Geometric methods in shape and pattern recognition Lecture: 3 Date: 0-0-006 Lecturer: Prof. dr. H. Alt Location: UU Utrecht Recorder: Mark Bouts 050761 1. Convex polygons (continued) Additional remark A convex polygon can be defined as being a simple, closed polygonal chain, which interior is convex. Confusion may rise whether the convex polygon are just the points at the boundary or that the points in the interior also are included. In the definition stated last week it has been defined that next to the boundary, the interior points are being included in the convex polygon. This leads to two important characterisations of the convex polygon. The convex polygon is always a convex hull of a finite set of points; an intersection of finitely many half planes. Where half-planes are planes that separate planes in two halves in which every point of the former whole plane lies on one side of a straight line. In figure 1.1 an example of a Figure 1.1 P convex hull is given with P in the convex hull. In figure 1. it is shown that the intersection of three different half planes can result in the creation of a convex polygon. Therefore Q lies at the intersection of the half-planes and is thus a convex polygon. Half plane II Q Half plane I Figure 1.a Convex polygon Half plane III Now that we know that a convex polygon can be defined by the intersection of finitely many half planes, is it also true to claim that infinitely many half planes form a convex polygon? This does however not hold! Consider an example in which two parallel lines are being used, no convex polygon can be created from that. (Except when it is supposed that the parallel straight lines touch in infinity). (Fig 1.3b) From this an empty set of just 1 line segments need not to be included for they don t add any value to the this subject. Q Half plane II Half plane III Half plane I Figure 1.b Non convex polygon
1.3 Planar graphs and voronoi diagrams 1.3.1 Planar graphs In this paragraph insight is gained on planar graphs, with examples and definitions the concept of planar graphs is explained. Definition: A graph is planar exactly if it can be embedded ( drawn ) in the plane so that no two edges intersect. Formally embedding map: V points in the plane E curves, so that the end points are at the respective vertices Figure 1.3 Planar graph Examples of graphs, with n vertices. Kn is a complete graph in which all vertices and all pairs form an edge. K 1 K 3 K K 4 Figure 1.4 Planar graph More nice How about K 5? Is there a planar graph possible in K 5? No! As an example figure 1.5 Missing, so not planar Figure 1.5 non planar graph
A cube is a classical example of a polyhedron, a three-dimensional shape that is made up of a finite number of polygonal faces which are parts of planes, the faces meet in edges which are straight-line segments, and the edges meet in points called vertices. These structures are always planar! The different polyhedra can be best viewed in 3D. Figure 1.6 shows some 3D version of different polyhedra. Figure 1.6 Polyhedra Non planar A graph is normally not planar when it contains substructures of a K 3,3 or a K 5 graph. As an example figure 1.7 is given. Where it is also shown what k n:m graphs mean. Not possible n m K 3,3 Not planar K n,m Not planar Figure 1.7 K 3,3 The Polish mathematician Kazimierz Kuratowski (1930) provided a characterization of planar graphs, now known as Kuratowski's theorem: Theorem: A finite graph is planar if and only if it does not contain a subdivision that is an expansion of K 5 (the complete graph on five vertices) or K 3,3 (complete bipartite graph on six vertices, three of which connect to each of the other three).
The questions that one can ask from this, is whether a planar graph can be presented by straight line segments. From the next theorem presented by Fáy in 1948, it can be concluded that this is possible. Theorem: Any planar graph has a straight line embedding. I.e. edges are represented by line segments. Facets Definition The embedding of a planar graph into connected regions are called facets. As an example the next two figures can be given: 1 4 3 Graph G 4 vertices 6 edges 4 facets Dual Graph G* 4 vertices 6 edges 4 facets Figure 1.8 Facets / Dual Graph Dual graph Definition Let G = (V,E) be a planar graph, embedded the dual graph G* has the facets of G as vertices, the edges of G as edges connecting the two vertices in G*, which are the facets of G separated by the edge. To make this definition a bit more clear figure 1.8 and figure 1.9 give an example of the graph G and the dual graph G*. The dual graph G* is constructed as follows: we choose one vertex in each face of G (including the outer face) and for each edge e in G we introduce a new edge in G* connecting the two vertices in G* corresponding to the two faces in G that meet at e. Furthermore, this edge is drawn so that it crosses e exactly once and that no other edge of G or G* is intersected. Then G* is again the embedding of a planar graph.
