  In this paper, we prove a compressive sensing guarantee for restricted
measurement domains on the rotation group, $\mathrm{SO}(3)$. We do so by first
defining Slepian functions on a measurement sub-domain $R$ of the rotation
group $\mathrm{SO}(3)$. Then, we transform the inverse problem from the
measurement basis, the bounded orthonormal system of band-limited Wigner
$D$-functions on $\mathrm{SO}(3)$, to the Slepian functions in a way that
limits increases to signal sparsity. Contrasting methods using Wigner
$D$-functions that require measurements on all of $\mathrm{SO}(3)$, we show
that the orthogonality structure of the Slepian functions only requires
measurements on the sub-domain $R$, which is select-able. Due to the
particulars of this approach and the inherent presence of Slepian functions
with low concentrations on $R$, our approach gives the highest accuracy when
the signal under study is well concentrated on $R$. We provide numerical
examples of our method in comparison with other classical and compressive
sensing approaches. In terms of reconstruction quality, we find that our method
outperforms the other compressive sensing approaches we test and is at least as
good as classical approaches but with a significant reduction in the number of
measurements.
