• International Journal of CAD/CAM
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• v.8 no.1
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• pp.21-28
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• 2009

In the present work, an attempt has been made to construct branching surface from 2-D contours, which are given at different layers and may have branches. If a layer having more than one contour and corresponds to contour at adjacent layers, then it is termed as branching problem and approximated by adding additional points in between the layers. Firstly, the branching problem is converted to single contour case in which there is no branching at any layer and the final branching surface is obtained by skinning. Contours are constructed from the given input points at different layers by energy-based B-Spline approximation. 3-D curves are constructed after adding additional points into the contour points for all the layers having branching problem by using energy-based B-Spline formulation. Final 3-D surface is obtained by skinning 3-D curves and 2-D contours. There are three types of branching problems: (a) One-to-one, (b) One-to-many and (c) Many-to-many. Oneto-one problem has been done by plethora of researchers based on minimizations of twist and curvature and different tiling techniques. One-to-many problem is the one in which at least one plane must have more than one contour and have correspondence with the contour at adjacent layers. Many-to-many problem is stated as m contours at i-th layer and n contours at (i+1)th layer. This problem can be solved by combining one-to-many branching methodology. Branching problem is very important in CAD, medical imaging and geographical information system(GIS).

• International Journal of CAD/CAM
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• v.12 no.1
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• pp.29-37
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• 2012

A new algorithm has been developed to construct surface from the contours having branches and the final smooth surface is obtained by the reversible Catmull-Clark subdivision. In branching, a particular layer has more than one contour that correspond with at least one contour at the adjacent layer. In the next step, three-dimensional composite curve is constructed from contours of a layer having correspondence with at least one contour at the adjacent layer by inserting points between them and joining the contours. The points are inserted in such a way that the geometric center of the contours should merge at the center of the contours at the adjacent layer. This process is repeated for all layers having branching problems. Polyhedra are constructed in the next step with the help of composite curves and the contours at adjacent layer. The required smooth surface is obtained in the proposed work by providing the level of smoothness.

• Journal of KIISE:Computer Systems and Theory
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• v.27 no.2
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• pp.123-134
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• 2000

This paper addresses a new triangulation method for constructing surface model from a set of wire-frame contours. The most important problem of contour triangulation is the branching problem, and we provide a new solution for the double branching problem, which occurs frequently in real data. The multiple branching problem is treated as a set of double branchings and an algorithm based on contour merging is developed. Our double branching algorithm is based on partitioning of root contour by Toussiant's polygon triangulation algorithml[14]. Our double branching algorithm produces quite natural surface model even if the branch contours are very complicate in shape. We treat the multiple branching problem as a problem of coarse section sampling in z-direction, and provide a new multiple branching algorithm which iteratively merge a pair of branch contours using imaginary interpolating contours. Our method is a natural and systematic solution for the general branching problem of contour triangulation. The result shows that our method works well even though there are many complicated branches in the object.

• The KIPS Transactions:PartB
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• v.14B no.6
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• pp.413-422
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• 2007

This paper addresses a new technique for constructing surface model from a set of wire-frame contours. The most difficult problem of this technique, called contour triangulation, arises when there are many branches on the surface, and causes lots of ambiguities in surface definition process. In this paper, the branching problem is reduced as the surface reconstruction from a set of virtual belts and virtual canyons. To tile the virtual belts, a divide-and-conquer strategy based tiling technique, called the BPA algorithm, is adopted. The virtual canyons are covered naturally by an iterative convex removal algorithm with addition of a center vertex for each branching surface. Compared with most of the previous works reducing the multiple branching problem into a set of tiling problems between contours, our method can handle the problem more easily by transforming it into more simple topology, the virtual belt and the virtual canyon. Furthermore, the proposed method does not involve any set of complicated criteria, and provides a simple and robust algorithm for surface triangulation. The result shows that our method works well even though there are many complicated branches in the object.

• Journal of the Korean Society for Precision Engineering
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• v.20 no.4
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• pp.174-182
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• 2003

Recently, medical imaging has taken interest on CAD based solution for anatomical part fabrication or finite element analysis of human body. In principle, contours representing object boundary are obtained through image processing techniques. Surface models are then approximated by a skinning method. For this, various methods should be applied to medical images and contours. The major bottleneck of the reconstruction is to remove shape inconsistency between contours and to generate the branching surface. In order to solve these problems, bi-directional smoothing and the composite contour generation method are proposed. Bi-directional smoothing has advantage of removing the shape inconsistency between contours and minimizing shrinkage effect with a large number of iterations. The composite contour by the proposed method ensures smooth transition in branching region.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1994.10a
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• pp.719-724
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• 1994

This paper describes a hybrid surface-based method for smooth 3D surface approximation to a sequence of 2D cross sections. The resulting surface is a hybrid G $^{1}$ surface represented by a mesh of triangular and rectangular Bezier patches defined on skinning, branching, or capping regions. Each skinning region is approximated with a closed B_spline surface, which is transformed into a mesh of Bezier patches. Triangular G $^{1}$ surfaces are constructed over brabching and capping regions such that the transitions between each capping regions such that the transitions between each triangular surface and its neighboring skinning surfaces are G $^{1}$ continuous. Since each skinning region is represented by an approximated rectangular C $^{2}$ suface instead of an interpolated trctangular G $^{[-1000]}$ surface, the proposed method can provide more smooth surfaces and realize more efficient data reduction than triangular surfacebased method.

