• Proceedings of the IEEK Conference
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• 2001.06a
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• pp.333-336
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• 2001

This paper reports that the Lambert W function applies to a non-iterative design of minimum mean square-error scalar quantizers for a Laplacian source. The contribution of the paper is in the reduction of the time needed for the design and the increased accuracy in resulting quantization points and thresholds, because the algorithm is non-iterative and the Lambert W function can be evaluated as accurately as desired.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.4
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• pp.393-402
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• 2006

We investigate relations between multiplicity of solutions and source terms in a elliptic equation. We have a concerne with a sharp result for multiplicity of a nonlinear Laplacian differential equation.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.36 no.6C
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• pp.384-390
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• 2011

The paper derives formulas for the mean-squared error distortion and resulting signal-to-noise (SNR) ratio of a fixed-rate scalar quantizer designed optimally in the minimum mean-squared error sense for a Gaussian density with the standard deviation ${\sigma}_q$ when it is mismatched to a Laplacian density with the standard deviation ${\sigma}_q$. The SNR formulas, based on the key parameter and Bennett's integral, are found accurate for a wide range of $p\({\equiv}\frac{\sigma_p}{\sigma_q}\){\geqq}0.25$. Also an upper bound to the SNR is derived, which becomes tighter with increasing rate R and indicates that the SNR behaves asymptotically as $\frac{20\sqrt{3{\ln}2}}{{\rho}{\ln}10}\;{\sqrt{R}}$ dB.

• Journal of the Korean Society for Precision Engineering
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• v.25 no.1
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• pp.160-170
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• 2008

Shape morphing is the process of transforming a source shape, through intermediate shapes, into a target shape. Two main problems to be considered in three dimensional shape morphing are vertex correspondence and path interpolation. In this paper, an approach which uses the linear interpolation of the Laplacian coordinates of the source and target meshes is introduced for the determination of more plausible path when two topologically identical shapes are morphed. When two shapes to be morphed are different in shape and topology, a new method which combines shape deformation theory based on Laplacian coordinate and mean value coordinate with distance field theory is proposed for the efficient treatment of vertex correspondence and path interpolation problems. The validity and effectiveness of the suggested method was demonstrated by using it to morph large and complex polygon models including male and female whole body models.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.31 no.10C
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• pp.991-999
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• 2006

This paper shows that the support growth of an optimum (minimum mean square-error) scalar quantizer for a Laplacian density is logarithmic with the number of quantization points. Specifically, it is shown that, for a unit-variance Laplacian density, the ratio of the support-determining threshold of an optimum quantizer to $\frac 3{\sqrt{2}}1n\frac N 2$ converges to 1, as the number of quantization points grows. Also derived is a limiting upper bound that says that the optimum support cannot exceed the logarithmic growth by more than a constant. These results confirm the logarithmic growth of the optimum support that has previously been derived heuristically.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.29 no.6C
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• pp.857-864
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• 2004

Many image compression standards such as JPEG, MPEG or H.263 are based on the discrete cosine transform (DCT) and quantization method. Quantization error. is the major source of image quality degradation. The current dequantization method assumes the uniform distribution of the DCT coefficients. Therefore the dequantization value is the center of each quantization interval. However DCT coefficients are regarded to follow Laplacian probability density function (pdf). The center value of each interval is not optimal in reducing squared error. We use mean of the quantization interval assuming Laplacian pdf, and show the effect of correction on image quality. Also, we compare existing quantization error to corrected quantization error in closed form. The effect of PSNR improvements due to the compensation to the real image is in the range of 0.2 ∼0.4 ㏈. The maximum correction value is 1.66 ㏈.

• Korean Journal of Mathematics
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• v.26 no.4
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• pp.683-700
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• 2018

We get one theorem which shows existence of solutions for one-dimensional jumping problem involving p-Laplacian and Dirichlet boundary condition. This theorem is that there exists at least one solution when nonlinearities crossing finite number of eigenvalues, exactly one solutions and no solution depending on the source term. We obtain these results by the eigenvalues and the corresponding normalized eigenfunctions of the p-Laplacian eigenvalue problem when 1 < p < ${\infty}$, variational reduction method and Leray-Schauder degree theory when $2{\leq}$ p < ${\infty}$.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.27 no.6A
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• pp.524-532
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• 2002

This paper reports that the Lambert W function applies to a non-iterative design of minimum mean square-error scalar quantizers for a Laplacian source. Specifically, it considers a non-iterative design algorithm for optimum quantizers for a Laplacian source; it finds that the solution of the recursive nonlinear equation in the non-iterative design is elegantly expressed in term of the principal branch of the Lambert W function in a closed form; and it proves that the non-iterative algorithm applies only to exponential or Laplacian sources. The contribution of the paper is in the reduction of the time needed for the design and the increased accuracy in resulting quantization points and thresholds, because the algorithm is non-iterative and the Lambert W function can be evaluated as accurately as desired. Also, numerical results show how optimal quantization distortion converges monotonically to the Panter-Dite constant and help derive an approximation formula for the key parameters of optimum quantizers.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.15 no.5
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• pp.1761-1777
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• 2021

An image with infrared features and visible details is obtained by processing infrared and visible images. In this paper, a fusion method based on Laplacian pyramid and generative adversarial network is proposed to obtain high quality fusion images, termed as Laplacian-GAN. Firstly, the base and detail layers are obtained by decomposing the source images. Secondly, we utilize the Laplacian pyramid-based method to fuse these base layers to obtain more information of the base layer. Thirdly, the detail part is fused by a generative adversarial network. In addition, generative adversarial network avoids the manual design complicated fusion rules. Finally, the fused base layer and fused detail layer are reconstructed to obtain the fused image. Experimental results demonstrate that the proposed method can obtain state-of-the-art fusion performance in both visual quality and objective assessment. In terms of visual observation, the fusion image obtained by Laplacian-GAN algorithm in this paper is clearer in detail. At the same time, in the six metrics of MI, AG, EI, MS_SSIM, Q_abf and SCD, the algorithm presented in this paper has improved by 0.62%, 7.10%, 14.53%, 12.18%, 34.33% and 12.23%, respectively, compared with the best of the other three algorithms.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.1003-1031
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• 2020

We consider the following Dirichlet initial boundary value problem with a gradient absorption and a nonlocal source $$\frac{{\partial}u}{{\partial}t}-div({\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)={\lambda}u^k{\displaystyle\smashmargin{2}{\int\nolimits_{\Omega}}}u^sdx-{\mu}u^l{\mid}{\nabla}u{\mid}^q$$ in a bounded domain Ω ⊂ ℝ^N, where p > 1, the parameters k, s, l, q, λ > 0 and µ ≥ 0. Firstly, we establish local existence for weak solutions; the aim of this part is to prove a crucial priori estimate on |∇u|. Then, we give appropriate conditions in order to have existence and uniqueness or nonexistence of a global solution in time. Finally, depending on the choices of the initial data, ranges of the coefficients and exponents and measure of the domain, we show that the non-negative global weak solution, when it exists, must extinct after a finite time.