Spatial Data Structures

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Spatial Data Structures
Jeffrey Schepers , Junior Student
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Principal DEM data structures

• Two data structures
• Rectangular grid (or elevation matrix),
• TIN (Triangulated Irregular Network, Delaunay
  triangulation). 
• Regular grids may not be adapted to the
  complexity of the relief, so that an excessive
  number of data points is needed to represent the
  terrain to a required level of accuracy.

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Analysis of surfaces
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Delaunay Triangulation
Named after Russian Mathematician Boris Delone
(Delaunay) (1934),
Satisfies a special Circumcircle property
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Example irregular set of points

• Delaunay triangulation is a uniqe triangulation
  which satisfy the empty circle property the
  circle passing through any three vertices of a
  Delaunay triangle does not contain any other site
  of P in its interior

Checking that if point 4 is inside the
circum-circle of triangle ?123.
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Applications of TIN

• Digital Elevation Models Drawing contour lines
  and interpolation and computation of earth work
  (cut and fill).
• Surface reconstruction Construction of three
  dimensional models for industrial applications,
  Producing animation for movies, games and
  simulators.

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Triangulation in Animation

• Animators use triangulation to create models of
  the characters. These models help them map the
  movements via computers. The models are then
  given a computer generated skin.

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Applications of TIN

• Delaunay triangulation provides a useful tool
  for chopping up space based on a set of points.
• Delaunay triangulation avoid triangles with
  extremely small angles to the extent possible.
• Can include break-lines Constrained
  triangulation.

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Advantages of Delaunay Triangulation

• Maximizes interior angles of the triangles in the
  network
• Avoids weak geometry and thin triangulation
• Creates a unique network (barring a minor
  exception)

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Constructing Delaunay triangulation

• The literature describes various techniques for
  the generation of a TIN on a computer.
• The triangulation methods can be divided into two
  groups static and dynamic triangulation.
• Static triangulation as the Radial sweep
  algorithm. Static triangulation means that the
  triangulation is not valid before every point
  from the data set is included in the network.
• Dynamic triangulation as the Incremental
  algorithm. Points are inserted one-by-one. When a
  new point is included in the network, the network
  is reorganized until the circle criterion is met.

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Radial Sweep algorithm
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Radial sweep algorithm

• Choose a point near the centroid of the points,
  and draw a line from this central point to all
  the other points.
• Every point on the boundary is listed in a
  boundary list. Each of these points is combined
  with the next two nodes on the list.
• If these points form a triangle outside the
  existing network, the triangle is included in the
  lattice and the boundary list is updated. The
  boundary of the area will now be a convex hull.
• Two neighbouring triangles make one
  quadrilateral. The shortest distance between two
  opposite vertices in the quadrilateral is chosen
  as the diagonal to improve the geometry. The
  diagonal swapping process is repeated until no
  more alterations appear.
• Another improvement is required to make it
  Deluanay triangulation.

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Example irregular set of points
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Example irregular set of points
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Example Building a triangulation
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Incremental algorithm
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Incremental algorithm

• An initial triangular network has to be created (
  Using three points at the borders).
• A new point is inserted, The point is connected
  to its enclosing triangle by three new triangle
  edges between the point and the vertices of the
  triangle
• The quadrilaterals, which have got the old''
  edges of the enclosing triangle as a diagonal,
  have to be tested by the Circum-circle rule. If
  they do not meet the criterion, their diagonals
  are swapped and the new, opposite edges to the
  inserted point will be examined as diagonals in
  their quadrilaterals.
• The Delaunay network now contains one more point.
  All the remaining points will be included in the
  network in exactly the same way.

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Example irregular set of points
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Example irregular set of points
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Example Building triangulation
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Triangulation with Break-lines

• An important, criterion for selecting an
  interpolation method is the degree to which
  structural features (Geomorphological
  information, drainage, ridge lines) can be taken
  into account.
• Constrained triangulation is the triangulation of
  a set S of N points and a set E of M
  non-intersecting edges on S (the constraints),
  which must appear in the triangulation.

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Triangulation with Break-lines

• Other methods include
• Re-sampling the lines with 0.25 times the contour
  interval namely adding the lines as a list of
  points and the problem is converted into a
  regular Delaunay triangulation.
• Deleting all the triangles which are crossed by
  given segments and re-triangulating the resulted
  quadrilateral.

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Voronoi diagram is the dual of the Delaunay
triangulation

• The Voronoi diagram delineates proximity (closest
  point)
• You can construct the diagram from the Delaunay
  triangulation by drawing the perpendicular
  bisectors.

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Example Building Voronoi diagram
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Drawing perpendicular bisectors
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Example Building Voronoi diagram
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Example
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A dried lake-bed, at Salar de Atacama in northern
Chile, As the water from recent rains has
evaporated, the surface layer of mud has dried
and contracted. Then moisture in the lower layers
of the mud, which has a high salt content, has
soaked upwards and evaporated, leaving behind
salt crystals outlining polygons which are almost
Voronoi cells
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Antrim in northern Ireland. Regular columns of
rock, created by the forces of contraction on the
cooling magma, which caused it to split into
shapes that are similar to Voronoi polygons.
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Questions?
Building the Voronoi diagram