• Nonlinear Functional Analysis and Applications
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• v.27 no.4
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• pp.855-867
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• 2022

This study finds three different solutions (3-Sol's) for the fourth order nonlinear discrete anisotropic equations (DAE) with real parameter. We use the variational method(VM) and 𝜙_p-Laplacian operator (𝜙_p-LO) to prove the main results. In the following paper, we take the parameters λ, 𝜇 such that λ > 0 and 𝜇 ≥ 0 into consideration.

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.507-519
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• 2017

We present an upper bound on the Cheeger constant of a distance-regular graph. Recently, the authors found an upper bound on the Cheeger constant of distance-regular graph under a certain restriction in their previous work. Our new bound in the current paper is much better than the previous bound, and it is a general bound with no restriction. We point out that our bound is explicitly computable by using the valencies and the intersection matrix of a distance-regular graph. As a major tool, we use the discrete Green's function, which is defined as the inverse of ${\beta}$-Laplacian for some positive real number ${\beta}$. We present some examples of distance-regular graphs, where we compute our upper bound on their Cheeger constants.

• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.135-163
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• 2010

Motivated by [Science in China (Ser. A Mathematics) 36 (2006), no. 7, 721?732], this article deals with the following discrete type BVP $\LARGE\left\{{{\;{\Delta}[{\phi}({\Delta}x(n))]\;+\;f(n,\;x(n\;+\;1),{\Delta}x(n),{\Delta}x(n + 1))\;=\;0,\;n\;{\in}\;[0,N],}}\\{\;{x(0)-{\sum}^m_{i=1}{\alpha}_ix(n_i) = A,}}\\{\;{x(N+2)-\;{\sum}^m_{i=1}{\beta}_ix(n_i)\;=\;B.}}\right.$ The sufficient conditions to guarantee the existence of at least three positive solutions of the above multi-point boundary value problem are established by using a new fixed point theorem obtained in [5]. An example is presented to illustrate the main result. It is the purpose of this paper to show that the approach to get positive solutions of BVPs by using multifixed-point theorems can be extended to treat nonhomogeneous BVPs. The emphasis is put on the nonlinear term f involved with the first order delta operator ${\Delta}$x(n).

• Communications of the Korean Mathematical Society
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• v.24 no.3
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• pp.467-479
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• 2009

This paper ultimately aims to develop noninvasive techniques to identify the inside of a given electrical network. Based on the theory of the partial differentiation equations and mathematical modeling, this paper devises the algorithms to find the locations of possible abnormalities. To ensure the certainty of the algorithms, this study restricted the forms of the network and the number of abnormalities, rendering it easy to prove the uniqueness of the position of the abnormalities.

• Proceedings of the Korean Information Science Society Conference
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• 2004.04b
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• pp.739-741
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• 2004

영상에서 Edge는 영역의 경계를 표현하며. 특징으로는 픽셀 밝기의 불연속점을 나타낸다. 이러한 Edge를 찾아내는 Edge detection은 설러 가지 영상 처리 기법에서 유용하게 사용되고 있다. 현재까지 많은 알고리즘들이 제안되었으며. 이 논문에서는 이러한 알고리즘들에 대한 장단점을 파악하고, 미분 연산자를 이용한 Sobel, Prewitt. Roberts. Laplacian, 그리고 Canny 마스크를 이용한 윤곽선 검출방법과 Discrete Sing에ar Convolution (DSC) 알고리즘을 이용한 윤곽선 검출방법을 백색 가우시안 잡음 환경과 비 잡음 환경에서 비교해 보았다.

• International Journal of Computer Science & Network Security
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• v.23 no.4
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• pp.39-47
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• 2023

This study compares various transformation techniques for multifocus image fusion. Multi-focus image fusion is a procedure of merging multiple images captured at unalike focus distances to produce a single composite image with improved sharpness and clarity. In this research, the purpose is to compare different popular frequency domain approaches for multi-focus image fusion, such as Discrete Wavelet Transforms (DWT), Stationary Wavelet Transforms (SWT), DCT-based Laplacian Pyramid (DCT-LP), Discrete Cosine Harmonic Wavelet Transform (DC-HWT), and Dual-Tree Complex Wavelet Transform (DT-CWT). The objective is to increase the understanding of these transformation techniques and how they can be utilized in conjunction with one another. The analysis will evaluate the 10 most crucial parameters and highlight the unique features of each method. The results will help determine which transformation technique is the best for multi-focus image fusion applications. Based on the visual and statistical analysis, it is suggested that the DCT-LP is the most appropriate technique, but the results also provide valuable insights into choosing the right approach.

