  In the framework of multidimensional Compressed Sensing (CS), we introduce an
analytical reconstruction formula that allows one to recover an $N$th-order
$(I_1\times I_2\times \cdots \times I_N)$ data tensor $\underline{\mathbf{X}}$
from a reduced set of multi-way compressive measurements by exploiting its low
multilinear-rank structure. Moreover, we show that, an interesting property of
multi-way measurements allows us to build the reconstruction based on
compressive linear measurements taken only in two selected modes, independently
of the tensor order $N$. In addition, it is proved that, in the matrix case and
in a particular case with $3$rd-order tensors where the same 2D sensor operator
is applied to all mode-3 slices, the proposed reconstruction
$\underline{\mathbf{X}}_\tau$ is stable in the sense that the approximation
error is comparable to the one provided by the best low-multilinear-rank
approximation, where $\tau$ is a threshold parameter that controls the
approximation error. Through the analysis of the upper bound of the
approximation error we show that, in the 2D case, an optimal value for the
threshold parameter $\tau=\tau_0 > 0$ exists, which is confirmed by our
simulation results. On the other hand, our experiments on 3D datasets show that
very good reconstructions are obtained using $\tau=0$, which means that this
parameter does not need to be tuned. Our extensive simulation results
demonstrate the stability and robustness of the method when it is applied to
real-world 2D and 3D signals. A comparison with state-of-the-arts sparsity
based CS methods specialized for multidimensional signals is also included. A
very attractive characteristic of the proposed method is that it provides a
direct computation, i.e. it is non-iterative in contrast to all existing
sparsity based CS algorithms, thus providing super fast computations, even for
large datasets.
