  In this paper, we introduce low-complexity multidimensional discrete cosine
transform (DCT) approximations. Three dimensional DCT (3D DCT) approximations
are formalized in terms of high-order tensor theory. The formulation is
extended to higher dimensions with arbitrary lengths. Several multiplierless
$8\times 8\times 8$ approximate methods are proposed and the computational
complexity is discussed for the general multidimensional case. The proposed
methods complexity cost was assessed, presenting considerably lower arithmetic
operations when compared with the exact 3D DCT. The proposed approximations
were embedded into 3D DCT-based video coding scheme and a modified quantization
step was introduced. The simulation results showed that the approximate 3D DCT
coding methods offer almost identical output visual quality when compared with
exact 3D DCT scheme. The proposed 3D approximations were also employed as a
tool for visual tracking. The approximate 3D DCT-based proposed system performs
similarly to the original exact 3D DCT-based method. In general, the suggested
methods showed competitive performance at a considerably lower computational
cost.
