Purpose: To investigate how an intrinsic speckle-tracking approach to speckle-based X-ray imaging can be used to extract an object's effective dark-field signal, which is capable of providing object information in three dimensions. Approach:The effective dark-field signal was extracted using a Fokker-Planck type formalism, which models the deformations of illuminating reference-beam speckles due to both coherent and diffusive scatter from the sample. We here assumed that (a) small-angle scattering fans at the exit surface of the sample are rotationally symmetric, and (b) the object has both attenuating and refractive properties. The associated inverse problem, of extracting the effective dark-field signal, was numerically stabilised using a "weighted determinants" approach.Results: Effective dark-field projection images are presented, as well as the dark-field tomographic reconstructions of the wood sample. Dark-field tomography was performed using a filtered-back projection reconstruction algorithm. The dark-field tomographic reconstructions of the wood sample provided complementary, and otherwise inaccessible, information to augment the phase-contrast reconstructions, which were also computed.Conclusions: An intrinsic speckle-tracking approach to speckle-based imaging can tomographically reconstruct an object's dark-field signal at a low sample exposure and with a simple experimental set-up. The obtained dark-field reconstructions have image quality comparable to alternative X-ray dark-field techniques.

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