1
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Lecture 6 – Geo-Spatial Data VisualizationOctober 6, 2010Georges Grinstein, University of Massachusetts Lowell
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Introduction to VisualizationTechniques for Spatial Data
Spatial Visualizations
Maps, GIS and Applications
What is 2D Spatial Data?
Data tied to a location in the real world
Sources:
Maps
Aerial Photographs
Field survey notes
GPS data
GIS
Satellite Systems
We Look for Data/Spatial Relationships
But in many applications it is important toseek relationships that involve geographiclocation
Examples
Telephone calls
Environmental records
Crime Data
Census demographics
...
Seeking Spatial RelationshipsExample (1)
Public Health and Safety Analysis
Customer
Analysis
Visualization
of AT&T long
distance call
volume
(AT&T
Swift3D)
Seeking Spatial RelationshipsExample (2)
2
Basic Definition
Result of accumulating
samples or readings of
some phenomena while
moving along 2d paths
in space
These are discrete samplesof a continuous phenomenon
Bodensee -
original topographic data
resolution 1:500'000
(Visual Spatial Data 2002)
Formalization (1)
A 2-dimensional spatial data item
is an ordered d-tuple of the form
with
(1)
(2) Without loss of generality we set
, the first 2 values, to be the 2d vector
representing the spatial dimensions
),(21
iixx
is a d-dimensional record
DBxi
( )d
iiixxx ,...,
1=
,...1 di
DDx
Formalization (2)
(3)1
1
vxi= :
2
2
vxi=
2
21 ),(;, RvvR
0:,221121=+ vvR
Then 1 = 2 = 0
in other words v1 and v2 are linearly independent
Data Analysis Techniques (1)
Example
Sales Transaction Application contains data about
Customers
Products
Quantities
Time
…
Data Analysis Techniques (2)
Many ways to approach analysis of the data,
including
Building statistical models
Clustering
Finding association rules
...
Earthquake
hazard
assessment
• Turkey
• Icon
Seeking Spatial RelationshipsExample (3)
3
2D Spatial Data Mining (1)
Goal
Extracting interesting knowledge or general
characteristics from large 2D spatial (location)
databases
Important task in the development of spatial
database systems
2D Spatial Data Mining (2)
Problem:
It is almost impossible for a user to examine the
huge amounts (usually terabytes) of spatial data
obtained from large databases (credit cards
payments, telephone calls, environmental
records,...) in detail and extract interesting
knowledge or general characteristics
WITHOUT SUPPORT
computational
human participation/performance
2D Spatial Data Mining (3) 2D Spatial Data Mining (4)
Key observation
The presentation of data in an interactive,
graphical form often fosters new insights,
encouraging the formation and validation of new
hypotheses to help in better problem-solving and
gaining deeper domain knowledge
That is the purpose of visualization
2D Spatial Data Mining (5) What are Maps? (1)
Definition of a map
A set of points, lines, and areas defined both by
1. position reference in a coordinate system
(spatial attributes) and
2. by their non-spatial attributes
From the U.S. Geological Survey (USGS))
4
What are Maps? (2)
Maps are the world reduced to points, lines,
and areas, using a variety of visual resources
size, shape, value, texture or pattern, color,
orientation, and shape
A thin line may mean something different from
a thick one, and similarly, red lines from blue
ones
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Symbolizing 2D Spatial Data
Spatial Dimension (1)
Point Phenomena (zero dimensional)
No spatial extent
Definition: (x, y) – (longitude, latitude)
z – statistical value (data value)
Examples
Locations of religious worship
Oil wells
Locations of nesting sites for eagles
Census Demographics
…
Spatial Dimension (2)
Linear Phenomena (one dimensional)
One dimensional in spatial extent
Have length, but essentially no width
Definition: unclosed series of (x, y) for each
phenomena
Examples:
Boundaries between countries
Path of a stunt plane during an air show
…
Spatial Dimension (3)
Area Phenomena (two dimensional)
Two dimensional in spatial extent
Have both length and width
Definition: series of (x, y)-coordinates that
completely enclose a region
with a statistical value for each phenomena
Examples:
Lakes
Political Units (States,…)
…
Spatial Dimension (4)
2 D Phenomena
Each point is defined by longitude, latitude, and an
associated value above a zero point
Examples
Surfaces
…
5
Smooth Statistical Surface Spatial Dimension (5)
True 3D Phenomena (most common)
Each point is defined by longitude, latitude, and
multiple associated values
Examples:
Longitude, Latitude, Height above sea level, CO2
Concentration, …
3D Model
3D Model of an
open-pit coal
mining site
Source:
Pennsylvania
Department of
Environmental
Protection
How are Maps used? (1)
Three ways
to provide specific information about particular
locations
to provide general information about spatial patterns
to compare patterns in one, two or more maps
How are Maps used? (2)
(1) to provide specific information about
particular locations
Approximately
2 millions
slaves were
transported
from Africa to
Spanish
America
between 1700
and 1870
How are Maps used? (3)
(2) to provide general information about
spatial patterns
Low percentage of
inhabitants voted for
Perot in the
southeastern part of the
United States
High percentage voted
for Perot in the central
and northwestern states
6
How are Maps used? (4)
(3) to compare spatial patterns in one or two
mapsSpatial patterns of the two
maps are quite different
Distribution for Corn:
concentrated in the
traditional corn belt region
of Midwest
Distribution for Wheat:
Concentrated on the Great
Plains
Spatial Visualization and Cartography (1)
How should the term “visualization“ be used in
the field of spatial data?
