  We introduce single-shot X-ray tomography that aims to estimate the target
image from a single cone-beam projection measurement. This linear inverse
problem is extremely under-determined since the measurements are far fewer than
the number of unknowns. Moreover, it is more challenging than conventional
tomography where a sufficiently large number of projection angles forms the
measurements, allowing for a simple inversion process. However, single-shot
tomography becomes less severe if the target image is only composed of known
shapes. Hence, the shape prior transforms a linear ill-posed image estimation
problem to a non-linear problem of estimating the roto-translations of the
shapes. In this paper, we circumvent the non-linearity by using a dictionary of
possible roto-translations of the shapes. We propose a convex program CoShaRP
to recover the dictionary-coefficients successfully. CoShaRP relies on
simplex-type constraint and can be solved quickly using a primal-dual
algorithm. The numerical experiments show that CoShaRP recovers shapes stably
from moderately noisy measurements.
