CPSC 335

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CPSC 335

• Representation and algorithms on spatial data.
• Dr. M. Gavrilova

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Presentation outline

• Types of spatial data
• Data representation
• Spatial data processing
• Algorithm design techniques
• Applications in GIS, modelling, computer
  graphics, compression and visualization

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Spatial data

• GIS (Geographic Information Systems)
• CAD (Computer Aided Design)
• VLSI (Very Large Scale Integration, IBM)
• Robotics
• Image Processing

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Spatial objects

• Stored
• Displayed
• Manipulated
• Queried

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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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Population Map
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Pollutants map
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Terrain map
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Data collection

• from GPS (Global Positioning Systems) BMP, GIF,
  GPEG, etc
• from existing maps, geometric (vector)
  representation
• from experiments (physical, biological,
  mechanical) - attributes
• generated for experiments data files, text,
  images)

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Spatial databases

• Spatial simulation systems (models of natural
  phenomena)
• Specific object representation, query languages,
  data formats.

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DBMS

• Any software system or DBMS

User-visible layer (interface)
Logical layer (internal language, net language)
Physical layer (files, ASCII, )
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Data representation

• Euclidean Space
• Other metrics
• Geometric entities
• Data types

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Geometric space

• Bounded Embedded space within some region
• Unbounded

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Euclidean d-dimensional space

• in d dimensions.
• Euclidean metric L2

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Other metrics

• Minkovski metric
• Manhattan metric
• Supremum metric

L1
L1
L?
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Metric spaces in 2d
L1
L2
L?
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Geometric objects

• point
• segment
• line
• circle
• sphere
• polygon
• Convex, concave
• Simple, Non-simple
• With holes, without holes
• polyhedron

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Polyline

• Polyline
• closed if 2 extreme points are identical
• simple if no pair of edges intersect
• monotone order of projections corresponds to
  the order of vertices

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Polygon

• Polygon is a region bounded by a closed polyline.
• Polygon is simple of its boundary is simple
• Polygon P is convex if for any 2 points A and B
  inside the polygon, the segment AB is fully
  enclosed in P. A non-convex polygon is concave.
• Polygon is monotone if it is simple and its
  bounds can be split into 2 monotone polylines.

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Planar subdivisions

• A straight-line planar embedding of a planar
  graph determines a partitioning of the plane
  called a planar subdivision.
• A planar subdivision is a triangulation if all
  its bounded regions are triangles.
• A triangulation of a finite set of points S is a
  planar graph with maximum number of edges and
  with nodes located at the points of S.

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Geometric data structures

• Regular grid (mesh)
• Grid file
• Quad tree
• k-d tree
• Interval tree
• Voronoi diagram and Delaunay triangulation

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Commonly used data structures

• 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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Algorithmic Strategies

• Transformation
• Incremental
• Divide-and-conquer
• Sweep-line/plane
• Dimension reduction
• Geometric decomposition

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Common Algorithm Techniques

• Transformation
• Problem A you know a lower bound
• Problem B it is unknown.
• If we transform A to B by a transformation step
  that costs less than solving A, then B has the
  same lower bound as A

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Convex Hull problem

• Convex Hull ? Sorting
• Sorting is O(n log n)
• Let x1,x2,,xn be a set of numbers (any order)
• Compute in linear time set of points O(xi,xi2)
• Build the convex hull of these points
• By scanning points from left to right obtain
  sorted array.
• Convex hull is O(n log n)

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Incremental Convex Hull

• Let Sp1,p2,,pn denote the set of points of
  CH.
• First, construct a triangle p1,p2, p3
• Add point pi to CHi-1 (existing)
• Two cases are possible
• pi ? CHi-1 detected in linear time by scanning
  the points of CHi-1 in clockwise order, pi is
  always on the right side
• pi ? CHi-1 add pi to CH in linear time, by
  scanning points of CHi-1, finding 2 tangential
  lines from pi to CH.
• Analysis O(n2)

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Convex Hull optimization

• Algorithm can be improved to O(n lg n) by
  presorting the input according to abscissas of
  points.
• Every time a point pi is added
• No need to check CHi-1, it is always outside
• To find the upper tangent, scan the upper chain,
  removing points that we scanned
• Lower tangent same
• Linear time to construct CH sorting O(nlg n)

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Convex Hull optimization
Algorithm can be improved using the property that
the list of points is sorted. If line through Pi
intersects CH, the vertex is discarded. Each
vertex is discarded only once.
Pi

• Pi-1

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Divide-and-conquer

• The half-plane intersection example result
  convex polygon
• Recursive
• Top-down subdivision to a smaller problem
• Bottom-up recursively merging the solution
• Similar to merge-sort, quick-sort

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Divide-and-conquer

• Create binary tree of half-planes
• Intersect each half-plane, starting from the
  leaves, with rectangle R, which contains the
  resulting polygon (by assumption) ? obtain convex
  polygons
• Compute intersection of convex polygons, bottom
  up on the tree

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Divide-and-conquer
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Divide-and-conquer complexity

• Convex polygon intersection is O(n)
• T(n) 2 T(n/2) cn
• i is of steps ? T(n) 2i T(n/2i)cniT(n)
  2lg n T(n/2lg n) c n lg n
• Assume T(n/2lg n) is O(1), when the size of the
  problem is small
• T(n) O(n lg n)

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Sweep-line (plane-sweep)

• Decompose the input into vertical strips
• Vertical line (sweep-line) allows to
  compute/maintain some information
• Sweep-line goes through a number of events
• Event list is usually represented by the queue
• The state of the sweep-line is also maintained.

