• East Asian mathematical journal
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• v.14 no.1
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• pp.63-76
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• 1998

we consider the hp-version to solve non-constant coefficients elliptic equations $-div(a{\nabla}u)=f$ with Dirichlet boundary conditions on a bounded polygonal domain $\Omega$ in $R^2$. In [6], M. Suri obtained an optimal error-estimate for the hp-version: ${\parallel}u-u^h_p{\parallel}_{1,\Omega}{\leq}Cp^{(\sigma-1)}h^{min(p,\sigma-1)}{\parallel}u{\parallel}_{\sigma,\Omega}$. This optimal result follows under the assumption that all integrations are performed exactly. In practice, the integrals are seldom computed exactly. The numerical quadrature rule scheme is needed to compute the integrals in the variational formulation of the discrete problem. In this paper we consider a family $G_p=\{I_m\}$ of numerical quadrature rules satisfying certain properties, which can be used for calculating the integrals. Under the numerical quadrature rules we will give the variational form of our non-constant coefficients elliptic problem and derive an error estimate of ${\parallel}u-\tilde{u}^h_p{\parallel}_{1,\Omega}$.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.1
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• pp.13-23
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• 2008

In [7, 8], they proposed a new singular function(NSF) method to compute singular solutions of Poisson equations on a polygonal domain with re-entrant angles. Singularities are eliminated and only the regular part of the solution that is in $H^2$ is computed. The stress intensity factor and the solution can be computed as a post processing step. This method was extended to the interface problem and Poisson equations with the mixed boundary condition. In this paper, we give NSF method for the Helmholtz equations ${\Delta}u+Ku=f$ with homogeneous Dirichlet boundary condition. Examples with a singular point are given with numerical results.

• International Journal of CAD/CAM
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• v.9 no.1
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• pp.25-29
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• 2010

Let $S_n^r(\Omega)$ be the spline space of degree n and smoothness r with respect to $\Omega$ where $\Omega$ is a triangulation of a planner polygonal domain. Dimensions of $S_n^r(\Omega)$ over the so-called unconstricted triangulation were given by Farin in [J. Comput. Appl. Math. 192(2006), 320-327]. In this paper, a counter example is given to show that the condition used in the main result in Farins paper is not correct, and then an improved necessary and sufficient condition is presented.

• Korean Journal of Computational Design and Engineering
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• v.7 no.2
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• pp.75-80
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• 2002

This paper details a robust and efficient algorithm for point classification with respect to a polygon in 2D real number domain. The concept of tolerance makes this algorithm robust and consistent. It enables to define‘on-boundary’ , which can be interpreted as either‘in-’or‘out-’side region, and to manage rounding errors in floating point computation. Also the tolerance is used as a measure of reliability of point classifications. The proposed algorithm is based on a ray-intersection technique known as the most efficient, in which intersections between a ray originating from a given test point and the boundary of a region are counted. An odd number of intersections indicates that the point is inside region. For practical examples the algorithm is most efficient because most edges of the polygon region are processed by simple bit operations.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2006.05a
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• pp.851-854
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• 2006

멀티미디어 데이터 중 3차원 멀티미디어 데이터의 저작권 보호를 위한 기술로 디지털 워터마킹에 대한 연구가 활발히 진행되고 있다. 워터마킹 기술은 공간영역에 워터마크를 삽입하는 것과 주파수 영역에 워터마크를 삽입하는 기술로 크게 나누어진다. 본 논문에서는 3차원 깊이정보로부터 다각형 모델링을 구현하고 깊이영상의 저작권보호를 위한 방법으로 먼저 3차원으로 획득된 깊이정보로부터 다각형메쉬(polygon mesh)를 구성하고 3차원 메쉬 데이터를 DCT변환을 이용하여 주파수 영역으로 변환한 후 변환된 주파수 영역에 적응적으로 워터마크를 삽입하고 검출하였다. 깊이영상의 저작권보호를 위한 비가시적이며 강인한 워터마킹 방법을 구현하였다.

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.467-489
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• 1994

Let $\Omega$ be a closed and bounded polygonal domain in R$^2$, or a closed line segment in R$^1$ with boundary $\Gamma$, such that there exists an invertible mapping T : $\Omega$ \longrightarrow $\Omega$ with the following correspondence: x$\in$$\Omega$ ↔ x = T(x) $\in$$\Omega$, (1.1) and (1.2) t $\in$ U$\sub$p/($\Omega$) ↔ t = to T$\^$-1/ $\in$ U$\sub$p/($\Omega$), where $\Omega$ denotes the corresponding reference elements I = [-1,1] and I ${\times}$ I in R$^1$ and R$^2$ respectively, (1.3) U$\sub$p/($\Omega$) = {t : t is a polynomial of degree $\leq$ p in each variable on $\Omega$}, and (1.4) U$\sub$p/($\Omega$) = {t : t = to T $\in$ U$\sub$p/($\Omega$)}.(omitted)

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.14 no.10
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• pp.4042-4059
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• 2020

The development of new multimedia techniques such as 3D printing is increasingly attracting the public's attention towards 3D objects. An optimized robust and imperceptible watermarking method based on Ant Colony Optimization (ACO) and Weber Law is proposed for 3D polygonal models. The proposed approach partitions the host model into smaller sub meshes and generates a secret watermark from the sub meshes using Weber Law. ACO based optimized strength factor is identified for embedding the watermark. The secret watermark is embedded and extracted on the wavelet domain. The proposed scheme is robust against geometric and photometric attacks that overcomes the synchronization problem and authenticates the secret watermark from the distorted models. The primary characteristic of the proposed system is the flexibility achieved in data embedding capacity due to the optimized strength factor. Extensive simulation results shows enhanced performance of the recommended framework and robustness towards the most common attacks like geometric transformations, noise, cropping, mesh smoothening, and the combination of such attacks.

• Journal of KIISE:Computer Systems and Theory
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• v.29 no.6
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• pp.307-317
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• 2002

In this paper we propose a new mesh reconstruction scheme that produces a displaced subdivision surface directly from unorganized points. The displaced subdivision surface is a new mesh representation that defines a detailed mesh with a displacement map over a smooth domain surface, but original displaced subdivision surface algorithm needs an explicit polygonal mesh since it is not a mesh reconstruction algorithm but a mesh conversion (remeshing) algorithm. The main idea of our approach is that we sample surface detail from unorganized points without any topological information. For this, we predict a virtual triangular face from unorganized points for each sampling ray from a parameteric domain surface. Direct displaced subdivision surface reconstruction from unorganized points has much importance since the output of this algorithm has several important properties: It has compact mesh representation since most vertices can be represented by only a scalar value. Underlying structure of it is piecewise regular so it ran be easily transformed into a multiresolution mesh. Smoothness after mesh deformation is automatically preserved. We avoid time-consuming global energy optimization by employing the input data dependant mesh smoothing, so we can get a good quality displaced subdivision surface quickly.