Presentation on theme: "TEL-AVIV UNIVERSITY FACULTY OF EXACT SCIENCES SCHOOL OF MATHEMATICAL SCIENCES An Algorithm for the Computation of the Metric Average of Two Simple Polygons."— Presentation transcript:
1 TEL-AVIV UNIVERSITY FACULTY OF EXACT SCIENCES SCHOOL OF MATHEMATICAL SCIENCES An Algorithm for the Computation of the Metric Average of Two Simple Polygons Shay Kels Nira Dyn Evgeny Lipovetsky
2 Approximation of set-valued functions (N. Dyn, E. Farkhi, A. Mokhov) Approximation of a set-valued function from a finite number of its samplesGiven a binary operation between sets, many tools fromapproximation theory can be adapted to set-valued functionsSpline subdivision schemesBernstein operators via de Casteljau algorithmSchoenberg operators via de Boor algorithmThe applied operation is termed the metric average (Z. Artstein )Repeated computations of the metric average are required.
3 Outline An algorithm that applies tools from computational geometry: segment Voronoi diagrams and planar arrangements to thecomputation of the metric average of two simple polygonsImplementation of the algorithm as a C++ software usingCGAL – Computational Geometry Algorithms LibraryConnectedness and complexity of the metric average.Extension to compact sets that are collections of simplepolygons with holes.
4 Preliminary definitions Let be the collection of nonempty compact subsets ofThe Euclidean distance from a point p to a set isThe Hausdorff distance between two sets isThe set of all projections of a point p on a set isThe linear Minkowksi combination of two sets is
5 The metric average Example in : Let and the one-sided t-weighted metric average of A and B isThe metric average of A and B isThe metric property:
6 Conic polygons with holes A conic segment is defined by:its base conic:its beginning pointits end pointA simple conic polygon is a region of theplane bounded by a single finite chain ofconic segments, that intersect only at theirendpoints.A simple conic polygon with holes is aconic polygon that contains holes, whichare simple conic polygons.
7 Planar arrangementsGiven a collection C of curves in the plane, the arrangement of C is the subdivision of the planeinto vertices,edgesand facesinduced by the curves in C.The overlay of two arrangements andis the arrangement produced by edges fromand
8 Segment Voronoi diagrams For a set S of n simple geometric objects(called sites) , the Voronoi diagram of S isthe subdivision of the plane into regions(called faces), each region being associated withsome site , and containing all points ofthe plane for which is closest among allthe sites in S.A segment is represented as three objects:an open segment and the endpoints.The diagram is bounded by a frame.All edges are conic segments.The diagram constitutes of an arrangement of conic segments.
9 Computation of the metric average - the algorithm The metric average can be written as:Computation of where A, B are simple polygons1. Compute the sets2. Compute3. Compute4. Compute
10 Computation of the metric average - the metric faces Let A, B be simple polygons and let F bea Voronoi face of VDB , we call a connected component of as a metric face originating from F.The metric faces are faces of the overlayof the arrangements representing VDBand A \ B, which are intersection ofthe bounded faces of the two originalarrangements.Each metric face “inherits” the Voronoi site ofthe face of VDB that contains it.
11 Computation of the metric average - the metric faces(1) By definition of the Voronoi diagram, forthusThe transformis a continuous and one-to-one functionfrom F toThe region bounded byisWe can compute the metric average only for boundaries of the metric faces and only relative to the corresponding Voronoi sites.
12 Computation of the metric average - the algorithm (1) Computation of the one-sided metric average1. Compute the segment Voronoidiagram induced by2 . Overlay with andobtain the metric faces with theircorresponding sites3. For each metric face in thecollection found in 2,compute
13 Computation of the metric average - the algorithm (2) for a metric face F1. For each conic segment ina. computeb. add the result of (a) to the collectionof conic segments already computed2. Return the resulting collection of conicsegments as boundary of a conic polygon( i.e. we computed )S
14 Computation of the metric average - the algorithm (3) Computation of for a conic segment and the corresponding point Voronoi site Sis the set of pointssatisfyingwhere and are collinear.1. Express in terms of p and S2. Substitute into the conic equation ofand by collecting the terms obtain the conic equation ofFor a segment Voronoi site S, computeand the computation is similar.
