  X-ray phase-contrast imaging has recently led to a revolution in resolving
power and tissue contrast in biomedical imaging, microscopy and materials
science. The necessary high spatial coherence is currently provided by either
large-scale synchrotron facilities with limited beamtime access or by
microfocus X-ray tubes with rather limited flux. X-rays radiated by
relativistic electrons driven by well-controlled high-power lasers offer a
promising route to a proliferation of this powerful imaging technology. A
laser-driven plasma wave accelerates and wiggles electrons, giving rise to
brilliant keV X-ray emission. This so-called Betatron radiation is emitted in a
collimated beam with excellent spatial coherence and remarkable spectral
stability. Here we present the first phase-contrast micro-tomogram revealing
quantitative electron density values of a biological sample using betatron
X-rays, and a comprehensive source characterization. Our results suggest that
laser-based X-ray technology offers the potential for filling the large
performance gap between synchrotron- and current X-ray tube-based sources.
