• Proceedings of the KIEE Conference
• /
• 1987.07b
• /
• pp.1364-1366
• /
• 1987

Laplacian of Gaussian, ${\nabla}^2G$ operator proposed by Marr and Hildreth is known as a rough bandpass filter. This paper shows how to detect the change of intensity corresponding to an arbitrary spatial frequency in an image using ${\nabla}^2G$ operator.

• Bulletin of the Korean Mathematical Society
• /
• v.59 no.5
• /
• pp.1255-1268
• /
• 2022

We consider the Schrödinger type operator 𝓛 = (-𝚫_ℍⁿ)² + V² on the Heisenberg group ℍⁿ, where 𝚫_ℍⁿ is the sub-Laplacian and the non-negative potential V belongs to the reverse Hölder class RH_s for s ≥ Q/2 and Q ≥ 6. We shall establish the (L^p, L^q) estimates for the Riesz transforms T_𝛼,𝛽,j = V^2𝛼𝛁^j_ℍⁿ𝓛^-𝛽, j = 0, 1, 2, 3, where 𝛁_ℍⁿ is the gradient operator on ℍⁿ, 0 < α ≤ 1-j/4, j/4 < 𝛽 ≤ 1, and 𝛽 - 𝛼 ≥ j/4.

• Proceedings of the IEEK Conference
• /
• 2005.11a
• /
• pp.979-982
• /
• 2005

This paper is to be pre-processing that decides the text recognizability and quality contained in natural image. Differentiated with the existing studies, In this paper, it suggests the application of partially unified color models, Coiflet Wavelet and text extraction algorithm that uses the closed curve edge features of LoG (laplacian of gaussian)operator. The text image included in natural image such as signboard has the same hue, saturation and value, and there is a certain thickness as for their feature. Each color element is restructured into closed area by LoG operator, the 2nd differential operator. The text area is contracted by Hough Transform, logical AND-OR operator of each color model and Minimum-Distance classifier. This paper targets natural image into which text area is added regardless of the size and resolution of the image, and it is confirmed to have more excellent performance than other algorithms with many restrictions.

• Journal of applied mathematics & informatics
• /
• v.32 no.1_2
• /
• pp.137-152
• /
• 2014

In this paper, we prove, in the spirit of [3, 12, 20, 22, 23], the existence of infinitely many small solutions to the following quasilinear elliptic equation $-{\Delta}_{p(x)}u+{\mid}u{\mid}^{p(x)-2}u={\mid}u{\mid}^{q(x)-2}u+{\lambda}f(x,u)$ in a smooth bounded domain ${\Omega}$ of ${\mathbb{R}}^N$. We also assume that $\{q(x)=p^*(x)\}{\neq}{\emptyset}$, where $p^*(x)$ = Np(x)/(N - p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountainpass lemma due to Kajikiya [22], and property of these solutions are also obtained.

• Proceedings of the Korean Institute of Intelligent Systems Conference
• /
• 2001.12a
• /
• pp.283-286
• /
• 2001

Based on a fuzzy system representation of gray scale images, we derive an edge detection algorithm whose convolution kernel is different from the known kernels such as those of Roberts', Prewitt's or Sobel's gradient. Our fuzzy system representation is an exact representation of the bicubic spline function which represents the gray scale image approximately. Hence the fuzzy system is a continuous function and it provides a natural way to define the gradient and the Laplacian operator. We show that the gradient at grid points can be evaluated by taking the convolution of the image with a 3 3 kernel. We also show that our gradient coupled with the approximate value of the continuous function generates an edge detection method which creates edge images clearer than those by other methods. A few examples of applying our methods are included.

• Bulletin of the Korean Mathematical Society
• /
• v.34 no.2
• /
• pp.155-171
• /
• 1997

Let $M^n$ be a connected subminifold of a Euclidean space $E^m$, equipped with the induced metric. Denoty by $\Delta$ the Laplacian operator of $M^n$ and by x the position vector. A well-known T. Takahashi's theorem [13] says that $\delta x = \lambda x$ for some constant $\lambda$ if and only if $M^n$ is either minimal subminifold of $E^m$ or minimal submanifold in a hypersphere of $E^m$. In [9], O. Garay studied the hypersurfaces $M^n$ in $E^{n+1}$ satisfying $\delta x = Dx$, where D is a diagonal matrix, and he classified such hypersurfaces. Garay's condition can be seen as a generalization of T.

• Journal of the Chungcheong Mathematical Society
• /
• v.22 no.1
• /
• pp.117-121
• /
• 2009

For $1{\leq}p$, $q{\leq}{\infty}$ and $s{\in}\mathbb{R}$, it is proved that there exists a constant C > 0 such that for any $f{\in}B^{s+2}_{p,q}(\mathbb{R}^n)$ $${\parallel}f{\parallel}_{B^{s+2}_{p,q}(\mathbb{R}^n)}{\leq}C{\parallel}f\;-\;{\Delta}f{\parallel}_{B^{s}_{p,q}(\mathbb{R}^n)}$$, which tells us that the operator $I-\Delta$ is $B^{s+2}_{p,q}$-coercive on the Besov space $B^s_{p,q}$.

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

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
• /
• 2013.05a
• /
• pp.892-894
• /
• 2013

Edge has been utilized in various application fields with development of technique of digital image processing. In conventional edge detection methods, there are some methods using mask including Sobel, Prewitt, Roberts and Laplacian operator. Those methods are that implement is simple but generates errors of edge detection in images added AWGN(additive white Gaussian noise). Therefore, to compensate the defect of those methods, in this paper, an edge detection algorithm using modified mask is proposed, and it showed superior edge detection property in AWGN.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
• /
• 2013.05a
• /
• pp.886-888
• /
• 2013

An edge includes information of objects such as magnitude, orientation, and location. Conventional edge detection methods to detect those edge are methods using Sobel, Prewitt, Roberts, Laplacian operator. Existing methods use fixed weighted mask to detect edges, and their edge detection characteristics are insufficient. Therefore, to remedy weakness of conventional methods, in this paper, an edge detection algorithm using weight with standard deviation and thresholds is proposed.