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.

本文介绍了低复杂度的多维离散余弦变换(DCT)近似算法，基于高阶张量理论，我们提出了几种$8	imes 8	imes 8$的近似方法，并将其嵌入到基于3D DCT的视频编码方案中，同时引入了改进的量化步骤，模拟结果表明所提出的3D近似算法与3D DCT方案的输出图像质量几乎相同，同时，所提出的3D近似算法还被应用于视觉跟踪领域，具有与精确3D DCT方案相似的表现且计算成本显著降低。

低复杂度的多维DCT近似