CPSC 335 Geometric Data Structures in Computer Modeling and GIS

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CPSC 335 Geometric Data Structures in Computer
Modeling and GIS
Dr. Marina L. Gavrilova
Assistant Professor Dept of Comp. Science,
University of Calgary, AB, Canada, T2N1N4
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Topics

• Survey of applications of computational geometry
  to modeling and simulation of natural processes
  and GIS
• Brief introduction to Voronoi diagrams, ROAM and
  progressive meshes
• Application to terrain modeling
• Application to image processing and visualization

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Computational geometry

• Originated in 1960 with invention of computers
• Describes algorithms and data structures for
  solving geometric problems with the use of
  computational methods
• Confirmed by practical implementation of methods

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Areas of applicability

• A spectrum of computational geometry
  applications
• Biology
• Molecular dynamics
• GIS (Geographical Information Systems)
• Physics
• Mechanics
• Chemistry

The Monter Carlo model of a lipid bilayer in
water, 8666 atoms
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Problems in CG

• Problems that are addressed include
• computing properties of particles arrangements,
  such as their volume and topology, in a general
  d-dimensional space
• testing intersections and collisions between
  particles
• finding offset surfaces (related to questions of
  accessibility of subregions)
• Solving nearest-neighbours problems
• Predicting time of the next collision between
  particles undergoing continuous motion
• Updating data structures accurately and
  efficiently
• Visualizing algorithms for rendering models

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Subjects of study in CG

• Spatial objects lines, polygons, planes
• Operations stored, displayed, manipulated,
  queried
• Methods querying algorithms, optimization
  algorithms, visualization methods

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Topological relationships

• Adjacent
• Overlapping
• Disjoined
• Inclusion
• Neighbor
• Closest neighbor

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Long ago

• Jorge Luis Borges
• "Map of a region was a size of a city,
• Map of an Empire was a size of a Province"

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Now

• Digital maps of geographical objects
• spatial, or geometric attribute (shape, location,
  orientation, size) in 2D or 3D
• non-spatial attribute (descriptive)

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Digital maps

• Geographical object
• spatial, or geometric attribute (shape, location,
  orientation, size) in 2D or 3D
• non-spatial attribute (descriptive) statistic,
  population, force, velocity, etc.

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What computer scientists are looking for

• Not only a goal, but a process itself!
• Data structure both inherent and coherent model
  properties, attributed geometric data structures
• Ex Surface representation
• Algorithms efficient, reliable, easy to update,
  portable, easy to implement, possible to visualize

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Commonly used data structures in GIS

• Space partitioning
• Planar subdivisions (regular and irregular)
• Tree-based data structure (segment trees, k-d
  trees, hierarchical octrees)
• Voronoi diagrams
• Triangulations

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Voronoi diagram

• Given a set of N sites (points) in the plane or a
  3D space
• Distance function d(x,P) between point x and site
  P is defined according to some metric
• Voronoi region Vor(P) is the set of all points
  which are closer to P than to any other site
• Voronoi diagram is the union of all Voronoi
  regions

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Delaunay triangulation
The generalized Delaunay tessellation (DT) is
the data structure, dual to the generalized VD,
containing proximity information for a set S.

• Definition 3. A Delaunay triangulation (DT) is
  the straight-line dual of the Voronoi diagram
  obtained by joining all pairs of sites whose
  Voronoi regions share a common Voronoi edge
  Delaunay 34.
• Follows from the definition
• If two Voronoi regions Vor(P) and Vor(Q) share an
  edge, then sites P and Q are connected by an edge
  in the Delaunay triangulation
• If a Voronoi vertex belongs to Vor(P), Vor(Q) and
  Vor(R), then DT contains a triangle (P,Q,R)

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VD and DT (for circles)
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Delaunay triangulation in 3D
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Delaunay triangulation in 3D, extra point
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Distances

• Distances Given a set of spherical objects
  (sites) S in Rd. Point x from Rd is the nearest
  neighbor of P from S iff d(x,P) lt d(x,Q), Q
  from S.
• Euclidean metric d(x,P) d(x,p) - rp
• Manhattan metric d(x,P) d(x,p) - rp x1 -
  p1 xd - pd-rp
• Supremum metric d(x,P) d(x,p) - rp max(x1 -
  p1xd - pd) - rp
• Laguerre geometry d(x,P) d(x,p)2 - r 2p (x1 -
  p1)2(xd- pd)2 - r2p

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Metric spaces in 2d
L1
L2
L?
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Power and Euclidean Voronoi diagrams biological
modeling
Euclidean bisector
Power bisector
Power diagram and Delaunay triangulation
Euclidean diagram and Delaunay triangulation
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Manhattan and Supremum VD city planning
Manhattan bisectors
Supremum bisectors
Manhattan diagram and Delaunay triangulation
Supremum diagram and Delaunay triangulation
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Example of Supremum VD and DT

• The supremum weighted Voronoi diagram (left) and
  the corresponding Delaunay triangulation (right)
  for 1000 randomly distributed sites .

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Metric spaces are used

• In computer modeling
• For optimization
• For point patter analysis
• For autocorrelation analysis

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3D Terrain Rendering

• Involves transformation of GIS data sources into
  a 3D coherent system
• Convert 2D map layers to 3D
• Using rendering methods to project the 3D data
  back to 2D
• Manipulate Camera and animation methods to obtain
  perspective view

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Sourcing data from real world
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DEM Digital Elevation Models

• Finite representation of an abstract modeled
  space.
• Represent a function of 2D space temperature,
  pressure, etc.
• Exist in sampled points in space, other points
  obtained by interpolation.

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DEM Digital Elevation Model

• Contains only relative Height
• Regular interval
• Pixel color determine height
• Discrete resolution

X
Kluanne National Park
Y
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TIN Triangulated Irregular Network

• TIN is a triangular partitioning of space.
• Set of points P, edges V and triangles T is
  stored in a quad-edge data structure.

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Tessellation in terrain modeling

• Tessellation is a cellular decomposition of the
  plane (space).

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Non-Photo-Realistic Real-time 3D Terrain
Rendering

• Uses DEM as input of the application
• Generates frame coherent animated view in
  real-time
• Uses texturing, shades, particles etc. for layer
  visualization

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Real-time terrain rendering

• Uses a graphics Engine/Library
• Central focus on efficient mesh representation
• View coherence and frame rate constancy
• Limited/Variable Level of Detail
• Speed optimization
• Representing layer data as textures or particles

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Converting Height field data into 3D topological
mesh

• Pixel value (z) is used as Height Map
• Vertices are generated as points in 3D
• A Mesh is triangulated

X
200
255
150
100
Y
100
255
255
200
200
150
200
100
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Mesh Representation

• Goals
• Speed
• Quality
• Constancy

• Representations
• Progressive Meshes
• ROAM Real-time Optimally Adapting Mesh

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Progressive Meshes
Developed by Hugues Hoppe, Microsoft Research
Inc. Published first in SIGGRAPH 1996.
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How PM works
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ROAM

• Can Subdivide when more details necessary
• Merge Split Queue
• Tree Structured

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Photo Realistic 3D Terrain
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Conclusions

• Computational geometry methods find a variety of
  applications in GIS and other areas for
  underlying data structures, algorithm
  implementation, querying, visualization,
  statistical analysis and optimization.