First Definition MacEachren et. al. (1992)
“Spatial (Geographic) Visualization will be defined
as the use of concrete visual representations to
make spatial contexts and problems visible, so as
to engage the most powerful human-processing
abilities, those associated with vision“
Spatial Visualization and Cartography (2)
Second Definition MacEachren et. al. (1994):
“Cartography-Cube Representation“
Visualization:
Private activity in which
unknowns are revealed in
highly interactive
environments
vs.
Communication:
Public activity in which
knowns are presented in non-
interactive environments
GIS – Relation to other spatial informationsystems
GIS – Structure and Relation toVisualization
Example: Repair Service
7
Organizing Spatial Data
• Reality
• Field based
• Object based
• Digital Landscape Model
• Raster Model
• Vector Model
• Digital Cartographic Model
• Raster Structure
• Vector Structure
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Visual Variables for Spatial Data
Visual Variables for Spatial Visualization
Spacing (Texture)
Changes in the distance between the symbols
Size
Change size of the entire symbols
point, linear phenomena
Change size of individual symbols
aerial, 2 1/2D, true 3D phenomena
Visual Variables for Spatial Visualization
Perspective Height
Refers to the perspective three-dimensional view ofthe phenomena
Cannot be used for true 3D phenomena becausethree dimensions are needed to locate thephenomena being mapped
Orientation
Refers for linear, aerial, true 3D phenomena tovarious directions of individual symbols
Refers for point phenomena to the direction of theentire symbol
Visual Variables for Spatial Visualization
Arrangement
for aerial and true 3D phenomena refers to how the
symbols are distributed (square pattern, randomly
distributed symbols, …)
for linear phenomena refers to how lines are broken
into a series of dots and dashes
for point phenomena refers to changing the position
of the white marker within the black symbol
Visual Variables for Spatial Visualization
Visual variables
for
black-and-white
maps matching
Bertin‘s original
seven visual
variables
8
Visual Variables for Black and White MapsVisual Variables for Color Maps
Choices during the visualization process Spatial and Non-spatial Information
Three
dimensional
space-time cube
in which time is
treated as the Z-
axis
Discrete and Continuous
Depictions of discrete and continuous phenomena
Absolute and Relative
9
Discrete vs. Continuous Data ModelSmooth vs. Abrupt Data Model
Discrete:
Presumed to occur
at distinct locations
Continuous:
Region of interest
Smooth:
Change in gradual
fashion
Abrupt:
Change suddenly
Discrete vs. Continuous VisualizationSmooth vs. Abrupt Visualization
Syntax of map
forms
(How Maps Work by
Alan MacEachren)
Subdivision
of map types
based on
measurement
scale,
graphical
variables and
continuity of
the data
(Cartography –
Visualization of
spatial data by
Kraak & Ormeling)
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Map Projections
Map Scales Comparison of Globe and Flat Map
Simple Way: Map earth
without distortion on a
globe
Disadvantages:
expensive to make
difficult to reproduce
cumbersome to handle
awkward to store
difficult to measure
difficult to draw
10
Choice of Map Projections (1)
Number of possible projections are unlimited
In practice
several projections combine useful characteristics
Geographer, Historian and Ecologist
more concerned with sizes of areas
Navigator, Meteorologist, Astronaut, Engineer
more concerned with angles and distances
Choice of Map Projections (2)
Problem: no specific method can be given thatwill lead to the right selection of the projection
General generalizations
