In this paper, we propose a new technique for interpolating shapes in order to upsample a sparsely acquired serial-section image stack. The method is based on a maximum a posteriori estimation strategy which models neighboring sections as observations of random deformations of an image to be estimated. We show the computation of diffeomorphic trajectories between observed sections and define estimated upsampled image sections as a Jacobian-weighted sum of flowing images at corresponding distances along those trajectories. We apply this methodology to upsample stacks of sparse 2D magnetic resonance cross-sections through live mouse hearts. We show that the proposed method results in smoother and more accurate reconstructions over linear interpolation, and report a Dice coefficient of 0.8727 against ground truth segmentations in our dataset and statistically significant improvements in both left ventricular segmentation accuracy and image intensity estimates.