A novel approach for ellipsoidal outer-approximation of the intersection region of ellipses in the plane

In this paper, a novel technique for tight outer-approximation of the intersection region of a finite number of ellipses in 2-dimensional space is proposed. First, the vertices of a tight polygon that contains the convex intersection of the ellipses are found in an efficient manner. To do so, the intersection points of the ellipses that fall on the boundary of the intersection region are determined, and a set of points is generated on the elliptic arcs connecting every two neighbouring intersection points. By finding the tangent lines to the ellipses at the extended set of points, a set of half-planes is obtained, whose intersection forms a polygon. To find the polygon more efficiently, the points are given an order and the intersection of the half-planes corresponding to every two neighbouring points is calculated. If the polygon is convex and bounded, these calculated points together with the initially obtained intersection points will form its vertices. If the polygon is non-convex or unbounded, we can detect this situation and then generate additional discrete points only on the elliptical arc segment causing the issue, and restart the algorithm to obtain a bounded and convex polygon. Finally, the smallest area ellipse that contains the vertices of the polygon is obtained by solving a convex optimization problem. Through numerical experiments, it is illustrated that the proposed technique returns a tighter outer-approximation of the intersection of multiple ellipses, compared to conventional techniques, with only slightly higher computational cost.

[1]  D Baltas,et al.  Optimized bounding boxes for three-dimensional treatment planning in brachytherapy. , 2000, Medical physics.

[2]  David Eberly,et al.  Intersection of Ellipses , 2002 .

[3]  Micha Sharir,et al.  A subexponential bound for linear programming , 1992, SCG '92.

[4]  E. Yaz Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[5]  Joseph O'Rourke,et al.  Finding minimal enclosing boxes , 1985, International Journal of Computer & Information Sciences.

[6]  A. Kurzhanski,et al.  Ellipsoidal Calculus for Estimation and Control , 1996 .

[7]  Chia-Tche Chang,et al.  Fast oriented bounding box optimization on the rotation group SO(3,ℝ) , 2011, TOGS.

[8]  Erik G. Ström,et al.  Distributed Bounding of Feasible Sets in Cooperative Wireless Network Positioning , 2013, IEEE Communications Letters.

[9]  Godfried T. Toussaint,et al.  A simple linear algorithm for intersecting convex polygons , 1985, The Visual Computer.

[10]  J. Norton,et al.  State bounding with ellipsoidal set description of the uncertainty , 1996 .

[11]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[12]  Emo Welzl,et al.  Smallest enclosing disks (balls and ellipsoids) , 1991, New Results and New Trends in Computer Science.

[13]  Leonidas J. Guibas,et al.  Arrangements of Curves in the Plane - Topology, Combinatorics and Algorithms , 2018, Theor. Comput. Sci..

[14]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[15]  F. L. Chernousko,et al.  Ellipsoidal bounds for sets of attainability and uncertainty in control problems , 2007 .

[16]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[17]  Godfried T. Toussaint,et al.  APPLICATIONS OF THE ROTATING CALIPERS TO GEOMETRIC PROBLEMS IN TWO AND THREE DIMENSIONS , 2014 .

[18]  Victor Klee,et al.  Finding the Smallest Triangles Containing a Given Convex Polygon , 1985, J. Algorithms.

[19]  Federico Thomas,et al.  An ellipsoidal calculus based on propagation and fusion , 2002, IEEE Trans. Syst. Man Cybern. Part B.

[20]  JOSEPH O’ROURKE,et al.  A new linear algorithm for intersecting convex polygons , 1982, Comput. Graph. Image Process..

[21]  Joseph O'Rourke,et al.  A new linear algorithm for intersecting convex polygons , 1982, Comput. Graph. Image Process..

[22]  N. Shor,et al.  New algorithms for constructing optimal circumscribed and inscribed ellipsoids , 1992 .

[23]  Benoît Champagne,et al.  Tight Two-Dimensional Outer-Approximations of Feasible Sets in Wireless Sensor Networks , 2016, IEEE Communications Letters.

[24]  David E. Muller,et al.  Finding the Intersection of n Half-Spaces in Time O(n log n) , 1979, Theor. Comput. Sci..

[25]  W. Kahan,et al.  Circumscribing an Ellipsoid about the Intersection of Two Ellipsoids , 1968, Canadian Mathematical Bulletin.

[26]  Mark J. Post,et al.  A minimum spanning ellipse algorithm , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[27]  D. Titterington,et al.  Minimum Covering Ellipses , 1980 .

[28]  Leonidas J. Guibas,et al.  Arrangements of Curves in the Plane - Topology, Combinatorics, and Algorithms , 1988, ICALP.

[29]  A. Vacavant,et al.  Reconstructions of Noisy Digital Contours with Maximal Primitives Based on Multi-Scale/Irregular Geometric Representation and Generalized Linear Programming , 2017 .

[30]  Herbert Freeman,et al.  Determining the minimum-area encasing rectangle for an arbitrary closed curve , 1975, CACM.

[31]  Aharon Ben-Tal,et al.  Lectures on modern convex optimization , 1987 .

[32]  M. Henk Löwner – John Ellipsoids , 2012 .

[33]  Henk Wymeersch,et al.  Tight 2-Dimensional Outer-approximations of Feasible Sets in Wireless Sensor Networks , 2016 .

[34]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.

[35]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .