  We describe the relationship between different forms of linearized
expressions for the spatial distribution of intensity of X-ray projection
images obtained in the Fresnel region. We prove that under the natural validity
conditions some of the previously published expressions can be simplified
without a loss of accuracy. We also introduce modified validity conditions
which are likely to be fulfilled in many relevant practical cases, and which
lead to a further significant simplification of the expression for the
image-plane intensity, permitting simple non-iterative linear algorithms for
the phase retrieval.
