• Journal of the Institute of Electronics Engineers of Korea SD
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• v.42 no.1
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• pp.57-67
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• 2005

Fast scalar multiplication of points on elliptic curve is important for elliptic curve cryptography applications. In order to vary field sizes depending on security situations, the cryptography coprocessors should support variable length finite field arithmetic units. To determine the effective variable length finite field arithmetic architecture, two well-known curve scalar multiplication algorithms were implemented on FPGA. The affine coordinates algorithm must use a hardware division unit, but the projective coordinates algorithm only uses a fast multiplication unit. The former algorithm needs the division hardware. The latter only requires a multiplication hardware, but it need more space to store intermediate results. To make the division unit versatile, we need to add a feedback signal line at every bit position. We proposed a method to mitigate this problem. For multiplication in projective coordinates implementation, we use a widely used digit serial multiplication hardware, which is simpler to be made versatile. We experimented with our implemented ECC coprocessors using variable length finite field arithmetic unit which has the maximum field size 256. On the clock speed 40 MHz, the scalar multiplication time is 6.0 msec for affine implementation while it is 1.15 msec for projective implementation. As a result of the study, we found that the projective coordinates algorithm which does not use the division hardware was faster than the affine coordinate algorithm. In addition, the memory implementation effectiveness relative to logic implementation will have a large influence on the implementation space requirements of the two algorithms.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.371-384
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• 2022

Fix an integer n ≥ 1. Then the simplex Π_n, Birkhoff polytope Ω_n, and Latin square polytope Λ_n each yield projective geometries obtained by identifying antipodal points on a sphere bounding a ball centered at the barycenter of the polytope. We investigate conditions for homogeneous coordinates of points in the projective geometries to locate exact vertices of the respective polytopes, namely crisp distributions, permutation matrices, and quasigroups or Latin squares respectively. In the latter case, the homogeneous conditions form a crucial part of a recent projective-geometrical approach to the study of orthogonality of Latin squares. Coordinates based on the barycenter of Ω_n are also suited to the analysis of generalized doubly stochastic matrices, observing that orthogonal matrices of this type form a subgroup of the orthogonal group.

• Korean Journal of Computational Design and Engineering
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• v.9 no.4
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• pp.397-405
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• 2004

Texture mapping is the process of covering 3D models with texture images in order to increase the visual realism of the models. For proper mapping the coordinates of texture images need to coincide with those of the 3D models. When projective images from the camera are used as texture images, the texture image coordinates are defined by a camera calibration method. The texture image coordinates are determined by the relation between the coordinate systems of the camera image and the 3D object. With the projective camera images, the distortion effect caused by the camera lenses should be compensated in order to get accurate texture coordinates. The distortion effect problem has been dealt with iterative methods, where the camera calibration coefficients are computed first without considering the distortion effect and then modified properly. The methods not only cause to change the position of the camera perspective line in the image plane, but also require more control points. In this paper, a new iterative method is suggested for reducing the error by fixing the principal points in the image plane. The method considers the image distortion effect independently and fixes the values of correction coefficients, with which the distortion coefficients can be computed with fewer control points. It is shown that the camera distortion effects are compensated with fewer numbers of control points than the previous methods and the projective texture mapping results in more realistic image.

• ETRI Journal
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• v.30 no.3
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• pp.365-376
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• 2008

This paper presents the design and implementation of a hyperelliptic curve cryptography (HECC) coprocessor over affine and projective coordinates, along with measurements of its performance, hardware complexity, and power consumption. We applied several design techniques, including parallelism, pipelining, and loop unrolling, in designing field arithmetic units, group operation units, and scalar multiplication units to improve the performance and power consumption. Our affine and projective coordinate-based HECC processors execute in 0.436 ms and 0.531 ms, respectively, based on the underlying field GF($2^{89}$). These results are about five times faster than those for previous hardware implementations and at least 13 times better in terms of area-time products. Further results suggest that neither case is superior to the other when considering the hardware complexity and performance. The characteristics of our proposed HECC coprocessor show that it is applicable to high-speed network applications as well as resource-constrained environments, such as PDAs, smart cards, and so on.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2002.11a
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• pp.226-230
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• 2002

This paper proposes a method of improvement of accuracy of projective transformation matrix for Image Mosaicing. Using shift theorem, we extracted global translation components between images and using translation components, we found matching points between images so we solve general matching point problem we extracted highly trusted matching point using RANSAC algorithm. we normalized matching point coordinates and improved accuracy of projective transformation matrix.

