Propagation-based, or in-line, phase contrast imaging is a simple way to achieve phase contrast with hard X-rays . It has gained increasing interest due to the potentially three orders of magnitude increase in sensitivity compared to attenuation based imaging. One of the most interesting aspects of in-line phase contrast, however, is that there exists a quantitative relationship between the mass density distribution in the imaged object and the recorded intensity. This allows the reconstruction of the phase shift induced by the object through a process known as phase retrieval. Since the phase can be considered a straight line projection, access to the phase shift allows 3D reconstruction of the complex refractive index. This talk outlines our contributions to the methodology of phase tomography on the level of the mathematical modelling [2, 3] and regularization [4, 5]. Further, we present several applications of the technique in biomedical imaging with synchrotron radiation. Special attention is given to the investigation of bone tissue at the micro and nano scale on images acquired at the ID19 and ID22 beamlines at the ESRF .
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 Guygay, J. P., Langer, M., Boistel, R., and Cloetens, P., "Mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region, Opt. Lett. Vol. 32, pp. 1617-1619, 2007.
 Davidoiu, V., Sixou, B., Langer, M., Peyrin, F., "Non-linear iterative phase retrieval based on Frechet derivative", Opt. Express Vol 19, pp. 22809-22819, 2011.
 Langer, M., Cloetens, P., and Peyrin, F., "Regularization of phase retrieval with phase-attenuation duality prior for 3D holotomography", IEEE Trans. Imag. Proces. Vol. 19, pp. 2428-2436, 2010.
 Langer, M., Cloetens, P., Pacureanu, A., and Peyrin, F., "X-ray in-line phase tomography of multimaterial objects", Opt. Lett. Vol 37, pp. 2151-2153, 2012.
 Langer, M., Pacureanu, A., Suhonen, H., Grimal, Q., Cloetens, P., and Peyrin, F., "X-ray phase nanotomography resolves the 3D human bone ultrastructure", PLoS One vol. 7, p. e35691, 2012.