Weighted Schatten $p$ -Norm Minimization for Image Denoising and Background Subtraction

By Yuan Xie, Shuhang Gu, Yan Liu, Wangmeng Zuo, Wensheng Zhang, Lei Zhang. IEEE Transactions on Image Processing, 2015

414 citations recorded by the source corpus.

Low rank matrix approximation (LRMA), which aims to recover the underlying low rank matrix from its degraded observation, has a wide range of applications in computer vision. The latest LRMA methods resort to using the nuclear norm minimization (NNM) as a convex relaxation of the nonconvex rank minimization. However, NNM tends to over-shrink the rank components and treats the different rank components equally, limiting its flexibility in practical applications. We propose a more flexible model, namely, the weighted Schatten p-norm minimization (WSNM), to generalize the NNM to the Schatten p-norm minimization with weights assigned to different singular values. The proposed WSNM not only gives better approximation to the original low-rank assumption, but also considers the importance of different rank components. We analyze the solution of WSNM and prove that, under certain weights permutation, WSNM can be equivalently transformed into independent non-convex lp-norm subproblems, whose global optimum can be efficiently solved by generalized iterated shrinkage algorithm. We apply WSNM to typical low-level vision problems, e.g., image denoising and background subtraction. Extensive experimental results show, both qualitatively and quantitatively, that the proposed WSNM can more effectively remove noise, and model the complex and dynamic scenes compared with state-of-the-art methods.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Media Technology

View Paper: https://www.semanticscholar.org/paper/410953002d72988e2befd58044a6444904e9ee2f

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In Computational Mechanics

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Citations in the corpus

Rank Minimization for Snapshot Compressive Imaging (2018, background): “Despite the theoretical guarantee of the singular value thresholding model [23], it has been observed that the recovery performance of such a convex relaxationwill degrade in the presence of noise, and the solution can seriously deviate from the original solution of the rank minimization problem [50”
Schatten Norms in Matrix Streams: Hello Sparsity, Goodbye Dimension (2019, methodology): “…are often used as surrogates for the spectrum, as explained in (Zhang et al., 2015; Kong and Valiant, 2016; Napoli et al., 2016; Khetan and Oh, 2019), and some speciﬁc cases have applications in optimization, image processing, and differential privacy etc. (Xie et al., 2016; Musco et al., 2018).”
Infrared Dim and Small Target Detection via Multiple Subspace Learning and Spatial-Temporal Patch-Tensor Model (2021, methodology): “Nevertheless, Xie et al. [27] proves that the variation of variables would converge to zero in WSNM based optimization problem, and the details can be found in Theorem 3 of work [27].”
Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset (2015, methodology): “– Matrix Schatten-α norm with 0 < α ≤ 2: The Schatten-α norm ||M ||Sα = ( ∑min(m,n) k=1 (σk(M)) ) where σk(M), denotes the k th singular values of M , can also be used as a surrogate of the nuclear norm like in [383][330][387][406].”