• Smart Structures and Systems
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• v.25 no.2
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• pp.135-153
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• 2020

There has been a sustained interest towards the non-contact structural displacement measurement by means of videogrammetric technique. On the way forward, one of the major concerns is the spurious image drift induced by temperature variation. This study therefore carries out an investigation into the temperature effect of videogrammetric technique, focusing on the exploration of the mechanism behind the temperature effect and the elimination of the temperature-caused measurement error. 2D videogrammetric measurement tests under monotonic or cyclic temperature variation are first performed. Features of measurement error and the casual relationship between temperature variation and measurement error are then studied. The variation of the temperature of digital camera is identified as the main cause of measurement error. An excellent linear relationship between them is revealed. After that, camera parameters are extracted from the mapping between world coordinates and pixels coordinates of the calibration targets. The coordinates of principle point and focal lengths show variations well correlated with temperature variation. The measurement error is thought to be an outcome mainly attributed to the variation of the coordinates of principle point. An approach for eliminating temperature-caused measurement error is finally proposed. Correlation models between camera parameters and temperature are formulated. Thereby, camera parameters under different temperature conditions can be predicted and the camera projective matrix can be updated accordingly. By reconstructing the world coordinates with the updated camera projective matrix, the temperature-caused measurement error is eliminated. A satisfactory performance has been achieved by the proposed approach in eliminating the temperature-caused measurement error.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.8 no.9
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• pp.3177-3195
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• 2014

In many moving object detection problems of an aerial video, accurate and robust stabilization is of critical importance. In this paper, a novel accurate image alignment algorithm for aerial electronic image stabilization (EIS) is described. The feature points are first selected using optimal derivative filters based Harris detector, which can improve differentiation accuracy and obtain the precise coordinates of feature points. Then we choose the Delaunay Triangulation edges to find the matching pairs between feature points in overlapping images. The most "useful" matching points that belong to the background are used to find the global transformation parameters using the projective invariant. Finally, intentional motion of the camera is accumulated for correction by Sage-Husa adaptive filtering. Experiment results illustrate that the proposed algorithm is applied to the aerial captured video sequences with various dynamic scenes for performance demonstrations.

• Journal of Korea Multimedia Society
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• v.15 no.6
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• pp.770-783
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• 2012

In this paper, we propose a novel structure recovery algorithm in the projective space using image feature points. We use normalized image feature coordinates for the numerical stability. To acquire an initial value of the structure and motion, we decompose the scaled measurement matrix using the singular value decomposition. When recovering structure and motion in projective space, we introduce matrix decomposition constraints. In the reconstruction procedure, a nonlinear iterative optimization technique is used. Experimental results showed that the proposed method provides proper accuracy and the error deviation is small.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2022.05a
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• pp.246-248
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• 2022

Binary Edwards curves (BEdC), a new form of elliptic curves proposed by Bernstein, satisfy the complete addition law without exceptions. This paper describes an efficient hardware implementation of point scalar multiplication on BEdC using projective coordinates. Modified Montgomery ladder algorithm was adopted for point scalar multiplication, and binary field arithmetic operations were implemented using 257-bit binary adder, 257-bit binary squarer, and 32-bit binary multiplier. The hardware operation of the BEdC crypto-core was verified using Zynq UltraScale+ MPSoC device. It takes 521,535 clock cycles to compute point scalar multiplication.

• 한국정보디스플레이학회:학술대회논문집
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• 2009.10a
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• pp.1014-1017
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• 2009

The analytical expression for quality function is found including the dependence on disparity. The problem is considered in the projective coordinates for which the forward and backward transformation matrices are found. The formation of side observer regions is considered. The probability of the pseudo stereo effect is also estimated. Testing patterns are improved in order to provide higher accuracy of measurements. This is confirmed in experiments.