A new approach is presented for hierarchical decomposition of third‐order tensors through their transformation into the generalized three‐dimensional (3D) spectrum space based on the inverse difference pyramid (IDP). For this, we choose the 3D Walsh–Hadamard transform (3D‐WHT). As result, each tensor is represented as a spectral tensor of m hierarchical levels which contains selected low‐frequency 3D‐WHT coefficients. Calculating sequentially the inverse 3D‐WHT for the coefficients from each pyramid level starting from its top, the tensor is approximated with increasing accuracy until its full restoration is achieved. To illustrate the new approach, given is the algorithm for hierarchical three‐level tensor decomposition based on the reduced IDP. The proposed approach permits simultaneous decorrelation of tensor elements in three mutually orthogonal directions. The energy of the tensor elements is concentrated in a small number of spectral coefficients which build the top of the inverse pyramid. The use of the 3D‐WHT permits to achieve minimum computational complexity, compared to deterministic 3D orthogonal transforms. The main applications of the new method for data mining in the contemporary intelligent systems are in the processing and analysis of large sets of different kinds of data/images/videos in the following areas: Compression of correlated image sequences, computer tomography, thermo vision, ultrasound and multichannel medical signals; search of 3D objects in image databases; extraction of features for recognition of 3D objects; multidimensional data denoising; multilayer watermarking of video sequences; and so on.

中文翻译：

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基于三维Walsh-Hadamard变换的逆差金字塔分层三阶张量分解及其在数据挖掘中的应用

提出了一种新的方法，用于通过将三阶张量转换为基于逆差金字塔（IDP）的广义三维（3D）频谱空间进行三阶张量的层次分解。为此，我们选择3D Walsh-Hadamard变换（3D-WHT）。结果，每个张量都表示为m的谱张量包含选定的低频3D-WHT系数的分层级别。从金字塔的顶部开始依次计算每个金字塔等级的系数的逆3D-WHT，以接近的精度估计张量，直到完全恢复。为了说明这种新方法，给出了基于简化IDP的分层三级张量分解算法。所提出的方法允许在三个相互正交的方向上同时对张量元素进行去相关。张量元素的能量集中在少数光谱系数中，这些光谱系数构成了反金字塔的顶部。与确定性3D正交变换相比，使用3D-WHT可以实现最低的计算复杂度。新的数据挖掘方法在现代智能系统中的主要应用是在以下领域处理和分析各种不同类型的数据/图像/视频：相关图像序列的压缩，计算机断层扫描，热成像，超声波和多通道医疗信号；在图像数据库中搜索3D对象；提取用于识别3D对象的特征；多维数据去噪；视频序列的多层水印；等等。提取用于识别3D对象的特征；多维数据去噪；视频序列的多层水印；等等。提取用于识别3D对象的特征；多维数据去噪；视频序列的多层水印；等等。