  In compressed sensing, it is often desirable to consider signals possessing
additional structure beyond sparsity. One such structured signal model - which
forms the focus of this paper - is the local sparsity in levels class. This
class has recently found applications in problems such as compressive imaging,
multi-sensor acquisition systems and sparse regularization in inverse problems.
In this paper we present uniform recovery guarantees for this class when the
measurement matrix corresponds to a subsampled isometry. We do this by
establishing a variant of the standard restricted isometry property for sparse
in levels vectors, known as the restricted isometry property in levels.
Interestingly, besides the usual log factors, our uniform recovery guarantees
are simpler and less stringent than existing nonuniform recovery guarantees.
For the particular case of discrete Fourier sampling with Haar wavelet
sparsity, a corollary of our main theorem yields a new recovery guarantee which
improves over the current state-of-the-art.
