  Tensor Compressive Sensing (TCS) is a multidimensional framework of
Compressive Sensing (CS), and it is advantageous in terms of reducing the
amount of storage, easing hardware implementations and preserving
multidimensional structures of signals in comparison to a conventional CS
system. In a TCS system, instead of using a random sensing matrix and a
predefined dictionary, the average-case performance can be further improved by
employing an optimized multidimensional sensing matrix and a learned
multilinear sparsifying dictionary. In this paper, we propose a joint
optimization approach of the sensing matrix and dictionary for a TCS system.
For the sensing matrix design in TCS, an extended separable approach with a
closed form solution and a novel iterative non-separable method are proposed
when the multilinear dictionary is fixed. In addition, a multidimensional
dictionary learning method that takes advantages of the multidimensional
structure is derived, and the influence of sensing matrices is taken into
account in the learning process. A joint optimization is achieved via
alternately iterating the optimization of the sensing matrix and dictionary.
Numerical experiments using both synthetic data and real images demonstrate the
superiority of the proposed approaches.