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Sweep-line example

• Given a set S of rectangles with sides parallel
  to the axes, report all their pair-wise
  intersections
• O(n2) for each rectangle, test intersections
  with others
• This is worst-case optimal, because there can be
  that many intersections. However, lets try to
  design an output-sensitive algorithm.

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Sweep-line example

• Observations
• Sweep-line undergoes changes in its status when
  it reaches left/right side of a rectangle (along
  X coordinate)
• If X projections intersect does not mean that
  rectangles intersect. When left endpoint reached
  add rectangle.
• When the sweep-line intersects the rectangles, Y
  projections of rectangles should be checked for
  intersection. If they too intersect, then
  rectangles intersect.
• When right endpoint is added remove rectangle.
• All events can be scheduled in advance!

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Sweep line complexity

• Initial sorting of interval endpoints O(n lg n)
• of events in the queue is 2n
• Time to process one event check for Y
  intersections is O(1)
• If the total of intersections is k, then the
  total complexity is O(n lg n k)

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Sweep-plane algorithm

• Algorithm description
• Throw pebbles in the water
• Intersection of waves gives the lines equidistant
  from the point where pebble touched the water
  (i.e. a edge of the Voronoi diagram)
• Add time as 3rd dimension waves transform to
  pyramids.
• Sweep pyramids with the sweep-plane to get the
  Voronoi diagram

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Sweep-plane algorithm

• Properties
• The complexity of the sweep-plane algorithm in
  generalized Manhattan metric is O(n log n).
• Sweep-plane method is not applicable to the power
  diagram construction.

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Sweep-plane algorithm

• An example of a sweep-plane construction of a
  Voronoi diagram in L1 metric

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Sweep-plane algorithm
Cones in L1
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Sweep-plane algorithm
Sweeping the cones
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Sweep-plane algorithm
sweep direction
Parabolas
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Sweep-plane algorithm
Site event
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Sweep-plane algorithm
Site event
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Sweep-plane algorithm
two half-bisectors are created one is growing
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Sweep-plane algorithm
The half-bisector changes its direction
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Sweep-plane algorithm
Site event 2 half-bisectors created
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Sweep-plane algorithm
The half-bisector Changes its direction
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Sweep-plane algorithm
Circle event Triangle added to DT 2
half-bisectors deleted 1 half-bisector created
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Sweep-plane algorithm
Half-bisector changes its direction
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Sweep-plane algorithm
No more events
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Sweep-plane algorithm
Resulting Voronoi diagram and Delaunay
triangulation
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Dimension reduction

• Typical problem build a VD for a set of points
  in the plane.
• Consider 3D, build a paraboloid, lift/project
  points (sites) onto the paraboloid
• Construct half-spaces tangent to paraboloid at
  those points.
• Construct intersection of these half-spaces
• Projection of these intersections down on the
  plane gives the VD of initial sites.

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Applications

• GIS terrain modeling
• Climate control weather patterns
• Environment study of migration, species
• City planning - maps
• Computer graphics filtering, multilevel
  resolution
• Computer vision - CAVE

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Digital Elevation Model (DEM)
Ariel photo (left), triangulated surface in 3D
(right)
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DEM
Terrain reconstructed using Delaunay Triangulation
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ArcView and Geology
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Digital Atmosphere 2000
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DA2000 Digital Satellite Photos
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AutoDesk Map 2002
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City planning

• This focuses on visualizing man-made structures
  such as roads.
• Used in a variety of areas such as urban
  development, residential planning, and GPS
  Navigation.
• Courtesy of http//ca.maps.yahoo.com/

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Multiresolution images

• Traditionally stacks of images
• 2n by 2n, 2n-1 by 2n-1, , 2 by 2, 1 by 1
• Does not have to be image data

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Simple Pyramid Example

• Lena, the typical example

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Divide and Conquer

• Creating a pyramid is a variation of the
  principle of divide and conquer
• Any point at a certain level is calculated from a
  small number of point on the previous level.

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Image Compression

• Usage Smaller images, especially useful for
  satellite or airphotos in GIS.
• Example .ecw (enchanced compressed wavelet)

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GIS LOD (Level of Detail)

• Usage A detail-in-context technique for GIS.
• Example The TerraVisionTM System

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Working Towards the Future

• Terrain model a CAVE Perspective _at_ the
  University of Illinois Urbana, Champagne