  In this paper we propose a motion-aware variational approach to reconstruct
moving objects from sparse dynamic data. The motivation of this work stems from
X-ray imaging of plants perfused with a liquid contrast agent, aimed at
increasing the contrast of the images and studying the phloem transport in
plants over time. The key idea of our approach is to deploy 3D shearlets as a
space-temporal prior, treating time as the third dimension. The rationale
behind this model is that a continuous evolution of a cartoon-like object is
well suited for the use of 3D shearlets. We provide a basic mathematical
analysis of the variational model for the image reconstruction. The numerical
minimization is carried out with primal-dual scheme coupled with an automated
choice of regularization parameter. We test our model on different measurement
setups: a simulated phantom especially designed to resemble a plant stem, with
spreading points to simulate a spreading contrast agent; a measured agarose gel
phantom to demonstrate iodide diffusion and geometry prior to imaging living
sample; a measured living tree grown \textit{in vitro} and perfused with a
liquid sugar-iodine-mix. The results, compared against a 2D static model, show
that our approach provides reconstructions that capture well the time dynamic
of the contrast agent onset and are encouraging to develop microCT as a tool to
study phloem transport using iodine tracer.
