Nonlinear approaches for phase retrieval in the Fresnel region for hard X-ray imaging ; Approches non linéaire en imagerie de phase par rayons X dans le domaine de Fresnel

The development of highly coherent X-ray sources offers new possibilities to image biological structures at different scales exploiting the refraction of X-rays. The coherence properties of the third-generation synchrotron radiation sources enables efficient implementations of phase contrast techniques. One of the first measurements of the intensity variations due to phase contrast has been reported in 1995 at the European Synchrotron Radiation Facility (ESRF). Phase imaging coupled to tomography acquisition allows threedimensional imaging with an increased sensitivity compared to absorption CT. This technique is particularly attractive to image samples with low absorption constituents. Phase contrast has many applications, ranging from material science, paleontology, bone research to medicine and biology. Several methods to achieve X-ray phase contrast have been proposed during the last years. In propagation based phase contrast, the measurements are made at different sample-to-detector distances. While the intensity data can be acquired and recorded, the phase information of the signal has to be "retrieved" from the modulus data only. Phase retrieval is thus an illposed nonlinear problem and regularization techniques including a priori knowledge are necessary to obtain stable solutions. Several phase recovery methods have been developed in recent years. These approaches generally formulate the phase retrieval problem as a linear one. Nonlinear treatments have not been much investigated. The main purpose of this work was to propose and evaluate new algorithms, in particularly taking into account the nonlinearity of the direct problem. In the first part of this work, we present a Landweber type nonlinear iterative scheme to solve the propagation based phase retrieval problem. This approach uses the analytic expression of the Fréchet derivative of the phase-intensity relationship and of its adjoint, which are presented in detail. We also study the effect of projection operators on the convergence properties of ...