A mathematical framework and accompanying numerical algorithm exploiting the continuity equation for 4D reconstruction of spatiotemporal attenuation fields from multi-angle full-field transmission measurements is presented. The algorithm is geared towards rotation-free dynamic multi-beam X-ray tomography measurements, for which angular information is sparse but the temporal information is rich. 3D attenuation maps are recovered by propagating an initial discretized density volume in time according to the advection equations using the Finite Volumes method with a total variation diminishing monotonic upstream-centered scheme (TVDMUSCL). The benefits and limitations of the algorithm are explored using dynamic granular system phantoms modelled via discrete elements and projected by an analytical ray model independent from the numerical ray model used in the reconstruction scheme. Three phantom scenarios of increasing complexity are presented and it is found that projections from only a few (unknowns:equations > 10) angles can be sufficient for characterisation of the 3D attenuation field evolution in time. It is shown that the artificial velocity field produced by the algorithm sub-iteration, which is used to propagate the attenuation field, can to some extent approximate the true kinematics of the system. Furthermore, it is found that the selection of a temporal interpolation scheme for projection data can have a significant impact on error build up in the reconstructed attenuation field.
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