Design of Spatial Information Systems

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DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF
JOENSUU JOENSUU, FINLAND

• Design of Spatial Information Systems
• Lecture 11
• Voronoi diagrams
• Alexander Kolesnikov

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The post office problem
The trading areas of the capitals of the 12
provinces in the Netherlands.
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Voronoi diagrams Definition

• Let Pp1, p2, , pn be a set of n distinct
  points in the
• plane these points are the sites.
• Voronoi diagram of P is the subdivision of the
  plane into n
• cells, one for each site in P, with the property,
  that a point
• q lies in the cell corresponding to a site pi, if
  and only if
• dist(q,pi) lt dist(q,pj)
  
• for each pj ?P with j?i.

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Voronoi diagrams Example
pj
q
pi
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Voronoi diagram Terms
Unbounded cell
Bounded cell
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Voronoi diagram for two sites
Perpendicular bisector for two sites.
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Voronoi diagram for collinear sites
Perpendicular bisector for 6 collinear sites.
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Voronoi diagram for non-collinear sites
.
?
.
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Voronoi diagram for three non-collinear sites.
Do the lines intersect at the one point?
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Voronoi diagram for non-collinear sites
Voronoi diagram for four non-collinear sites
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Algorithms Survey
1) Plane intersect Complexity O(n2
logn) 2) Incremental algorithm Worst case
O(n2) 3) Divide-and-conquer Divide Step
O(n logn) Conquer Step O(n logn)
Total complexity O(n logn) 4) Plane sweep or
Fortunes algorithm (1985) Complexity
O(n logn)
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Plane intersect algorithm Main idea
Follow the mathematical description for
generating a Voronoi cell merge all half-planes
generated by the bisecting line between the site
pi and all other sites pj in the plane.
Vor(pi) merge H(pi,pj) for i!j where H(pi,pj)
is the half-plane between site pi and site pj.
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Plane intersect algorithm Example (1)
The 1st half-plane is found. The 2nd
half-plane is found.
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Plane intersect algorithm Example (2)
The 3rd half-plane is found. The 4th
half-plane is found.
Repeat the procedure for all sites.
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Incremental algorithm Main idea

• After the straightforward approach in the
  previous algorithm, lets try to achieve some
  efficiency by cutting out some redundancy.
• Sort all sites along the increasing x-axis and
  generate the first diagram.
• Afterwards the sites are subsequently inserted
  into the fringe of the diagram, minimizing
  necessary cutting operations.

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1. Incremental Algorithm

• Suppose that we have already built the Voronoi
  diagram Vp-1, and would like to add a new sites
  sp.
• First, find the site, say si, whose Voronoi
  polygon contains sp, and draw the perpendicular
  bisector between sp and si, denoted by B(sp, si).

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2. Incremental Algorithm

• The bisector crosses the boundary of V(si) at two
  points, point x1 and point x2. Site sp is to the
  left of the directed line segment x1x2.
• The line segment x1x2 divides the Voronoi polygon
  V(si) into two pieces, the one on the left
  belonging to the Voronoi polygon of sp.
• Thus, we get a Voronoi edge on the boundary of
  the Voronoi polygon of si .

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3. Incremental Algorithm

• Starting with the edge x1x2, expand the boundary
  of the Voronoi polygon of sl by the following
  procedure. The B(si, sl) crosses the boundary of
  V(si) at x2, entering the adjacent Voronoi
  polygon, say V(sj).
• Therefore, next draw the B(si, sj), and find the
  point, x3, at which the bisector crosses the
  boundary of V(sj).
• Similarly, find the sequence of segments of
  perpendicular bisectors of s and the neighboring
  sites until we reach the starting point x1.

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4. Incremental Algorithm

• Let this sequence be (x1x2, x2x3, , xm-1xm,
  xmx1). This sequence forms a CCW boundary of the
  Voronoi polygon of the new site s.
• Finally, we delete from Vp-1 the substructure
  inside the new Voronoi polygon, and thus get Vp.

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. Example
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Example
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Example
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Example
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Example
O(n) per point insertion Total complexity O(n2)
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Fortunes algorithm Main idea
1) A cone with 45 degree angle is constructed
centered on each site in the plane. 2) A
slanted plane (45 degree angle) is now dragged
across the coordinate system. 3) The
intersection between the cones, when projected
onto the x-y plane defines the Voronoi lines.
When the angled plane moves across the x axis,
the intersection of the plane with the
individual cones (projected onto the x-y plane)
defines the intersection points of the
individual parabola curves, and thus the Voronoi
lines between the sites in the x-y
plane. Complexity O(n logn)
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Fortunes algorithm Main idea
A cone with 45 degree angle is constructed
centered on each site in the plane.
A slanted plane (45?) is now dragged across the
coordinate system
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Fortunes algorithm Demo
Beach line
http//www.diku.dk/hjemmesider/studerende/duff/For
tune
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Voronoi diagram ? ...
Once you have a Voronoi diagram for point set
P 1) How to get Convex Hull for the point set
P? Vor(P) ? CH(P) 2) Draw an edge
between each pair of points that share a Voronoi
edge Vor(P) ? Delaunay
triangulation
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Other names of ...

• Voronoi diagram, tessellation, polygons
• Dirichlet diagram, regions, tessellation
• Thiesen polygons, polytopes
• Descartes cosmic fragmentation
• Wigner-Seitz cells (metallurgy)
• Blum's transform (biological shape and visual
  science)
• Brown area potentially available to a tree
• Mead plant polygons