• KSII Transactions on Internet and Information Systems (TIIS)
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• v.13 no.12
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• pp.6043-6062
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• 2019

Low-rank matrix decomposition has shown its capability in many applications such as image in-painting, de-noising, background reconstruction and defect detection etc. In this paper, we consider the texture background of rail track images and the sparse foreground of the defects to construct a low-rank matrix decomposition model with block sparsity for defect inspection of rail tracks, which jointly minimizes the nuclear norm and the 2-1 norm. Similar to ADM, an alternative method is proposed in this study to solve the optimization problem. After image decomposition, the defect areas in the resulting low-rank image will form dark stripes that horizontally cross the entire image, indicating the preciselocations of the defects. Finally, a two-stage defect extraction method is proposed to locate the defect areas. The experimental results of the two datasets show that our algorithm achieved better performance compared with other methods.

• Journal of the Institute of Electronics Engineers of Korea SP
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• v.46 no.5
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• pp.15-24
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• 2009

Compressed sensing addresses the recovery of a sparse vector from its few linear measurements. Recently, the success for the vector case has been extended to the matrix case. Compressed sensing of low-rank matrices solves the ill-posed inverse problem with fie low-rank prior. The problem can be formulated as either the rank minimization or the low-rank approximation. In this paper, we survey recently proposed efficient algorithms to solve these two formulations.

• Journal of Communications and Networks
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• v.12 no.4
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• pp.289-307
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• 2010

The problem of model selection arises in a number of contexts, such as subset selection in linear regression, estimation of structures in graphical models, and signal denoising. This paper studies non-asymptotic model selection for the general case of arbitrary (random or deterministic) design matrices and arbitrary nonzero entries of the signal. In this regard, it generalizes the notion of incoherence in the existing literature on model selection and introduces two fundamental measures of coherence-termed as the worst-case coherence and the average coherence-among the columns of a design matrix. It utilizes these two measures of coherence to provide an in-depth analysis of a simple, model-order agnostic one-step thresholding (OST) algorithm for model selection and proves that OST is feasible for exact as well as partial model selection as long as the design matrix obeys an easily verifiable property, which is termed as the coherence property. One of the key insights offered by the ensuing analysis in this regard is that OST can successfully carry out model selection even when methods based on convex optimization such as the lasso fail due to the rank deficiency of the submatrices of the design matrix. In addition, the paper establishes that if the design matrix has reasonably small worst-case and average coherence then OST performs near-optimally when either (i) the energy of any nonzero entry of the signal is close to the average signal energy per nonzero entry or (ii) the signal-to-noise ratio in the measurement system is not too high. Finally, two other key contributions of the paper are that (i) it provides bounds on the average coherence of Gaussian matrices and Gabor frames, and (ii) it extends the results on model selection using OST to low-complexity, model-order agnostic recovery of sparse signals with arbitrary nonzero entries. In particular, this part of the analysis in the paper implies that an Alltop Gabor frame together with OST can successfully carry out model selection and recovery of sparse signals irrespective of the phases of the nonzero entries even if the number of nonzero entries scales almost linearly with the number of rows of the Alltop Gabor frame.