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<p>Tensor completion is a fundamental task in computer vision and image processing, with wide-ranging applications from recommendation systems to medical imaging. Prevailing methods, predominantly based on the singular value decomposition (SVD) of real-valued tensors and nuclear norm minimization, face significant challenges: they often fail to preserve the intrinsic correlation among color channels and lack robustness against variations across video frames. Moreover, their reliance on computationally intensive SVD operations limits their scalability and practical utility for large-scale data. To address these limitations, we design the accelerated SVD decomposition algorithm: QTSVD-QR, and propose a novel quaternion tensor (matrix) completion method that utilizes QR decomposition and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L_{2,1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>-norm minimization (QTLNM-TQR), which can effectively balance model generalization ability with computational efficiency, resulting in a notable enhancement in completion performance. Numerical experiments conducted on color images, videos, and PET-CT provide compelling evidence of the proposed method’s effectiveness.</p>

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Efficient Quaternion Tensor Completion via \(L_{2,1}\)-Norm and QR Decomposition

  • 期刊论文

  • 发表日期：

    2026年7月29日

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  • Jian Sun,
  • Xin Liu,
  • Liang Xiao,
  • Hui Luo,
  • Yang Zhang

摘要

Tensor completion is a fundamental task in computer vision and image processing, with wide-ranging applications from recommendation systems to medical imaging. Prevailing methods, predominantly based on the singular value decomposition (SVD) of real-valued tensors and nuclear norm minimization, face significant challenges: they often fail to preserve the intrinsic correlation among color channels and lack robustness against variations across video frames. Moreover, their reliance on computationally intensive SVD operations limits their scalability and practical utility for large-scale data. To address these limitations, we design the accelerated SVD decomposition algorithm: QTSVD-QR, and propose a novel quaternion tensor (matrix) completion method that utilizes QR decomposition and \(L_{2,1}\) L 2 , 1 -norm minimization (QTLNM-TQR), which can effectively balance model generalization ability with computational efficiency, resulting in a notable enhancement in completion performance. Numerical experiments conducted on color images, videos, and PET-CT provide compelling evidence of the proposed method’s effectiveness.