• Geophysics and Geophysical Exploration
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• v.10 no.2
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• pp.124-130
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• 2007

Least squares ($l^2$ norm) solutions of seismic inversion tend to be very sensitive to data points with large errors. The $l^1$ norm minimization gives more robust solutions, but usually with higher computational cost. Iteratively reweighted least squares (IRLS) method gives efficient approximate solutions of these $l^1$ norm problems. I propose an efficient implementation of the IRLS method for a hybrid $l^1/l^2$ minimization problem that behaves as $l^2$ norm fit for small residual and $l^1$ norm fit for large residuals. The proposed algorithm shows more robust characteristics to the decision of the threshold value than the l1 norm IRLS inversion does with respect to the threshold value to avoid singularity.

• Journal of Software Assessment and Valuation
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• v.16 no.2
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• pp.153-162
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• 2020

Digital image forgery detection is one of very important fields in the field of digital forensics. As the forged images change naturally through the advancement of technology, it has made it difficult to detect forged images. In this paper, we use passive forgery detection for copy paste forgery in digital images. In addition, it detects copy-paste forgery using the L₀ Norm-based LE operator, and compares the detection accuracy with the forgery detection using the existing L₂, L₁ Norm-based LE operator. In comparison of detection rates, the proposed lower triangular(Ayalneh and Choi) window was more robust to BAG mismatch detection than the conventional window filter. In addition, in the case of using the lower triangular window, the performance of image forgery detection was measured increasingly higher as the L₂, L₁ and L₀ Norm LE operator was performed.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.12 no.7
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• pp.3194-3216
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• 2018

Slow Feature Discriminant Analysis (SFDA) is a supervised feature extraction method inspired by biological mechanism. In this paper, a novel method called Two Dimensional Slow Feature Discriminant Analysis via $L_{2,1}$ norm minimization ($2DSFDA-L_{2,1}$) is proposed. $2DSFDA-L_{2,1}$ integrates $L_{2,1}$ norm regularization and 2D statically uncorrelated constraint to extract discriminant feature. First, $L_{2,1}$ norm regularization can promote the projection matrix row-sparsity, which makes the feature selection and subspace learning simultaneously. Second, uncorrelated features of minimum redundancy are effective for classification. We define 2D statistically uncorrelated model that each row (or column) are independent. Third, we provide a feasible solution by transforming the proposed $L_{2,1}$ nonlinear model into a linear regression type. Additionally, $2DSFDA-L_{2,1}$ is extended to a bilateral projection version called $BSFDA-L_{2,1}$. The advantage of $BSFDA-L_{2,1}$ is that an image can be represented with much less coefficients. Experimental results on three face databases demonstrate that the proposed $2DSFDA-L_{2,1}/BSFDA-L_{2,1}$ can obtain competitive performance.

• Smart Structures and Systems
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• v.34 no.2
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• pp.97-116
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• 2024

The standard l₁-norm regularization is recently introduced for impact force identification, but generally underestimates the peak force. Compared to l₁-norm regularization, l_p-norm (0 ≤ p < 1) regularization, with a nonconvex penalty function, has some promising properties such as enforcing sparsity. In the framework of sparse regularization, if the desired solution is sparse in the time domain or other domains, the under-determined problem with fewer measurements than candidate excitations may obtain the unique solution, i.e., the sparsest solution. Considering the joint sparse structure of impact force in temporal and spatial domains, we propose a general l_p-norm (0 ≤ p < 1) regularization methodology for simultaneous identification of the impact location and force time-history from highly incomplete measurements. Firstly, a nonconvex optimization model based on l_p-norm penalty is developed for regularizing the highly under-determined problem of impact force identification. Secondly, an iteratively reweighed l₁-norm algorithm is introduced to solve such an under-determined and unconditioned regularization model through transforming it into a series of l₁-norm regularization problems. Finally, numerical simulation and experimental validation including single-source and two-source cases of impact force identification are conducted on plate structures to evaluate the performance of l_p-norm (0 ≤ p < 1) regularization. Both numerical and experimental results demonstrate that the proposed l_p-norm regularization method, merely using a single accelerometer, can locate the actual impacts from nine fixed candidate sources and simultaneously reconstruct the impact force time-history; compared to the state-of-the-art l₁-norm regularization, l_p-norm (0 ≤ p < 1) regularization procures sufficiently sparse and more accurate estimates; although the peak relative error of the identified impact force using l_p-norm regularization has a decreasing tendency as p is approaching 0, the results of l_p-norm regularization with 0 ≤ p ≤ 1/2 have no significant differences.

• Journal of IKEEE
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• v.20 no.3
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• pp.226-234
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• 2016

Electrical resistance tomography (ERT) is an imaging technique where the internal resistivity distribution inside an object is reconstructed. The ERT image reconstruction is a highly nonlinear ill-posed problem, so regularization methods are used to achieve desired image. The reconstruction outcome is dependent on the type of regularization method employed such as l2-norm, l1-norm, and total variation regularization method. That is, use of an appropriate regularization method considering the flow characteristics is necessary to attain good reconstruction performance. Therefore, in this paper, regularization methods are tested through numerical simulations with different flow conditions and the performance is compared.

• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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• pp.125-130
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• 2007

We propose a simple and intuitive method to derive the exact convergence rate of global $L_{2}-norm$ error for strong numerical approximation of stochastic differential equations the result of which has been reported by Hofmann and $M{\"u}ller-Gronbach\;(2004)$. We conclude that any strong numerical scheme of order ${\gamma}\;>\;1/2$ has the same optimal convergence rate for this error. The method clearly reveals the structure of global $L_{2}-norm$ error and is similarly applicable for evaluating the convergence rate of global uniform approximations.

• Kyungpook Mathematical Journal
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• v.48 no.3
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• pp.387-393
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• 2008

In this note, we present some inequalities for the norm and numerical radius of 2-homogeneous polynomials from the 2-dimensional real space $l_p^2$, (1 < p < $\infty$) to itself in terms of their coefficients. We also give an upper bound for n^{(k)}(l_p^2), (k = 2, 3, $\cdots$).

• The Journal of Korea Institute of Information, Electronics, and Communication Technology
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• v.12 no.5
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• pp.521-528
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• 2019

Epilepsy is one of the most prevalent neurological diseases. Electroencephalogram (EEG) signals are widely used for monitoring and diagnosis tool for epileptic seizure. Typically, a huge amount of EEG signals is needed, where they are visually examined by experienced clinicians. In this study, we propose a simple automatic seizure detection framework using intracranial EEG signals. We suggest a sparse approximation based classification (SAC) scheme by solving overdetermined system. L1-norm minimization algorithms are utilized for efficient sparse signal recovery. For evaluation of the proposed scheme, the public EEG dataset obtained by five healthy subjects and five epileptic patients is utilized. The results show that the proposed fast L1-norm minimization based SAC methods achieve the 99.5% classification accuracy which is 1% improved result than the conventional L2 norm based method with negligibly increased execution time (42msec).

• The Pure and Applied Mathematics
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• v.9 no.1
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• pp.81-90
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• 2002

The least informative (favorable) distributions, minimizing Fisher information for a multivariate location parameter, are derived in the parametric class of the exponential-power spherically symmetric distributions under the following characterizing restrictions; (i) a bounded variance, (ii) a bounded value of a density at the center of symmetry, and (iii) the intersection of these restrictions. In the first two cases, (i) and (ii) respectively, the least informative distributions are the Gaussian and Laplace, respectively. In the latter case (iii) the optimal solution has three branches, with relatively small variances it is the Gaussian, them with intermediate variances. The corresponding robust minimax M-estimators of location are given by the $L_2$-norm, the $L_1$-norm and the $L_{p}$ -norm methods. The properties of the proposed estimators and their adaptive versions ar studied in asymptotics and on finite samples by Monte Carlo.

• Journal of applied mathematics & informatics
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• v.5 no.3
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• pp.735-748
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• 1998

We analyze the error in the p version of the of the finite element method when the effect of the quadrature error is taken in the load vector. We briefly study some results on the $H^{1}$ norm error and present some new results for the error in the $L^{2}$ norm. We inves-tigate the quadrature error due to the numerical integration of the right hand side We present theoretical and computational examples showing the sharpness of our results.