  Signal-to-noise ratio and spatial resolution are quantitatively analysed in
the context of in-line (propagation based) X-ray phase-contrast imaging. It is
known that free-space propagation of a coherent X-ray beam from the imaged
object to the detector plane, followed by phase retrieval in accordance with
Paganin's method, can increase the signal-to-noise in the resultant images
without deteriorating the spatial resolution. This results in violation of the
noise-resolution uncertainty principle and demonstrates "unreasonable"
effectiveness of the method. On the other hand, when the process of free-space
propagation is performed in software, using the detected intensity distribution
in the object plane, it cannot reproduce the same effectiveness, due to the
amplification, during free-space propagation, of photon shot noise in the
object-plane intensity. We show that the performance of Paganin's method is
determined by just two dimensionless parameters: the Fresnel number and the
ratio of the phase shift to the logarithm of intensity in the object plane. The
relevant theoretical analysis is performed first, followed by computer
simulations and then by a brief test using experimental images collected at a
synchrotron beamline. More extensive experimental tests will be presented in
the second part of this paper.