Spherical Geometry Symmetry and Deformationcharacteristics (atlas maker)
Equivalence, Conformality and Azimuthally(temperature distribution over large areas)
Overall Shape
Projection Classification
Widely used theoretical surfaces on which the earth‘s surface
can be projected
Major Projections
Azimuthal Projections
Cylindrical Projections
Conical Projections
Other Projections
Equivalent Projections
World Projections
Interruption and Condensing
Conformal Projections
Equal Area Projections
Used for presentations
attempts to give a correct visual impression of the
relative sizes
Two important factors
Size of the involved area
Distribution of the angular deformation
Azimuthal Projections
Azimuthal Projections
11
Azimuthal Projections
Theoretical positions of the points for the class of azimuthal
projections: (1) gnomonic, (2) stereographic, (3) equidistant, (4)
equivalent, (5) orthographic
Azimuthal Projections
Nearly same areas: left France, right
Madagascar
Using non-equal area projections France
appears larger than it should in comparison
to Madagascar
Azimuthal patterns of distortion:
Contour shows the arrangement of
relative values
Spacing of the contours indicates the
gradients or rates of change
Example: Albers ProjectionU
sed u
nd
er perm
ission
from
F. M
ansm
ann
Cylindrical Projections
Example: Lambert Projection
Used
un
der p
ermissio
n fro
m F
. Man
sman
n
Conical Projections
12
Example: Conical equidistant projection
Source: Hans Havlicek
Other Projection Systems
Equal Area Projection
Azimuthal
equal-area
projection
Source: Hans Havlicek
Aitoff-Hammer Projection
Used
un
der p
ermissio
n fro
m F
. Man
sman
n
Mollweide Projection
Used
un
der p
ermissio
n fro
m F
. Man
sman
n
Cosinusodial Projection
Used
un
der p
ermissio
n fro
m F
. Man
sman
n
13
Some Comparisons Some Comparisons
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Visualization Strategies
Visualization Strategy
Map the spatial (location) attributes of data
with two spatial dimensions directly to the
spatial attributes of the screen
The result will be some of the following
visualizations
Visualization Strategy
(1) Dot Map
if the data contains point phenomena(zero dimensional)
Then point objects (road or stream) arerepresented as pixels on the screen
Dot Map
A simple dot map of commercial wireless antennas in the USA
14
Map of the United
States shows the
total dissolved
solid (TDS)
content of water
from oil and gas
wells in the USGS
produced-water
database
(Source:
US Department of
the Interior)
Dot Map Visualization Strategy
(2) Network/Transportation Map
if the data contains linear phenomena (onedimensional), as well as point objects
Then linear phenomena (roads or streams)are represented as a sequence ofconnected coordinates, which are plottedas a series of line segments
WorldCom Networks
Network/Transportation Map Network/Transportation Map
Lower Manhattan Subway Map
Visualization Strategy
(3) Thematic Map
if the data contains aerial phenomena (two
dimensional), as well as point objects
Then area features (lake or political
boundaries) are generally represented as a
closed contour, a set of coordinates where the
first and last points are the same
Thematic Map
15
Thematic Map Visualization Strategy
(3) Choropleth Maps
Greeks words: Choro = area
pleth = value
Similar to thematic maps
Uses enumeration units to represent different
magnitudes of a variable
Choropleth Maps Visualization Strategy
(4) Isovalue Map
if the data contains 2 D phenomena
Then boundary information is extracted from
an image and depicts a continuous
phenomena, such as elevation or temperature
Iso-Value Map
Monthly
average of ozone
during the
spring warming
left: Southern
Hemisphere
right: Northern
Hemisphere
(Source:
IBM T.J.
