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Coherent wave-propagation in the near-field Fresnel-regime is the underlyingcontrast-mechanism to (propagation-based) X-ray phase contrast imaging (XPCI),an emerging lensless technique that enables 2D- and 3D-imaging of biologicalsoft tissues and other light-element samples down to nanometer-resolutions.Mathematically, propagation is described by the Fresnel-propagator, aconvolution with an arbitrarily non-local kernel. As real-world detectors mayonly capture a finite field-of-view, this non-locality implies that therecorded diffraction-patterns are necessarily incomplete. This raises thequestion of stability of image-reconstruction from the truncated data -- evenif the complex-valued wave-field, and not just its modulus, could be measured.Contrary to the latter restriction of the acquisition, known as thephase-problem, the finite-detector-problem has not received much attention inliterature. The present work therefore analyzes locality of Fresnel-propagationin order to establish stability of XPCI with finite detectors.Image-reconstruction is shown to be severely ill-posed in this setting -- evenwithout a phase-problem. However, quantitative estimates of the leakedwave-field reveal that Lipschitz-stability holds down to a sharp resolutionlimit that depends on the detector-size and varies within the field-of-view.The smallest resolvable lengthscale is found to be 1/F times the detector'saspect length, where F is the Fresnel number associated with the latter scale.The stability results are extended to phaseless imaging in the linearcontrast-transfer-function regime.

Locality estimates for Fresnel-wave-propagation and stability of X-ray phase contrast imaging with finite detectors