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The in-line X-ray phase contrast imaging technique relies on the measurement of Fresnel diffraction intensity patterns due to the phase shift and the absorption induced by the object. The recovery of both phase and absorption is an ill-posed non-linear inverse problem. In this work, we address this problem with an iterative algorithm based on a primal-dual method, which allows us to introduce the non-linearity of the forward operator. We used a variational approach with different regularizations for the phase and absorption, in order to take into account the specificities of each quantity. Assuming the solution to be piecewise constant, the functional used involves the Total Generalized Variation (TGV) as well as the classical Total Variation (TV), which enables a compromise between sharp discontinuities and smoothness in the solution. This optimization problem is solved efficiently using primal-dual approach such as Primal-Dual Hybrid Gradient Method (PDHGM). From this approach, we propose an algorithm called PDGHGM-CTF, which is based on the linearized Contrast Transfer Function model, that we generalize for the nonlinear problem to get the Non-Linear Primal-Dual Hybrid Gradient Method (NL-PDHGM). The proposed iterative algorithms are able to recover simultaneously the phase and absorption from a single diffraction pattern without homogeneity assumption or support constraint, and the nonlinear version is valid without restriction on the object. Moreover, we show that the approach is robust with respect to the initialization. While giving a good approximation as starting point reduced the convergence time, it did not improve the reconstruction results. We demonstrate the potential of the proposed algorithms on simulated datasets. We show that it produces reconstructions with fewer artifacts and improved normalized mean squared error compared to a gradient descent scheme. We evaluate the robustness of the proposed algorithm by evaluating the reconstruction on simulated images of 1 000 random objects, given the same hyperparameters.
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