scholarly journals Hyperspectral Image Recovery Using Non-Convex Low-Rank Tensor Approximation

2020 ◽  
Vol 12 (14) ◽  
pp. 2264
Author(s):  
Hongyi Liu ◽  
Hanyang Li ◽  
Zebin Wu ◽  
Zhihui Wei
Keyword(s):  
Hyperspectral Image ◽  
Convex Relaxation ◽  
Principal Component ◽  
Nuclear Norm ◽  
Low Rank ◽  
Low Rank Approximation ◽  
Rank Tensor ◽  
Alternating Direction ◽  
Tensor Approximation ◽  
Tensor Nuclear Norm

Low-rank tensors have received more attention in hyperspectral image (HSI) recovery. Minimizing the tensor nuclear norm, as a low-rank approximation method, often leads to modeling bias. To achieve an unbiased approximation and improve the robustness, this paper develops a non-convex relaxation approach for low-rank tensor approximation. Firstly, a non-convex approximation of tensor nuclear norm (NCTNN) is introduced to the low-rank tensor completion. Secondly, a non-convex tensor robust principal component analysis (NCTRPCA) method is proposed, which aims at exactly recovering a low-rank tensor corrupted by mixed-noise. The two proposed models are solved efficiently by the alternating direction method of multipliers (ADMM). Three HSI datasets are employed to exhibit the superiority of the proposed model over the low rank penalization method in terms of accuracy and robustness.

scholarly journals Hyperspectral Image Denoising via Framelet Transformation Based Three-Modal Tensor Nuclear Norm

2021 ◽  
Vol 13 (19) ◽  
pp. 3829
Author(s):  
Wenfeng Kong ◽  
Yangyang Song ◽  
Jing Liu
Keyword(s):  
Fourier Transform ◽  
Image Quality ◽  
Hyperspectral Image ◽  
Convex Relaxation ◽  
Nuclear Norm ◽  
Low Rank ◽  
Tensor Rank ◽  
Spatial Dimensions ◽  
Value Decomposition ◽  
Tensor Nuclear Norm

During the acquisition process, hyperspectral images (HSIs) are inevitably contaminated by mixed noise, which seriously affects the image quality. To improve the image quality, HSI denoising is a critical preprocessing step. In HSI denoising tasks, the method based on low-rank prior has achieved satisfying results. Among numerous denoising methods, the tensor nuclear norm (TNN), based on the tensor singular value decomposition (t-SVD), is employed to describe the low-rank prior approximately. Its calculation can be sped up by the fast Fourier transform (FFT). However, TNN is computed by the Fourier transform, which lacks the function of locating frequency. Besides, it only describes the low-rankness of the spectral correlations and ignores the spatial dimensions’ information. In this paper, to overcome the above deficiencies, we use the basis redundancy of the framelet and the low-rank characteristics of HSI in three modes. We propose the framelet-based tensor fibered rank as a new representation of the tensor rank, and the framelet-based three-modal tensor nuclear norm (F-3MTNN) as its convex relaxation. Meanwhile, the F-3MTNN is the new regularization of the denoising model. It can explore the low-rank characteristics of HSI along three modes that are more flexible and comprehensive. Moreover, we design an efficient algorithm via the alternating direction method of multipliers (ADMM). Finally, the numerical results of several experiments have shown the superior denoising performance of the proposed F-3MTNN model.

scholarly journals Low-Rank Tensor Completion via Tensor Nuclear Norm With Hybrid Smooth Regularization

IEEE Access ◽  
2019 ◽  
Vol 7 ◽  
pp. 131888-131901
Author(s):  
Xi-Le Zhao ◽  
Xin Nie ◽  
Yu-Bang Zheng ◽  
Teng-Yu Ji ◽  
Ting-Zhu Huang
Keyword(s):  
Nuclear Norm ◽  
Low Rank ◽  
Tensor Completion ◽  
Rank Tensor ◽  
Tensor Nuclear Norm

scholarly journals Tensor theta norms and low rank recovery

2020 ◽  
Author(s):  
Holger Rauhut ◽  
Željka Stojanac
Keyword(s):  
Convex Relaxation ◽  
Nuclear Norm ◽  
Low Rank ◽  
Nuclear Norm Minimization ◽  
Computational Algebraic Geometry ◽  
Matrix Recovery ◽  
Norm Minimization ◽  
The Matrix ◽  
Higher Order Tensors ◽  
Tensor Nuclear Norm

AbstractWe study extensions of compressive sensing and low rank matrix recovery to the recovery of tensors of low rank from incomplete linear information. While the reconstruction of low rank matrices via nuclear norm minimization is rather well-understand by now, almost no theory is available so far for the extension to higher order tensors due to various theoretical and computational difficulties arising for tensor decompositions. In fact, nuclear norm minimization for matrix recovery is a tractable convex relaxation approach, but the extension of the nuclear norm to tensors is in general NP-hard to compute. In this article, we introduce convex relaxations of the tensor nuclear norm which are computable in polynomial time via semidefinite programming. Our approach is based on theta bodies, a concept from real computational algebraic geometry which is similar to the one of the better known Lasserre relaxations. We introduce polynomial ideals which are generated by the second-order minors corresponding to different matricizations of the tensor (where the tensor entries are treated as variables) such that the nuclear norm ball is the convex hull of the algebraic variety of the ideal. The theta body of order k for such an ideal generates a new norm which we call the θk-norm. We show that in the matrix case, these norms reduce to the standard nuclear norm. For tensors of order three or higher however, we indeed obtain new norms. The sequence of the corresponding unit-θk-norm balls converges asymptotically to the unit tensor nuclear norm ball. By providing the Gröbner basis for the ideals, we explicitly give semidefinite programs for the computation of the θk-norm and for the minimization of the θk-norm under an affine constraint. Finally, numerical experiments for order-three tensor recovery via θ1-norm minimization suggest that our approach successfully reconstructs tensors of low rank from incomplete linear (random) measurements.

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