  Compressed sensing (CS) exploits the sparsity of a signal in order to
integrate acquisition and compression. CS theory enables exact reconstruction
of a sparse signal from relatively few linear measurements via a suitable
nonlinear minimization process. Conventional CS theory relies on vectorial data
representation, which results in good compression ratios at the expense of
increased computational complexity. In applications involving color images,
video sequences, and multi-sensor networks, the data is intrinsically of
high-order, and thus more suitably represented in tensorial form. Standard
applications of CS to higher-order data typically involve representation of the
data as long vectors that are in turn measured using large sampling matrices,
thus imposing a huge computational and memory burden. In this chapter, we
introduce Generalized Tensor Compressed Sensing (GTCS)--a unified framework for
compressed sensing of higher-order tensors which preserves the intrinsic
structure of tensorial data with reduced computational complexity at
reconstruction. We demonstrate that GTCS offers an efficient means for
representation of multidimensional data by providing simultaneous acquisition
and compression from all tensor modes. In addition, we propound two
reconstruction procedures, a serial method (GTCS-S) and a parallelizable method
(GTCS-P), both capable of recovering a tensor based on noiseless and noisy
observations. We then compare the performance of the proposed methods with
Kronecker compressed sensing (KCS) and multi-way compressed sensing (MWCS). We
demonstrate experimentally that GTCS outperforms KCS and MWCS in terms of both
reconstruction accuracy (within a range of compression ratios) and processing
speed. The major disadvantage of our methods (and of MWCS as well), is that the
achieved compression ratios may be worse than those offered by KCS.
