This paper extends the theory of equivalence canonical forms from quaternion matrices to quaternion tensors under the Einstein product. Motivated by recent results on the simultaneous decomposition of two specific configurations of five quaternion matrices, we establish a comprehensive framework for the corresponding configurations of five quaternion tensors. The core approach leverages bijective transformation maps that establish isomorphisms between quaternion tensor spaces and matrix spaces, allowing us to systematically construct invertible transformation tensors that simultaneously reduce the given tensor quintuples to canonical forms consisting solely of binary entries (0 and 1). A detailed structural analysis of the resulting canonical tensor forms is provided, including explicit dimension formulas for all identity blocks derived from precise rank conditions. To demonstrate practical utility, we integrate the proposed tensor decomposition with the discrete wavelet transform to construct a color video encryption and decryption system. Experimental results confirm perfect reconstruction (PSNR exceeding 300 dB, SSIM equal to 1) and strong security performance: NPCR of 49.8%, UACI of 49.6%, information entropy of 0.9986 bits per pixel, adjacent pixel correlation below 0.03 in absolute value, and a key space exceeding 2512. The developed theory significantly extends the existing literature on quaternion tensor decompositions and provides powerful tools for multidimensional signal processing.
He et al. (Tue,) studied this question.
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