Tensor Compressive Sensing (TCS) is a multidimensional framework ofCompressive Sensing (CS), and it is advantageous in terms of reducing theamount of storage, easing hardware implementations and preservingmultidimensional structures of signals in comparison to a conventional CSsystem. In a TCS system, instead of using a random sensing matrix and apredefined dictionary, the average-case performance can be further improved byemploying an optimized multidimensional sensing matrix and a learnedmultilinear sparsifying dictionary. In this paper, we propose a jointoptimization approach of the sensing matrix and dictionary for a TCS system.For the sensing matrix design in TCS, an extended separable approach with aclosed form solution and a novel iterative non-separable method are proposedwhen the multilinear dictionary is fixed. In addition, a multidimensionaldictionary learning method that takes advantages of the multidimensionalstructure is derived, and the influence of sensing matrices is taken intoaccount in the learning process. A joint optimization is achieved viaalternately iterating the optimization of the sensing matrix and dictionary.Numerical experiments using both synthetic data and real images demonstrate thesuperiority of the proposed approaches.
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