Watson
Research
Center)
Visualization Strategy
(5) Smooth Statistical Surface
if the data contains 2 D phenomena
Then statistical values are mapped to height
above sea level (or some other zero
boundary)
16
Smooth Statistical Surface Visualization Strategy
(6) 3D Model
if the data contains true 3D phenomena
Then boundary information is extractedfrom an image and depicts a continuousphenomena, such as elevation ortemperature
3D Model
3D Model of an
open-pit coal
mining site
(Source:
Pennsylvania
Department of
Environmental
Protection)
Transformation
possibilities
among maps
(Cartography –
Visualization of spatial
data, Kraak & Ormeling)
http://www.fao.org/docrep/003/T0446E/T0446E06.htm
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Part III: GIS - ComputationalApproach
GIS – Computational Approach
• A computationalapproach tovisualize,manipulate, analyze,and display spatialdata to study theworld
• “Smart Maps”linking a database tothe map, creatingdynamic displays
17
GIS – Relation to other spatialinformation systems
GIS – Structure and Relation toVisualization
Example: Repair Service Example: Environmental DiseaseAnalysis for industrial locations
Organizing Spatial Data
• Reality
Field based
Object based
• Digital Landscape Model
Raster Model
Vector Model
• Digital Cartographic Model
Raster Structure
Vector Structure
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Models of Spatial Data
18
Reality
Field Objects
Collection of
spatial
distributions
Geo-Objects
Discrete geo-
referenced
entities
Field-Objects
Grid of Rows and Columns: Tessellated
Basic data unit is the cell (spatial)
(x,y) implicit
Entity information must be explicitly encoded
Geo-Objects
Geometric Data Objects (Geo-objects) are
related to a spatial reference point in a
coordinate system
Geo-objects are at least two-dimensional
Geo-objects in general have other spatial and
non-spatial attributes (height information,
name, etc)
X
Y
2-dim. Points 2-dim. Polygons
Wheat
Barley
Sugar beets
Geo-Objects
Geo-Objects
Attribute classes
Spatial Attributes
(x,y)-Coordinates / Spatial Data
Extent of a polygon
Area of the polygon
Topological Attributes
Polygon Pol1 is a neighbor ofpolygon Pol2
Polygon Pol1 and Pol2 have acommon edge
Topic Tables Attributes
In polygon Pol1 barley iscultivated
At the point P2 is the post office78457
K1
Pol1
Pol2
Pol3
X
Y
P1
P2
Digital Landscape Model
Vector Model
Geo-objects aredescribed by theirborders
Border is defined by a setof points
Raster Model
Geo-objects aredescribed by their innerdefinition
Inside is defined as ancollection of pixels fromwithin a grid
P1
P2
P3
P4
P5
(P1, P2, P3, P4, P5)
19
Reality Raster Model Raster Resolution Problem
Advantages
Easy to conceptualize
Overlay operations are easy
Represents a two-dimensional array (easy to
implement)
Raster Resolution Problem
Small Grid: Coarse Resolution but limited storage
space
Large Grid: Fine Resolution but large storage space
Raster Resolution Problem Raster vs. Vector
Raster vs. VectorInstitute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Visualization of Point DataDot maps
20
Dot Map
A simple dot map of commercial wireless antennas in the USA
Map of the United
States shows the
total dissolved
solid (TDS)
content of water
from oil and gas
wells in the USGS
produced-water
database
(Source:
US Department of
the Interior)
Dot Map
Highly non-uniformly distribution
Input: (longitude,lattitude,s) –
geographic location and statistical
value
Observation: spatial data are highly
non-uniformly distributed in real world
data sets
Example: State New York Median
Household Income Year 2000
Database (1% of the data)
8.3 Visualization of Point Data 8.3.1 Dot Maps
Problem Definition
Visualization strategy:
Map the spatial (location) attributes of data with two spatialdimensions directly to the spatial attributes of the screen.
Mapping
to colored
pixels
8.3 Visualization of Point Data 8.3.1 Dot Maps
Visual Exploration Goals
• No Overlap “normal” screen resolutions
Provide effective
visualizations
No lack of information
Provide visualization with
general geo-spatial
relationships
Make interesting geo-
patterns visible
Achieve Pixel-Coherence
Make small clusters visible
• Clustering
• Position Preservation
8.3 Visualization of Point Data 8.3.1 Dot Maps
Simple Mapping – Dot Map
• High degree of overlap in the highly
populated areas
• Low populated areas are virtually empty
8.3 Visualization of Point Data 8.3.1 Dot Maps
21
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Visualization of Point DataPixel maps
PixelMaps
KDE DetermineKernelDenEst3D(D)
Smoothness of the kernel function needs to vary depending on the
density in the x-y dimension
Kernel function also has to be different for the x-y and third dimension
P DeterminePeaks(KDE)
Determine the peaks at a given point required to make a scan over the
database
C DetermineClusters(P)
Hill-Climbing, ...
