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.

翻译：本文提出了低复杂度多维离散余弦变换（DCT）近似方法。基于高阶张量理论，形式化定义了三维离散余弦变换（3D DCT）近似。该公式可扩展至任意长度的更高维度。提出了多种无需乘法的$8\times 8\times 8$近似方法，并讨论了通用多维情况下的计算复杂度。对所提方法的复杂度进行了评估，与精确3D DCT相比，其算术运算量显著降低。将所提近似方法嵌入基于3D DCT的视频编码方案中，并引入改进的量化步骤。仿真结果表明，与精确3D DCT方案相比，近似3D DCT编码方法在输出视觉质量上几乎相同。所提出的3D近似方法还被用作视觉跟踪工具。基于近似3D DCT的提出系统性能与原始精确3D DCT方法相近。总体而言，所提方法在显著降低计算成本的同时，展现了具有竞争力的性能。