目的 各类终端设备获取的大量数据往往由于信息丢失而导致数据不完整,或经常受到降质问题的困扰。为有效恢复缺损或降质数据,低秩张量补全备受关注。张量分解可有效挖掘张量数据的内在特征,但传统分解方法诱导的张量秩函数无法探索张量不同模式之间的相关性;另外,传统张量补全方法通常将全变分约束施加于整体张量数据,无法充分利用张量低维子空间的平滑先验。为解决以上两个问题,提出了基于稀疏先验与多模式张量分解的低秩张量恢复方法。方法 在张量秩最小化模型基础上,融入多模式张量分解技术以及分解因子局部稀疏性。首先对原始张量施加核范数约束,以此捕获张量的全局低秩性,然后,利用多模式张量分解将整体张量沿着每个模式分解为一组低维张量和一组因子矩阵,以探索不同模式之间的相关性,对因子矩阵施加因子梯度稀疏正则化约束,探索张量子空间的局部稀疏性,进一步提高张量恢复性能。结果 在高光谱图像、多光谱图像、YUV（也称为YCbCr）视频和医学影像数据上,将本文方法与其他8种修复方法在3种丢失率下进行定量及定性比较。在恢复4种类型张量数据方面,本文方法与深度学习GP-WLRR方法（global prior refined weighted low-rank representation）的修复效果基本持平，本文方法的MPSNR（mean peak signal-to-noise ratio）在所有丢失率及张量数据上的总体平均高0.68dB,MSSIM（mean structural similarity）总体平均高0.01;与其他6种张量建模方法相比,本文方法的MPSNR及MSSIM均取得最优结果。结论 提出的基于稀疏先验与多模式张量分解的低秩张量恢复方法,可同时利用张量的全局低秩性与局部稀疏性,能够对受损的多维视觉数据进行有效修复。

Objective The large amount of data obtained by various terminal devices often results in incomplete data due to missing information or is frequently plagued by degradation issues.Low-rank tensor completion has received significant attention for recovering contaminated data.Tensor decomposition can effectively explore the essential features of tensors,but the tensor rank function induced by traditional tensor decomposition methods cannot explore the correlation between different modes of tensors.In addition,traditional tensor completion methods typically impose the total variational constraint on the overall tensor data,which cannot fully utilize the smoothing prior for the low-dimensional subspace of tensors.To address the above two problems,this study proposes a low-rank tensor recovery algorithm using sparse prior and multimode tensor factorization.The traditional low-rank tensor completion models based on tensor rank minimization restore tensors by directly minimizing the tensor rank,in which the tensor rank can be Tucker rank and tensor nuclear norm(TNN).However,extensive research has shown that a correlation exists among different modes of tensor data.The Tucker rank induced by Tucker decomposition and the TNN induced by tensor singular value decomposition cannot flexibly handle multimode correlations within tensors.Therefore,we consider introducing multimode tensor decomposition via mode-n product,incorporating multimode tensor decomposition into the tensor rank minimization model.In the process of continuous iteration to complete the overall tensor,our model can effectively explore the characteristics of mutual correlation between different modes of the tensor,which can address the limitation of traditional TNN in inadequately capturing the intermode correlations within the tensor.Each factor matrix obtained from the multimode tensor decomposition framework encapsulates latent information corresponding to its respective mode,revealing valuable correlated auxiliary information within and across modes,such as the local sparsity exhibited by natural tensor data.By showing that the majority of factor gradients in the factor gradient histogram are zero or close to zero,we can demonstrate that the factors in multimode tensor decomposition exhibit local sparsity.Therefore,on the basis of the assumption of tensor subspace,we consider introducing the local sparsity prior to preserve the similarity in local segments.Method The method incorporates multimode tensor factorization and local sparsity of decomposed factors based on the tensor rank minimization model.First,the nuclear norm constraint is imposed on the original tensor to capture the global low rankness of the tensor,which makes the model robust when dealing with tensor completion tasks.Second,multimode tensor factorization is used to decompose the tensor into a series of lowdimensional tensors and a series of factor matrices along each mode,which explores the correlation between different modes.The factor gradient sparsity regularization constraint is imposed on the factor matrices to explore the local sparsity of the tensor subspace,which further improves the tensor recovery performance.Specifically,after tensor decomposition,first-order differencing is applied,and the norm smoothness constraint is leveraged.Combining multimode tensor decomposition with tensor subspace sparsity,a robust tensor completion model is developed.The proposed model is optimized through the alternating direction method of multipliers(ADMM)framework,which is achieved by iteratively updating various variables to accomplish tensor completion and tensor decomposition simultaneously.Result The method in this paper is quantitatively and qualitatively compared with eight other restoration methods at three loss rates on hyperspectral images,multispectral images,YUV(also known as YCbCr)videos,and medical imaging data.The restoration effect of our method is basically the same as that of the deep learning GP-WLRR method,but it has no computational burden at all.Compared with six other tensor modeling methods,our method achieves the best results in terms of mean peak signal-to-noise ratio (MPSNR)and mean structural similarity(MSSIM)metrics.It exhibits superior recovery performance even at high loss rates up to 95%.This finding demonstrates the effectiveness of the proposed model in tensor data recovery.Conclusion The low-rank tensor completion algorithm based on sparse prior and multimode tensor decomposition proposed in this paper can simultaneously exploit the global low rankness and local sparsity of a tensor and effectively recover contaminated multichannel visual data.