For our evaluation we implemented a PixelMap-Algorithm
based on the DenClue-Clustering Algorithm
Only usable for a subset of the data (<50000 ~ 30h)
An Efficient Implementation
Basic Idea: Rescaling of the map
regions to fit the 3D point clouds on the
screen better
Fast-PixelMap: heuristic to approximate
the result of the PixelMap algorithm
Combines the advantages of gridfile and
quadtree
Density-based Distortion
Basic Idea:
Rescaling of the
map regions to fit
better the 3D
points clouds on
the screen
Data Structure: Combination of the gridfile and quadtree
KDE-Display for USA Rescale Operations Series
22
Array based Clustering
Constant number of classes NoOfClasses
Partioning the third dimensions in NoOfClasses
intervals beginning with minimal value
Endpoints of each interval are stored in an arrayBinary search to finding the interval for each data point
Pixel Placement
New York Census Year 2000
Traditional
Dot Map
PixelMap
showing local
pattern
New York Census Year 2000
Dot Map of
the USA
US Census Year 2000 Analysis
PixelMap of
the USA
US Census Year 2000 Analysis
23
US Census ExamplesInstitute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Visualization of Point DataOther distortion techniques
Other Distortion Techniques
Basic Idea:
Rescaling of the
map regions to fit
better the 3D
points clouds on
the screen
Traditional map of the U.S. The data set is U.S. Year 2000 Median Household Income.
This scatter-plot map shows artifacts, such as spiral effects, especially in dense areas
such as Los Angeles County,Cook County and Manhattan.
Other Distortion Techniques
HistoScale
HistoScale using the U.S. Year 2000 Median Household Income dataset.
The problem is the non-suffient distortion especially for sparse areas which are in the
same bin with highly dense areas, such as the areas north of Los Angeles.
Other Distortion Techniques
HistoScale RadialScale AngularScale
US median household income shown by different distortion techniques.
Advantages and disadvantages of the different techniques are observable e.g. Florida,
Atlanta, Chicago, and the coastal areas, when an analysis of income distribution is
conducted or high/low income-areas are compared.
Other Distortion Techniques
Generalized Scatter Plots
Generalized Scatter Plots of Census Data.The color map indicates the household
income (green: low; blue: medium, and red: high):
(1) Traditional Scatter Plot without distortion and data-induced overlap
(2) 50% distortion and 50% overlap
(3) 90% distortion and 5% overlap
24
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Visualization of Line DataNetwork maps
Network Map
if the data contains linear phenomena
(one dimensional), as well as point objects
Linear phenomena (road or stream) are represented as a
sequence of connected coordinates, which are plotted as a
series of line segments
WorldCom Networks
Network Map Network Map
Lower
Manhattan
Subway
Map
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Visualization of Line DataFlow maps
Flow Maps
(a) Minard’s 1864 flow map of wine exports from France
(b) Tobler’s computer generated flow map of migration from California from
1995 - 2000.
(c) A flow map produced by our system that shows the same migration data.
Doantam Phan, Ling Xiao, Ron Yeh, Pat Hanrahan, and Terry Winograd, Flow Map Layout,Proceedings of the IEEE Symposium on Information Visualization 2005 (INFOVIS’05)
25
Flow Maps
Input to the Process
Enforce minimum separation distance among the
nodes while preserving their relative positions to
one another.
Agglomerative hierarchical clustering
Modify the produced tree to make a particular node
(the source of the flow) the root of the tree.
Recursively lay out the tree of flows rooted at the
source in a way that edge crossings can be
minimized.
Lay out the edges. Make sure they do not intersect
nodes.
Render the whole map.
Layout Adjustment
minimum separation distance among the nodes in the horizontal and vertical
directions
= maximum width of a flow line
Algorithm: Misue et al.‘s force scan algorithm
Advantage: stable layout (no randomness)
For further details on the algorithm see:Kazuo Misue, Peter Eades, Wei Lai, Kozo Sugiyama. Layout adjustment and the Mental Map.
Journal of Visual Languages and Computing. 1995
Primary Clustering
Spatial representation of the primary hierarchical clustering (PHC) andits equivalent tree.
Flow Maps
Rooted hierarchicalclustering (RHC) modifies
the PHC to produce aflow map for
a particular root.
(b) The RHC for a flowmap from C. The (A,B)
and the ((D,E),F) clustersare kept.
(c) The RHC for a flowmap from D. Only the
(A,B) cluster is preserved.
Spatial Layout
The binary structure of the rooted clustering allows to generate the layout recursively.
Branching points are always placed on the line between the start node and thedestination that has more weight (or flow).
Edge Routing
Spatial layout may
cause anintersection
by placing b3 in away that intersects
c1. The algorithmfinds the
intersection of b1-b3 with c1, and
adds a new nodeand adjusts the
position of b3 to
avoid c1 ifnecessary.
