• Journal of the Korean Mathematical Society
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• v.31 no.4
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• pp.569-608
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• 1994

The subject matter of this survey has to do with holomorphic maps from a compact Riemann surface to projective space, which are also called algebrac curves; the theory we survey lies at the crossroads of function theory, projective geometry, and commutative algebra (although we should mention that the present survey de-emphasizes the algebraic aspect). Algebraic curves have been vigorously and continuously investigated since the time of Riemann. The reasons for the preoccupation with algebraic curves amongst mathematicians perhaps have to do with-other than the usual usual reason, namely, the herd mentality prompting us to follow the leads of a few great pioneering methematicians in the field-the fact that algebraic curves possess a certain simple unity together with a rich and complex structure. From a differential-topological standpoint algebraic curves are quite simple as they are neatly parameterized by a single discrete invariant, the genus. Even the possible complex structures of a fixed genus curve afford a fairly complete description. Yet there are a multitude of diverse perspectives (algebraic, function theoretic, and geometric) often coalescing to yield a spectacular result.

• The Pure and Applied Mathematics
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• v.21 no.4
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• pp.257-270
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• 2014

In this article, we consider a homogeneous function of degree four in quaternionic vector spaces and $S^{4n+3}$ which is invariant under $S^3$ and U(n + 1)-action. We show it is an isoparametric function providing isoparametric hypersurfaces in $S^{4n+3}$ with g = 4 distinct principal curvatures and isoparametric hypersurfaces in quaternionic projective spaces with g = 5. This extends study of Nomizu on isoparametric function on complex vector spaces and complex projective spaces.

• Proceedings of the Korean Society of Broadcast Engineers Conference
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• 1998.06b
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• pp.141-146
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• 1998

Currently virtual studio has used the cromakey method in which an image is captured, and the blue portion of that image is replaced by a graphic image or a real image. The replaced image must be changed according to the camera motion. This paper proposes a novel method to extract camera parameters using the recognition of pentagonal patterns which are painted on the blue screen. The corresponding parameters are position, direction and focal length of the camera in the virtual studio. At first, pentagonal patterns are found using invariant features of the pentagon. Then, the projective transformation of two projected images and the camera parameters are calculated using the matched points. Simulation results indicate that camera parameters are more easily calculated compared to the conventional methods.

• Journal of the Korean Mathematical Society
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• v.61 no.3
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• pp.547-564
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• 2024

We study totally real and complex subsets of a right quarternionic vector space of dimension n + 1 with a Hermitian form of signature (n, 1) and extend these notions to right quaternionic projective space. Then we give a necessary and sufficient condition for a subset of a right quaternionic projective space to be totally real or complex in terms of the quaternionic Hermitian triple product. As an application, we show that the limit set of a non-elementary quaternionic Kleinian group 𝚪 is totally real (resp. commutative) with respect to the quaternionic Hermitian triple product if and only if 𝚪 leaves a real (resp. complex) hyperbolic subspace invariant.

• Journal of the Institute of Electronics Engineers of Korea SP
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• v.48 no.6
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• pp.115-123
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• 2011

Most of cameras being used in practice induce lens distortion; the amount of distortion depends on the specific applications as well as the camera cost. Up to now, many methods of lens distortion correction have relied on invariant properties of projective geometry to find distortion parameters. A common property is "the straight line in scene is straight line in image". However, if the straight lines are also parallel together, the previous works have missed this restriction in determining lens distortion parameters. In this paper, we propose a method that leads to guarantee of the restrictions simultaneously for the determination. Therefore, corrected image will close to an ideal image taken by the pinhole camera model. The effectiveness of the proposed method is verified by our experiments on both synthetic images and real images.

• Journal of the Korea Society of Computer and Information
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• v.10 no.3 s.35
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• pp.173-182
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• 2005

In this paper, we propose a landmark recognition method which is irrelevant to the camera viewpoint on the navigation for localization. Features in previous research is variable to camera viewpoint, therefore due to the wealth of information, extraction of visual landmarks for positioning is not an easy task. The proposed method in this paper, has the three following stages; first, extraction of features, second, learning and recognition, third, matching. In the feature extraction stage, we set the interest areas of the image. where we extract the corner points. And then, we extract features more accurate and resistant to noise through statistical analysis of a small eigenvalue. In learning and recognition stage, we form robust feature models by testing whether the feature model consisted of five corner points is an invariant feature irrelevant to viewpoint. In the matching stage, we reduce time complexity and find correspondence accurately by matching method using similarity evaluation function and Graham search method. In the experiments, we compare and analyse the proposed method with existing methods by using various indoor images to demonstrate the superiority of the proposed methods.

