  We develop a speckle-tracking method for x-ray phase-contrast imaging, based
on the concept of geometric flow. This flow is a conserved current associated
with deformation of illuminating x-ray speckles induced by passage through a
sample. The method provides a rapid, efficient, and accurate algorithm for
quantitative phase imaging. It is highly photon efficient and able to image
dynamic objects, since a single radiograph of the sample is sufficient for the
phase recovery. We experimentally quantify the resolution and contrast of the
approach with both two-dimensional and three-dimensional phase-imaging
applications using x-ray synchrotron radiation. Finally, we discuss adaptations
of the method to imaging with compact x-ray sources that have a large source
size and significant spectral bandwidth.
