited by continuous mathematical formulations.

Purpose of the Study:

  • To investigate the rederivation of phase retrieval algorithms using discrete mathematics for propagation-based phase-contrast X-ray imaging.
  • To improve the precision of phase retrieval for single- and multi-material objects and spectral image decomposition.
  • To experimentally validate the impact of discrete mathematics on spatial resolution in X-ray imaging.

Main Methods:

  • Developed discrete mathematical formulations for phase retrieval algorithms in X-ray imaging.
  • Performed experimental measurements using computed tomography (CT) reconstructions of phantoms and biological tissues.
  • Evaluated spatial resolution using detectors with varying point spread functions (PSFs), including single-pixel PSFs from photon-counting detectors.

Main Results:

  • The discrete mathematical derivation offers more precise phase retrieval for various objects and spectral decomposition.
  • Improvement in spatial resolution is limited when the point spread function (PSF) significantly suppresses high spatial frequencies.
  • A significant spatial resolution improvement (up to 17%) was achieved with detectors characterized by a single-pixel PSF.

Conclusions:

  • Rederving phase retrieval algorithms using discrete mathematics enhances precision in X-ray imaging.
  • The benefits of discrete derivation are most pronounced with high-resolution detectors like photon-counting X-ray detectors.
  • This approach offers a pathway to improved spatial resolution in critical X-ray imaging applications.