15 Complexity boundsProposition: Let A,B be simple polygons and let n be the sum of the number of vertices in A and the number of vertices in B. Let k be the combinatorial complexity (the sum of the number of vertices, the number of edges, and the number of faces) of the overlay of the arrangements representing the sets and.Then:k isThe combinatorial complexity of with isThen the run-time complexity of the computation of themetric average is
16 The metric average of two simple convex polygons with ExamplesThe metric average of two simple convex polygons with
17 Examples (1)The metric average of two simple polygons with
18 Connectedness of the metric average The metric average of two intersecting simple polygonscan be a union of several disjoin conic polygons.The connectedness problem ismodel by a graph.Vertices:connected components of- metric faces of- metric faces of
19 Connectedness of the metric average (1) are called metric connected if and only if the set is connected.There is an edge on the graph between each twovertices corresponding to metric connected elements.is connected iff the metric connectivity graph is connected
20 Connectedness of the metric average (2) Several propositions considering metric connectedness, for example:Proposition: Let be simple polygons and be metric faces , are metric connected if and only if there are points and , satisfying:andIn terms of metric faces and the corresponding Voronoi sites:condition 1condition 2condition 3
21 The metric average of two simple polygonal sets A set consisting of pairwise disjoint polygons with holes is termed a simple polygonal set.The segment Voronoi diagram induced by the boundary of a simple polygonal set is well defined.Let be simple polygonal sets and F a face of , a connected component of is termed a metric face originating from F.The metric faces are conic polygons with holes.
22 The metric average of two simple polygonal sets (1) Let the metric face F be a conic polygon P with holesThe operation isa continuous and one-to-one function fromF tois a conic polygonwith holesThe computation is similar to the computation of the metric average and the modified metric average of two simple polygons.The implementation is supported by CGAL.
23 Examples , where A is a polygon and B consists of two polygons contained in A.
24 Examples (1),where A, B are simple polygonal sets.
25 Future workAn algorithm for the computation of the metric average of two-dimensional compact sets with boundaries consisting of spline curvesAn algorithm for the computation of the metric average oftwo polyhedra.Research for new set averaging operations with the ‘metric property’ relative to some distanceWork in progress: new set averaging operation with superior geometric features and the ‘metric property’ relative to the measure of the symmetric difference distance
26 ReferencesZ. Artstein, “Piecewise linear approximation of set valued maps”, Journal of Approximation Theory, vol. 56, pp , 1989.F. Aurenhammer, R Klein, "Voronoi Diagrams" in Handbook of Computational Geometry, J. R. Sack, J. Urrutia, Eds., Amsterdam: Elsevier, 2000, ppN. Dyn, E. Farkhi, A. Mokhov, “Approximation of univariate set-valued functions - an overview”, Serdica, vol. 33, pp , 2007.D. Halperin, "Arrangements", in Handbook of Discrete and Computational Geometry, J. E. Goodman, J. O’Rourke, Eds., Chapman & Hall/CRC, 2nd edition, 2004, pp 529–562.The CGAL project homepage.
27 Appendix A: Computation of the metric average with Voronoi diagrams – the mathematics Let A, B be simple polygons, the set A\ B can be written asand thereforeFor a point p on the interior of a Voronoi face Ftherefore(*)or in terms of metric faces(**)(*) and (**) can be extended to any two compact sets A, B in
28 Appendix A: Computation of the metric average with Voronoi diagrams – the mathematics (1) For a site S(F) of the segment Voronoi diagram and a point p in R2the set is a singleton.can be regarded as a functionwhich is continuous and one-to-one.The boundary of a metric face is a simple closed curve, so is itsmapping under G, and thereforestands for the region bounded byThe metric average can be computed aswhere