• 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 applied mathematics & informatics
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• v.3 no.2
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• pp.245-252
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• 1996

We described a relationship of the $L_2$ norm of the $L_2$norm of a Bzier curve and l2 norm of its confrol points. The use of Bezier curves finds much application in the general description of curves and surfaces and provided the mathematical basis for many computer graphics system. We define the $L_2$ norm for Bezier curves and find a upper and lower bound for many computer graphics system. We define the $L_2$ norm for Bezier curves and find a upper and lower bound for the $L_2$ norm with respect to the $L_2$ norm for its control points for easy computation.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.32 no.11C
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• pp.1073-1078
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• 2007

In this paper, we propose an iterative mixed norm image restoration algorithm using multi regularization parameters. A functional which combines the regularized $l_2$ norm functional and the regularized $l_4$ norm functional is proposed to efficiently remove arbitrary noise. The smoothness of each functional is determined by the regularization parameters. Also, a regularization parameter is used to determine the relative importance between the regularized $l_2$ norm functional and the regularized $l_4$ norm functional using kurtosis. An iterative algorithm is utilized for obtaining a solution and its convergence is analyzed. Experimental results demonstrate the capability of the proposed algorithm.

• 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.

• Proceedings of the IEEK Conference
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• 2003.11a
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• pp.489-492
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• 2003

In this paper, we propose an iterative mixed norm image restoration algorithm using multi regularization parameters. A functional which combines the regularized l$_2$ norm functional and the regularized l$_4$ functional is proposed. The smoothness of each functional is determined by the regularization parameters. Also, a regularization parameter is used to determine the relative importance between the regularized l$_2$ functional and the regularized l$_4$ functional. An iterative algorithm is utilized for obtaining a solution and its convergence is analyzed.

• 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.

• Journal of the Institute of Convergence Signal Processing
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• v.19 no.4
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• pp.147-152
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• 2018

Underwater transient signals are complex, variable and nonlinear, resulting in a difficulty in accurate modeling with reference patterns. We analyze one type of underwater transient signals, in the form of whale sounds, using the MFCC(Mel-Frequency Cepstral Constant) and synthesize them from the MFCC and the weighted $L_2$-norm minimization techniques. The whales in this experiments are Humpback whales, Right whales, Blue whales, Gray whales, Minke whales. The 20th MFCC coefficients are extracted from the original signals using the MATLAB programming and reconstructed using the weighted $L_2$-norm minimization with the inverse MFCC. Finally, we could find the optimum weighted factor, 3~4 for reconstruction of whale sounds.

• 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 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.