• Korean Journal of Veterinary Research
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• v.47 no.1
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• pp.7-17
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• 2007

The left main descending artery (LMDA) of left coronary artery (LCA) in rats runs around the left side of conus arteriosus after arising from the aortic sinus and descends to the apex of heart with branching several branches into the wall of left ventricle (LV). The ligation site of LMDA for myocardial infarction (MI) is the 2~4 mm from LCA origin, between the pulmonary trunk and left auricle. The characteristics that rat heart has no interventricular groove on the surface and its coronary arteries run intramyocardially with branching several branches give the difficulty in surgery for MI which resulted in expected size. This study was aimed to elucidate the branching patterns of the left coronary artery for analysis of MI size and for giving the basic data to producing small MI intentionally in 2 male species that are widely used, Sprague-Dowley (SD) and Wistar-Kyoto (WKY), in the world. Red latex casting was followed by the microdissection in 27 and 28 hearts of SD and WKY male rats, respectively. The branching patterns of LMDA were classified into 3 major types and others based on the left ventricular branches (L). The Type I, Type II, Type III and others are shown in 55.6%, 22.2%, 14.8%, and 7.4% in SD, 60.7%, 10.7%, 7.1%, and 21.5% in WKY, respectively. The branching number of the first left ventricular branch (L1) that are distribute the upper one third of LV was 1.2~1.5, and its branching sites were ranging 0.9~2.1 ᒠfrom LCA origin. L2, the second left ventricular branch distributing middle one third of LV, was the number of 1.2~1.4 and branching out ranging 5.1~5.7 mm. L3, the third left ventricular branch of LMDA distributing lower one third of LV, was the number of 1~1.5 and branching out ranging 7.0~9.3 mm from LCA origin. The common branch of L1 and L2 was branched from LMDA with the number of 1.1, and its site was located in the distance of mean of 1.5 mm and 2.8 mm in SD and WKY, respectively. The common branch of L2 and L3 was branched from LMDA with the number of 1, and its site was located in the distance of mean of 7.2 mm and 2.9 mm in SD and WKY, respectively. The right ventricular branches (R) of LMDA were short and branched in irregularly compared with L. The number of 1~4 of R were branched from LMDA. With regarding to the distribution area of L and the ligation site for MI, moderate MI (25~35% of LV) might be resulted in 70.4% and 60.7% in SD and WKY rats. Small MI might be produced intentionally if the ligation would be located at the 4~6 mm from LCA origin in the left side of LMDA. These data wold be helpful to expect the size of MI and to reproduce of small MI, intentionally, in rat hearts.

• Bulletin of the Korean Chemical Society
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• v.20 no.8
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• pp.897-904
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• 1999

We have presented the thermodynamic, structural and dynamic properties of liquid pentane isomers - normal pentane, isopentane, and neopentane - using an expanded collapsed atomic model. The thermodynamic properties show that the intermolecular interactions become weaker as the molecular shape becomes more nearly spherical and the surface area decreases with branching. The structural properties are well predicted from the site-site radial, the average end-to-end distance, and the root-mean-squared radius of gyration distribution func-tions. The dynamic properties are obtained from the time correlation functions - the mean square displacement (MSD), the velocity auto-correlation (VAC), the cosine (CAC), the stress (SAC), the pressure (PAC), and the heat flux auto-correlation (HFAC) functions - of liquid pentane isomers. Two self-diffusion coefficients of liquid pentane isomers calculated from the MSD's via the Einstein equation and the VAC's via the Green-Kubo relation show the same trend but do not coincide with the branching effect on self-diffusion. The rotational re-laxation time of liquid pentane isomers obtained from the CAC's decreases monotonously as branching increases. Two kinds of viscosities of liquid pentane isomers calculated from the SAC and PAC functions via the Green-Kubo relation have the same trend compared with the experimental results. The thermal conductivity calculated from the HFAC increases as branching increases.

• The Pure and Applied Mathematics
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• v.17 no.4
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• pp.299-306
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• 2010

In this article we consider axes of a complete embedded minimal surface in $R^3$ of finite total curvature, and then prove that there is no planar ends at which the Gauss map have the minimum branching order if the minimal surface has a single axis.

• Journal of the Korean Society for Precision Engineering
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• v.10 no.4
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• pp.81-93
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• 1993

The three dimensional(3D) shape reconstruction from two dimensional(2D) cross-sections can be completed through three main phases : the input compilation, the triangular grid formation, and the smooth surface construction. In the input compilation phase, the cross-sections are analyzed to exctract the input data required for the shape reconstruction. This data includes the number of polygonized contours per cross-section and the vertices defining each polygonized contour. In the triangular grid formation phase, a triangular grid, leading to a polyhedral approximations, is constructed by extracting all the information concerning contour links between two adjacent cross- sections and then performing the appropriate triangulation procedure for each contour link. In the smooth surface construction phase, a smooth composite surface interpolating all vertices on the triangular grid is constructed. Both the smooth surface and the polyhedral approximation can be used as reconstructed models of the object. This paper proposes a new method for reconstructing the geometric model of a 3D objdect from a sequence of planar contours representing 2D cross-sections of the objdect. The method includes the triangular grid formation algorithms for contour closing, one-to-one branching, and one-to-many braanching, and many-to-many branching. The shape reconstruction method has been implemented on a SUN workstation in C.