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An Efficient Node Localization in Wireless Sensor Networks Using a Hybrid Metaheuristics Optimization Model
Published in Amit Kumar Tyagi, Ajith Abraham, A. Kaklauskas, N. Sreenath, Gillala Rekha, Shaveta Malik, Security and Privacy-Preserving Techniques in Wireless Robotics, 2022
Panimalar Kathiroli, S. Kanmani
Nelder Mead algorithm is a heuristic optimization technique that belongs to family of global optimization known as direct search method as in Figure 15.7. by John Ashworth Nelder and Roger Mead during the year 1965. The NM algorithm also called Downhill simplex or Amoeba method is simple, easy to use and entails low memory. It is managed to unravel the multidimensional unconstrained problem of curtailing a given nonlinear function of variables. This sort of nondirective search method solves problems that have degree of randomness Simplex minimizes objective function without constraints. Here NM begins with a randomly generated simplex as initial condition, then a polytype is built with additional points. It runs with the constant problem but bring in different results, i.e., for every iteration, the simplex point is reshaped or shifted towards a best result in the search space. The polytype is a form of simplices on any dimension. For instance, a simplex in 1-D is a line, a simplex in 2-D is a triangle, where as a simplex in 3-D is a tetrahedron, and at the same time a simplex in 4-D is a 5-cell. A simplex is the convex hull of n+1 vertex in D-dimensional space. The convex hull is the smallest rounded polygon including all given points. The NM method uses four elementary geometric transformations called reflection, expansion, contraction, and shrinkage as in Figure 15.6. The objective function is calculated at the simplex vertices until minimum stopping condition is met.
The Convex Hull of Two Closed Implicit Surfaces
Published in Wasim Ahmed Khan, Ghulam Abbas, Khalid Rahman, Ghulam Hussain, Cedric Aimal Edwin, Functional Reverse Engineering of Machine Tools, 2019
Algorithms for computing convex hulls of various objects have a broad range of applications in pattern recognition, collision detection [1,2], image processing, statistics, and geographical information system (GIS). Computing the convex hull is to construct a non-ambiguous and efficient representation of the required convex shape. In computational geometry [3], numerous algorithms are proposed for computing the convex hull of a finite set of points, polygons, and polyhedra with various computational complexities [3–5]. For spline surface, we first take the convex hull of its control points as an approximation to its actual convex hull and subdivide the surface into smaller pieces. Then the union created by the control points of these smaller pieces will be a better approximation to its actual convex hull. Repeat the process until the approximating convex hull converges to the exact convex hull within a certain reasonable bound. Seong et al. [6] once presented an algorithm for computing the convex hull of a rational surface without using the subdivisions, usually a troublesome process, in which the problem of computing convex hull is reduced to one of finding the zero-sets of polynomial equations. Considering a special case, Geismann et al. [7] developed an algorithm by reducing the convex hull problem of a set of ellipsoids to that of computing a cell in an arrangement of quadrics.
Domain-Modeling Techniques
Published in Alexandru Telea, Data Visualization, 2014
The best-known triangulation method is the Delaunay algorithm [de Berg et al. 00]. This method generates triangular cells ci for a set of 2D points pi ∈ ℝ2 and tetrahedra for a set of 3D points pi ∈ ℝ3. A Delaunay triangulation of a point set consists of a set of triangles that covers the convex hull of the point set. An important property of a Delaunay triangulation is that no point from the input point set {pi} lies in the circumscribed circle of any triangle in the triangulation. Triangulations that obey this property are called conforming Delaunay triangulations. Given a set of scattered points with data values recorded at the point locations, using the Delaunay triangulation is the most “natural” way to create a 𝒞1, piecewise linear, interpolation of the data values over the convex hull of the points. To do this, we define piecewise linear basis functions over the triangles contained in the unstructured grid generated by the Delaunay triangulation, and use these functions to interpolate the vertex data values, as explained in Section 3.3. Figure 8.3(a) shows a Delaunay triangulation of a random point cloud containing 600 points. The point density is higher in the center, which causes the creation of smaller triangles in that area. Another example of Delauney triangulation is shown in Figure 3.12 (middle).
Micro Expression Recognition Using Delaunay Triangulation and Voronoi Tessellation
Published in IETE Journal of Research, 2022
Rashmi Adyapady R., B. Annappa
The DT and the VD have been extensively studied and applied in a variety of fields [31,32]. DT is a mesh generation technique invented by Delaunay with the capability for autonomous operation [33]. The compactness of the produced mesh is a criterion for an automatic mesh generating algorithm. According to Euler-Poincare, the association between the vertices, edges, and faces should be precise. At the end of the creation process, there should not be any loose vertices, gangling edges, or faces. The DT technique is employed for filling a volume V with three-dimensional triangles (i.e. tetrahedra is one of the Delaunay-based techniques). Given a finite collection of “P” points, a triangulation of “P” is a simplicial complex that tessellates the convex hull of “P” and whose vertices belong to “P”. Also, multiple triangulations are created using the same set of points.
Tolerance estimation and metrology for reverse engineering based remanufacturing systems
Published in International Journal of Production Research, 2022
Instead of defining a specific plane as the datum, multiple valid datums could be established from a datum feature. For example, if a planar feature is chosen as the datum feature, then the set containing all the planes that are defined by at least three points on the feature and above which the entire point cloud lies can be defined as a candidate datum set. Obviously, the facet subset of the convex hull of the datum feature, which satisfies the above condition, is the minimum candidate datum set. The QuickHull (Barber et al. 1996) algorithm can be applied to find out the minimum convex hull. Then, the candidate datum set could be established through iteratively screening the facets by checking whether the point-to-facet direction is opposite to the out-of-material direction of the datum feature. Next, following the minimax and maximin conditions defined by the target tolerance, a specific datum, and its corresponding dimension could be found.
A new method to optimise finite dimensions thermodynamic models: application to an irreversible Stirling engine
Published in International Journal of Ambient Energy, 2018
F. Lanzetta, A. Vaudrey, P. Baucour
Obtaining the long-awaited ‘power vs. efficiency’ curve, like the one presented in Figure 7, requires computing the convex hull (or convex envelope) of a set of points. The convex hull could be defined as the minimal set of points that will define a polygon which includes all the points of the initial set. With a large number of points (around 2 150 in our case), the convex hull is computed thanks to Graham's algorithm (Graham 1972; de Berg et al. 2000). This algorithm can be described as follows: Find the centre of gravity, G, of the initial set of points.Compute the counter-clockwise angle from G to all other points of the set.Sort the points by angle.Construct the boundary by scanning the points in the sorted order and performing only right turns (trim off left turns).