26
Examples
Outgoing migration map from Colorado from 1995-
2000, generated without layout adjustment or edge
routing.
Outgoing migration map from Colorado for 1995-
2000 generated using edge routing but no layout
adjustment.
Examples
Outgoing migration map
from Colorado for 1995-
2000 generated using edge
routing and layout
adjustment.
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Visualization of Area DataThematic maps
if the data contains areal phenomena
(two dimensional), as well as point
objects
Area features (lake or political boundary) are generally
represented as a closed contour, a set of coordinates
where the first and last points are the same
Thematic Map
Thematic Map Thematic Map
27
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Visualization of Area DataChloropleth maps
Greeks words: Choro = area
pleth = value
thematic maps that use enumeration units to
represent different magnitudes of a variable
Choropleth Maps
Choropleth MapsInstitute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Visualization of Area DataCartograms
thematic maps that try to represent a statistical
value by the area of the polygons
Cartograms
28
Cartograms
Cartogram Solvability
Given:
Goal:
,
Checker Board Example Impossible Cartogram Example
29
Continuous Cartogram DrawingProblem
The contiguous Cartogram problem may
be defined as a transformed set of
polygons for which
(Area) (Shape) (Global Shape)
CartoDraw Algorithm
Reduction Algorithm
Cartogram Construction Algorithm
Manual Placement
Automatic Placement
Cartogram Construction Algorithm
Basic IdeasIncremental Repositioning of Vertices
Scanlines Define Path of Action
Automatic Scanlines versus Manual Scanlines
Algorithm Controlled by Area and Shape Error
Shape Error Defined by Fourier Transform
Area Error
Shape Error
1. Parameterized Representation
of the Polygons
2. Normalized Curvature
3. Compute Fourier-Coefficients
4. Compute Euclidean Distance of Fourier-Coefficients
8.5.3 Cartograms
Cartogram Construction Algorithm
30
Cartogram Construction Algorithm Cartogram Construction Algorithm
Cartogram Construction Algorithm Cartogram Construction Algorithm
Cartogram Construction Algorithm Cartogram Construction Algorithm
31
Cartogram Construction Algorithm Cartogram Construction Algorithm
Cartogram Construction Algorithm Cartogram Construction Algorithm
Cartogram Construction Algorithm Algorithm
32
Automatic and Manual Scanlines Using Medial Axes as Skeleton
Area vs ShapeInstitute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Visualization of Area DataRecMaps
Circular Map Approxiamation
Daniel Dorling. Area Cartograms: Their Use and Creation. Department
of Geography, University of Bristol, England, 1st edition, 1996.
Rectangular Map Approximations
Erwin Raisz. Principles of Cartography. McGraw-Hill, New York,1962.
33
Problem Definition
Objective function: Area Objective function: Shape
Objective function: Topology
The Figures show the pseudo dual of the U.S. map (left) and the pseudo dual
of the map partition (right). The red colored segments demonstrate the
topology error as defined in equation 3.
Objective function: Relative polygonpositions
34
Objective function: Empty space Formulation of the optimization problem
Variant 1: No Empty Space Variant 2: Preserve aspect ratio
User control over the weighting functionof MP2
Cartograms P resulting from different weights for the components
of f for Problem Variant MP2
Results of RecMap (Variant 2) using
US-Census population data on U.S. state
Results of RecMap (Variant 1) using US-
Census population data on U.S. state
Examples for MP1 and MP2
35
Basics of Genetic Algorithm
Population I is a set of individualsThe vth population is also called vthgenerationAn individual is characterized by threeaspects
The genotype
A construction algorithm
The phenotype
Basics of Genetic Algorithm(con‘t)
Three aspects of individual
Basics of Genetic Algorithm(con‘t)
Genetic Algorithms are based on three
components:
Selection (survival of the fittest)
Replication (combination of father and mother
individuals)
Mutation (alter its genotype and hence its
phenotype)
Basics of Genetic Algorithm(con‘t)
Creation of the next generation within three steps (with m = 6)
Genetic algorithm for Rectangular MapApproximation
Compute a candidate
cartogram
The RecMap Construction Heuristics TwoSolutions
Quadtree-based
Layout-based
36
Quadtree-Algorithm QT-Algorithm – Important Observation
The splits do not preservethe aspect ratio of thepolygons andneighborhood of thepolygons
Solution:Using different splitsequences which can beevaluated by the objectivefunctions
The QT-algorithm applied to the U.S.
