  The dual-tree complex wavelet transform (DT-$\mathbb{C}$WT) is extended to
the 4D setting. Key properties of 4D DT-$\mathbb{C}$WT, such as directional
sensitivity and shift-invariance, are discussed and illustrated in a
tomographic application. The inverse problem of reconstructing a dynamic
three-dimensional target from X-ray projection measurements can be formulated
as 4D space-time tomography. The results suggest that 4D DT-$\mathbb{C}$WT
offers simple implementations combined with useful theoretical properties for
tomographic reconstruction.