8 6 1 5 3 1 3 4 5 4 7 6 Graph G 8 vertices 1 edges 6 facets Figure 1.9 Facets / Dual Graph Dual graph G * 6 vertices 1 edges 8 facets The figures 1.8 and 1.9 clearly show the interrelationship between the dual graph and the graph itself (shown by the arrows). The numbers in the figures show the facets that can be discerned either the graph or its corresponding dual graph. From these some remarks need to be made in order to stress the interrelationship between the Dual graph and the normal planar graph. The only graph which has the same K value for both the Graph and its corresponding dual graph is K 4 ; G ** = G. In other words the dual graph of the dual graph is the graph itself ; Since the graph and the dual graph are interrelated it is not difficult to derive that the both the graph as the corresponding dual graph are both planar. I e. G is planar G * is planar. The second picture in figure 1.9 can be better translated to a 3D model and then we see that this figure is also known as an octahedron. Figure 1.10 Octahedron
Now we know what dual graphs are and what their relation is with the original graph, we can start looking at the relation between the number of facets, vertices and edges. In order to define that relationship Euler s formula is given. Euler s formula Euler's formula states that if a finite connected planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer infinitely large region), then v e + f =, i.e. the Euler characteristic is. Euler's formula can be proven as follows: if the graph isn't a tree, then remove an edge which completes a cycle. This lowers both e and f by one, leaving v e + f constant. Repeat until you arrive at a tree; trees have v = e + 1 and f = 1, yielding v - e + f =. In other words Euler s formula can be proven by induction. From this it follows that: 1 3 e 3n 6 f n 4 n f 4 From these it can be the derived that the and 3 show the relation between the G and G * and their relation ship as has been shown above. We can ask ourselves now the question how many edges there are for a non planar graph. n 1 From e 3n 6 this can normally be e 3n 6 = n ( n 1) = (n ) For planar graphs this is (n). A simple graph is called maximal planar if it is planar but adding any edge would destroy that property. All faces (even the outer one) are then bounded by three edges 3e, explaining the alternative term triangular for these graphs. If a triangular graph has v vertices with v >, then it has precisely 3v-6 edges and v-4 faces.
1.3. Voronoi Diagrams In the this paragraph the concept of voronoi diagrams is explained. Definition s R in which R is a finite set and in which p S Voronoi cell The voronoi cell of p (VC(p)) is the set of all points closer to P than any other point in S, in which closer is the shortest normal Euclidian distance. As an example of voronoi cells, we can think of emergency districts in which the ambulance needs to travel distance to a certain emergency site. By dividing the are in several districts in which the Euclidian distance is shortest from the centre point, it is guaranteed that the ambulance will travel the shortest distance to a certain emergency site. The voronoi is cell is finite when all the points lie inside the convex hull and a voronoi cell is of course infinite set of points when the points are not in the convex hull. The division of the points in a voronoi diagram can be done O(log n), provided that the points are pre-processed. S = {p 1,,p n } in which P i P j for i + j. VC(p i ) = {P p p i < p p j for j+i} is the intersection of half planes or in other terms {{P p p i < p p j } which is always a half plane. In a figure this looks like: j i Bisector Bisector P k P l P i Pj Bisector Figure 1.11 Halfplane bisection The planes are bisected by the half planes which can be called bisectors this context.
From this definition we can introduce some new terms to explain something more of voronoi diagrams. So a voronoi cell is a convex polygon or unbounded (you cannot draw a box around it), but its boundary consists of line segments, rays or lines. These are called voronoi edges (1 D) of S and their end points are called voronoi vertices (0 D). p i v e Characterization: Each point on a voronoi edge has: equal distance and is closest to a certain point and point P lies on a voronoi edge exactly if: there are at least nearest sites. Voronoi edges (see the figure) are part of bisectors between sites with n = θ( n ) Figure 1.1 Halfplane bisection n. 3 As can be seen in the next figure 1.1. With 3 = θ( n ) A Voronoi vertex is the center of an empty circle touching 3 or more sites. Both are linear, but are rough estimates. Instead we better can consider voronoi diagrams as a planar graph with one artificial end point at infinity where all unbounded points end. Drawing a voronoi diagram Figure 1.13 Drawing voronoi diagrams For drawing a voronoi diagram the next algorithm can be observed: 1. Draw a point in the plane;. Draw a bisector between any points; 3. Draw a new point and cut any non participating bisectors; 4. If not finished with all points go back to point 1; Figure 1.13 shows the three first steps of this process. This results in a running time of O(n log n).
The graph has n facets therefore: 3n 6 edges and n 4 vertices in which the n of the first equation is the n that applies on G* and the n of the second equation applies on the n used in voronoi diagrams