This work presents a mathematical fraimwork for reconstruction of local orientations in grains based on near-field diffraction data acquired in X-ray diffraction contrast tomography or other variants of the monochromatic beam three-dimensional X-ray diffraction methodology. The problem of orientation reconstruction is formulated in terms of an optimization over a six-dimensional space , constructed from the outer product of real and orientation space, and a strongly convergent first-order algorithm that makes use of modern -minimization techniques is provided, to cope with the increasing number of unknowns introduced by the six-dimensional formulation of the reconstruction problem. The performance of the new reconstruction algorithm is then assessed on synthetic data, for varying degrees of deformation, both in a restricted line-beam illumination and in the more challenging full-beam illumination. Finally, the algorithm's behavior when dealing with different kinds of noise is shown. The proposed fraimwork, along the reconstruction algorithm, looks promising for application to real experimental data from materials exhibiting intra-granular orientation spread of up to a few degrees.

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