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The development of phase contrast methods for diagnostic x-ray imaging is inspired by the potential of seeing the internal structures of the human body without the need to deposit any harmful radiation. An efficient class of x-ray phase contrast imaging and scatter correction methods share the idea of using structured illumination in the form of a periodic fringe pattern created with gratings or grids. They measure the scatter and distortion of the x-ray wavefront through the attenuation and deformation of the fringe pattern via a phase stepping process. Phase stepping describes image acquisition at regular phase intervals by shifting a grating in uniform steps. However, in practical conditions the actual phase intervals can vary from step to step and also spatially. Particularly with the advent of electromagnetic phase stepping without physical movement of a grating, the phase intervals are dependent upon the focal plane of interest. We describe a demodulation algorithm for phase stepping at arbitrary and position-dependent (APD) phase intervals without assuming  a priori  knowledge of the phase steps. The algorithm retrospectively determines the spatial distribution of the phase intervals by a Fourier transform method. With this ability, grating-based x-ray imaging becomes more adaptable and robust for broader applications.

Flexible Retrospective Phase Stepping in X-Ray Scatter Correction and Phase Contrast Imaging Using Structured Illumination