• Korean Journal of Remote Sensing
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• v.21 no.3
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• pp.243-252
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• 2005

Wavelet transform and its inverse processing provide the effective framework for data fusion. The purpose of this study is to investigate applicability of wavelet transform using texture images for the urban remote sensing application. We tried several experiments regarding image fusion by wavelet transform and texture imaging using high resolution images such as IKONOS and KOMPSAT EOC. As for texture images, we used homogeneity and ASM (Angular Second Moment) images according that these two types of texture images reveal detailed information of complex features of urban environment well. To find out the useful combination scheme for further applications, we performed DWT(Discrete Wavelet Transform) and IDWT(Inverse Discrete Wavelet Transform) using texture images and original images, with adding edge information on the fused images to display texture-wavelet information within edge boundaries. The edge images were obtained by the LoG (Laplacian of Gaussian) processing of original image. As the qualitative result by the visual interpretation of these experiments, the resultant image by each fusion scheme will be utilized to extract unique details of surface characterization on urban features around edge boundaries.

• Journal of the Korean Mathematical Society
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• v.56 no.4
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• pp.857-867
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• 2019

Suppose that G = (V, E) is a connected locally finite graph with the vertex set V and the edge set E. Let ${\Omega}{\subset}V$ be a bounded domain. Consider the following quasilinear elliptic equation on graph G $$\{-{\Delta}_{pu}={\lambda}K(x){\mid}u{\mid}^{p-2}u+f(x,u),\;x{\in}{\Omega}^{\circ},\\u=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}^{\circ}$ and ${\partial}{\Omega}$ denote the interior and the boundary of ${\Omega}$, respectively, ${\Delta}_p$ is the discrete p-Laplacian, K(x) is a given function which may change sign, ${\lambda}$ is the eigenvalue parameter and f(x, u) has exponential growth. We prove the existence and monotonicity of the principal eigenvalue of the corresponding eigenvalue problem. Furthermore, we also obtain the existence of a positive solution by using variational methods.

• Journal of the Korean Mathematical Society
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• v.59 no.5
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• pp.911-926
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• 2022

Let u be a function on a locally finite graph G = (V, E) and Ω be a bounded subset of V. Let 𝜀 > 0, p > 2 and 0 ≤ λ < λ₁(Ω) be constants, where λ₁(Ω) is the first eigenvalue of the discrete Laplacian, and h : V → ℝ be a function satisfying h ≥ 0 and $h{\not\equiv}0$. We consider a perturbed Yamabe equation, say $$\{\begin{array}{lll}-{\Delta}u-{\lambda}u={\mid}u{\mid}^{p-2}u+{\varepsilon}h,&&\text{ in }{\Omega},\\u=0,&&\text{ on }{\partial}{\Omega},\end{array}$$ where Ω and ∂Ω denote the interior and the boundary of Ω, respectively. Using variational methods, we prove that there exists some positive constant 𝜀₀ > 0 such that for all 𝜀 ∈ (0, 𝜀₀), the above equation has two distinct solutions. Moreover, we consider a more general nonlinear equation $$\{\begin{array}{lll}-{\Delta}u=f(u)+{\varepsilon}h,&&\text{ in }{\Omega},\\u=0,&&\text{ on }{\partial}{\Omega},\end{array}$$ and prove similar result for certain nonlinear term f(u).

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.7 no.7
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• pp.1671-1689
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• 2013

The key issue of block-based multi-focus image fusion is to determine the size of the sub-block because different sizes of the sub-block will lead to different fusion effects. To solve this problem, this paper presents a novel genetic algorithm (GA) based multi-focus image fusion method, in which the block size can be automatically found. In our method, the Sum-modified-Laplacian (SML) is selected as an evaluation criterion to measure the clarity of the image sub-block, and the edge information retention is employed to calculate the fitness of each individual. Then, through the selection, crossover and mutation procedures of the GA, we can obtain the optimal solution for the sub-block, which is finally used to fuse the images. Experimental results show that the proposed method outperforms the traditional methods, including the average, gradient pyramid, discrete wavelet transform (DWT), shift invariant DWT (SIDWT) and two existing GA-based methods in terms of both the visual subjective evaluation and the objective evaluation.