Coherent wave-propagation in the near-field Fresnel-regime is the underlying
contrast-mechanism to (propagation-based) X-ray phase contrast imaging (XPCI),
an emerging lensless technique that enables 2D- and 3D-imaging of biological
soft tissues and other light-element samples down to nanometer-resolutions.
Mathematically, propagation is described by the Fresnel-propagator, a
convolution with an arbitrarily non-local kernel. As real-world detectors may
only capture a finite field-of-view, this non-locality implies that the
recorded diffraction-patterns are necessarily incomplete. This raises the
question of stability of image-reconstruction from the truncated data -- even
if the complex-valued wave-field, and not just its modulus, could be measured.
Contrary to the latter restriction of the acquisition, known as the
phase-problem, the finite-detector-problem has not received much attention in
literature. The present work therefore analyzes locality of Fresnel-propagation
in order to establish stability of XPCI with finite detectors.
Image-reconstruction is shown to be severely ill-posed in this setting -- even
without a phase-problem. However, quantitative estimates of the leaked
wave-field reveal that Lipschitz-stability holds down to a sharp resolution
limit 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's
aspect length, where F is the Fresnel number associated with the latter scale.
The stability results are extended to phaseless imaging in the linear
contrast-transfer-function regime