• The Transactions of the Korea Information Processing Society
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• v.6 no.9
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• pp.2540-2547
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• 1999

Chromakey method is one of key technologies for realizing virtual studio, and the blue portions of a captured image in virtual studio, are replaced with a computer generated or real image. The replaced image must be changed according to the camera parameter of studio for natural merging with the non-blue portions of a captured image. This paper proposes a novel method to extract camera parameters using the recognition of pentagonal patterns that are painted on a blue screen. We extract corresponding points between a blue screen. We extract corresponding points between a blue screen and a captured image using the projective invariant features of a pentagon. Then, calculate camera parameters using corresponding points by the modification of Tsai's method. Experimental results indicate that the proposed method is more accurate compared to conventional method and can process about twelve frames of video per a second in Pentium-MMX processor with CPU clock of 166MHz.

• Journal of the Institute of Electronics Engineers of Korea SP
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• v.46 no.4
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• pp.97-107
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• 2009

In image-based gait recognition systems, physical factors, such as the camera angle and the lens distortion, and environmental factors such as illumination determines the performance of recognition. In this paper we present a robust gait recognition method by compensating various types of image distortions. The proposed method is compared with existing gait recognition algorithm with consideration of both physical and environmental distortion factors in the input image. More specifically, we first present an efficient compensation algorithm of image distortion by using the projective transform, and test the feasibility of the proposed algorithm by comparing the recognition performances with and without the compensation process. Proposed method gives universal gait data which is invariant to both distance and environment. Gained data improved gait recognition rate about 41.5% in indoor image and about 55.5% in outdoor image. Proposed method can be used effectively in database(DB) construction, searching and tracking of specific objects.

• Journal of the Institute of Electronics Engineers of Korea SC
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• v.43 no.2 s.308
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• pp.11-19
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• 2006

In this paper, we propose a method to estimate camera extrinsic parameter based on projective and permutation invariance point features. Because feature informations in previous research is variant to c.:men viewpoint, extraction of correspondent point is difficult. Therefore, in this paper, we propose the extracting method of invariant point features, and new matching method using similarity evaluation function and Graham search method for reducing time complexity and finding correspondent points accurately. In the calculation of camera extrinsic parameter stage, we also propose two-stage motion parameter estimation method for enhancing convergent degree of LM algorithm. In the experiment, we compare and analyse the proposed method with existing method by using various indoor images to demonstrate the superiority of the proposed algorithms.

• Bulletin of the Korean Mathematical Society
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• v.22 no.1
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• pp.11-15
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• 1985

Manddelberg [9] has shown that a Clifford algebra of a free quadratic space over an arbitrary semi-local ring R in Brawer-Wall group BW(R) is determined by its rank, determinant, and Hasse invariant. In this paper, we prove a corresponding result when R is a full ring.Throughout this paper, unless otherwise specified, we assume that R is a commutative ring having 2 a unit. A quadratic space (V, B, .phi.) over R is a finitely generated projective R-module V with a symmetric bilinear mapping B: V*V.rarw.R which is non-degenerate (i.e., the natural mapping V.rarw.Ho $m_{R}$(V,R) induced by B is an isomorphism), and with a quadratic mapping .phi.: V.rarw.R such that B(x,y)=1/2(.phi.(x+y)-.phi.(x)-.phi.(y)) and .phi.(rx) = $r^{2}$.phi.(x) for all x, y in V and r in R. We denote the group of multiplicative units of R by U9R). If (V, B, .phi.) is a free rank n quadratic space over R with an orthogonal basis { $x_{1}$,.., $x_{n}$}, we will write < $a_{1}$,.., $a_{n}$> for (V, B, .phi.) where the $a_{i}$=.phi.( $x_{i}$) are in U(R), and denote the space by the table [ $a_{ij}$ ] where $a_{ij}$ =B( $x_{i}$, $x_{j}$). In the case n=2 and B( $x_{1}$, $x_{2}$)=1/2 we reserve the notation [a $a_{11}$, $a_{22}$] for the space. A commutative ring R having 2 a unit is called full [10] if for every triple $a_{1}$, $a_{2}$, $a_{3}$ of elements in R with ( $a_{1}$, $a_{2}$, $a_{3}$)=R, there is an element w in R such that $a_{1}$+ $a_{2}$w+ $a_{3}$ $w^{2}$=unit.TEX>=unit.t.t.t.