using population data
The Quadtree-based heuristic
Adapted construction algorithm of
Quadtree-based heuristic
Construction algorithm of quadtree-based
heuristic
Example for the U.S. Population Data
Layout-based construction
Basic Idea:Initial Step:
Finding the core polygon pc
Main Step:
Construct a sequence of partial layouts or partialcartograms starting with pc
R-1 polygons are placed around pc one after theother until we found a (complete) cartogram
Extended adjacency graph
Insert an additional
node R+1 for the outer
region
pc equals to a polygon
which has the maximal
BFS number running
from node R+1
37
The Layout-based constructionheuristics Example for the U.S. Population Data
Efficiency
The scatterplots display the errors over time for the U.S.-state partition where both
algorithm variants were used. Note that the time axes were logarithmic scaled. The
whole computation time after 10 iteration took 0.33 seconds for MP1 60 seconds
for MP2.
Conclusion
Fully automatic (+)
Scalable (+)
explicit control of all visualization constrains: (+)
no area error
explicit control of shape
topology
empty space
relative position
Computational Complexity (O(n)*number of iterations)
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Spatial Data Types
Spatial Data Types
.
What should be modeled?
Point LineArea
Partition Network
38
.
..
.
...
..
.
..
. ..
... ..
.
..
.
...
..
.
..
. ..
... ..
Euclidean Plane Issues (very general)
Point p=(r1,r2); ri are real
numbers
The Intersection Problem
The coordinates of the
intersection point are
rounded to the nearest
computed precision
Computers cannot represent
arbitrary real numbers
build a grid of possible
points
Intersection point may
not be on one of the lines
Let a finite discrete space with
N-Realm is a set R with
P PN (R-Points), S SN (R-Segments)
" s S : s = (p,q) p P q P
" p P " s S : ¬ ( p in s)
" s, t S, s t : ¬ ( s and t intersect)
NN
{ }1,...,0= nN
A
SPR =
Spatial Data Types - Realms
Realm Structures
R-cycle
R-face
R-faceR-block
• The following are based on an interpretation of a Realm as graph
An R-cycle is a loop in the graph of R
An R-face is an R-cycle, which includes additional disjunctive R-
cycles
An R-block is a connected component in the graph of R
Realm-based Spatial Data Types
. .. . .. . . ... ..
..
.
..
..
.
..
. ..
... ..
.
..
.
.
.. ... ..
... ..
.
..
.
.
.. ... ..
... ....
... ..
. ..
... ..
.
..
.
.....
...
. ..
... ..
.
..
.
...
..
.
..
. ..
... ..
Elements of the point type
Set of R-points
Elements of the line type
Set of different R-blocks
Elements of the area type
Set of different R-faces
.. .
. .
.. ..
Spatial Data Types
Types:
EXT = {lines, regions}
GEO = {points, lines, regions}
OBJ = {cities, highways, . . .}
(depends on the application scenario)
set(OBJ) is a spatial database
Second Order Signature
Spatial Data Types
Topological predicates:
39
Spatial Data Types
Important Queries:
Spatial Data Management Models
Hierarchical Database Structure
Spatial Data Management Models
Network Database Structure
Spatial Data Management Models
Relational Database Structure
Spatial Data Types
Queries using relational algebra:
(1) select cities [center inside Bavaria]
(2) select rivers [route intersects Window]
(3) select cities [dist(center,Hagen) < 100 and
population > 500,000]
(4) join cities states [center inside area]
(5) join cities rivers [dist(center,route) < 50]
Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Signs and Labeling
40
Ucar‘s typology of map signs
Diagrammtitel
written words
dimensions of the plane
symbols
figure icons plan icons
icons
artificial sign with representation function
non-linguistic signs
map sign
(How Maps Work by Alan MacEachren)
Categories of map sign aspects
designate appraise
indicate label
relate
apprise
prescribe emotive
connote aesthetic
stimulate
Sign Aspects
(How Maps Work by Alan MacEachren)
Types of Map Signs
Pictorial,
associative,
geometric
nominal point
symbols
Types of Map Signs
Example of the mimetic to an arbitrary continuum
of map marks for a city
Nyerges‘s meaning triangle Nyerges‘s meaning triangle
41
Generalization Geometric Generalization
Conceptual Generalization Effect of parameter binning and boundariesorShould we believe what we see?
Different sets of bin boundaries applied to the same data yield
different-looking choropleth maps
Should we believe what we see? Should we believe what we see?
42
Should we believe what we see?Institute for Visualization and Perception Research
information visualized
knowledge discovereddecisions made
Spatial Index Structures
Spatial Index Structures
Spatial index structure
(MUR, , ...) (MUR, , ...) ......
R-Tree
Quadtree
(EB)
• Normal index structures are not usable for organizing spatial data
• So organize simplified approximations to the geo-objects
- Minimal Bounding Rectangle (MBR)
- Pointer to the detailed description (EB) of the geo-object
P
R
Point Query Window Query
Spatial Index Structures
• The following types of queries must be efficiently supported
- Given a query point P or a query rectangle R
- Point Query: Find the Geo-Objects Obj: P (Obj.MBR)
- Window Query: Find the Geo-Objects Obj: R (Obj.MUR)
Spatial Index Structures
• Given two sets with minimal bounding rectangles
M1 = {MUR1,1, MUR1,2, …, MUR 1,m} and M2 = {MUR2,1, MUR2,2, …, MUR2,n}
• Spatial Join:
{ ( MUR1, MUR2 ) | MUR1 M1, MUR2 M2 and MUR1 MUR2 }
B1
A2
A3
A4A5
A6
A1
B2
B3
Result Set:
(A5, B1)(A4, B1)
(A1, B2)
(A6, B2)(A2, B3)
Spatial-Join
Spatial Index Structures
+ =
Ground load Disease numbers Combination
Support
through
Spatial Join
a1
a3
a4
a2
b3
b2b1
(a1, b1)
(a4, b2)
(a2, b2)
43
Principles of Spatial Data Organization
X
A5A1
A4A3
A6A2
Y
A2
A3
A4A5
A6
A1
3 Classes (Pages):
• Partitioning the minimal bounding rectangles in disjunctive
classes
• Each class is stored exclusively in a page
R-Tree
Principles of Spatial Data Organization
• Median-Split operation in each dimension (x/y-median splits)
• Points = Pixels in a discrete data space
00 00 00 00
00 01 00 00
00 00 01 0000 00 01 10
00 00 00 00
10 00 00 01
00 00 00 0101 00 01 00
NW NE
SW SE
Principles of Spatial Data Organization
NW
NE SW
SE
Quadtree
00 00 00 00
00 01 00 00
00 00 01 00
00 00 01 10
00 00 00 00
10 00 00 01
00 00 00 01
01 00 01 00
NW NE
SW SE
Point Query supported by R-Tree
A5A1A4A3A6A2
RST
Point Query
X
Y
A2
A3
A4A5
A6
A1
R S
T
.
Result Set:
Paths to be observed
[]
PointQuery (Page, Point);
FOR ALL Entries Page DO
IF Point IN Entry.Rectangle THEN
IF Page = DataPage THEN
Write (Entry.Rectangle)
ELSE
PointQuery (Entry.Subtree^, Point);
Window Query supported by R-Tree
A5A1A4A3A6A2
RSTWindow Query
X
Y
A2
A3
A4A5
A6
A1
R
S
TResult Set:
[A1, A2]
Paths to be observed
Window Query (Page, Window);
FOR ALL Entries Page DO
IF Window INTERSECTS Entry.Rectangle THEN
IF Page = DataPage THEN
Write (Entry.Rectangle)
ELSE
WindowQuery (Entry.Subtree^, Window);
Literature
Jaccques Bertin, Graphische Darstellungen und die graphische Weiterverarbeitungder Information, 1982, Walter de Gruyter, Chap. B5
Leland Wilkinson, The Grammar of Graphics, Springer, Chap. 10
D. A. Keim, S. C. North, C. Panse: CartoDraw: A Fast Algorithm for GeneratingContiguous Cartograms, IEEE Transactions on Visualization and Computer Graphics(TVCG), Vol. 10, No. 1, pp. 95-110, 2004.
D. A. Keim, S. C. North, C. Panse, M. Sips: Visual Data Mining in Large Geo-SpatialPoint Sets, Special Issue: Visual Analytics, IEEE Computer Graphics and Applications(CG&A), September-October 2004, pp. 36-44, IEEE Press, September, 2004.
D. A. Keim, S. C. North, C. Panse, M. Sips: Pixel based Visual Mining of Geo-SpatialData, Computers & Graphics (CAG), Vol. 28, No. 3, pp. 327-344, Elsevier Science,June, 2004.
Natalia Andrienko, Gennady Andrienko, Exploratory analysis of spatial and temporaldata: A systematic approach, 2006, Springer
J. Dykes, A.M. MacEachren, M.-J. Kraak, Exploring geovisualization, 2005, Elsevier