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Effective Density Visualization of Multiple
Overlapping Axis-aligned Objects
York University, Toronto, Canada
MSc. Thesis of Niloy Eric Costa
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Background
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Density-based visualization
2012 US Election
Activity Map
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Many data analytics problems
need to visualize the density of
axis-aligned objects
Observation
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Axis-aligned geometric objects
1-D line segments/intervals 2-D rectangles
3-D boxes/cuboids
Need for effective density
visualization of multiple
overlapping axis-aligned
objects
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1. How to detect multiple overlaps?
i. How many overlapping elements?
ii. Which rectangles are overlapping?
iii. Size of the overlaps?
Research questions
2. How to evaluate the efficiency of the methods?
3. What are the real-world use cases for these
methods?
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Object intersection problem
A
BC
(A,B)(B,C)
Inputa set of axis-aligned geometric objects
Outputpairs of intersecting objects
size of overlap
how can we address this problem?
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Sweep-line algorithm
L
A
Ax0 x0B x0
C x1B Ax1 x1
C
BC
Sweep direction
Ay0
By0
Cy0
By1
Ay1
Cy1
an efficient one pass computational geometry algorithm
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Multiple Object Intersection Problem
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The problem
Inputa set of regions in R2
Outputenumeration of all intersecting regions
size of each common region
position of each common region
(A,B)(A,C)(A,D)(B,C)(B,D)(C,D)(D,E)(A,B,C)(A,B,D)(A,C,D)(B,C,D)(A,B,C,D)
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Many applications
simulationsspatial databases
task scheduling
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Baseline Methods
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Baseline 1: naive algorithm
iteratively check all possible ways that n objects can intersect
(-) limitation
there are 2n ways, so exponential computational cost
Baseline 2: grid-based approach
create a grid, perform orthogonal queries to find objects intersecting
with each grid cells, assign value based on intersections
(-) limitation
trade-off between accuracy and time-performance based on grid-cell
sizes
Sensible baseline algorithms
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1. Use R-tree* to create a grid
2. Search in the tree for finding z-index scores
3. Color each grid-cells based on the corresponding z-
index scores
Grid-based approach
Input data-set 1. 4 X 4 grid 2. z-index scores of
cells
3. 4 X 4 grid heat-
map
*R-tree is a depth balanced tree, provides aid in
faster spatial queries
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Grid-based approach trade-off
Trade-off
• 4 X 4 grid is less
accurate, but z-indexes
calculated quickly
• 8 X 8 grid is more
accurate, took longer to
calculate each z-index
score
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Our Approach (OverLap-HeatMap)
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Observation 1
intersection graph:
⦁ vertex: represents an object
⦁ edge: represents that two objects
intersect
intersections of n-dimensional objects (1-D, 2-D, 3-D, …) can be
universally modeled as an intersection graph
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a k-clique in the intersection graph, corresponds to k objects that
are simultaneously intersecting and share a common region
Observation 2
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a k-clique is a complete subgraph of size k (i.e., a subset of k
vertices that are all connected to each other)
k-clique
2-cliques: all edges
3-cliques: ABC, ABD, ACD, BCA
4-cliques: ABCD (maximal clique)
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OL-HeatMap* algorithm (sketch)
1. Apply sweep-line to find intersecting pairs
2. Construct the rectangle intersection graph (RIG)
3. Apply a clique enumeration algorithm on graph
*OL-HeatMap is an extended version of SLIG -
Sweep-Line (with an auxiliary) Intersection Graph
By Tilemachos et al.
(A,B)(A,C)(A,D)(B,C)(B,D)(C,D)(D,E)(A,B,C)(A,B,D)(A,C,D)(B,C,D)(A,B,C,D)
(1) (2) (3)
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OL-HeatMap: Other metrics computed
z-index
The number of simultaneously
overlapping objects in a set
size of overlap(|S|)
For more dimensions, |S| is the product of the common region lengths in each dimension |So|
zABCD = 4zDE = 2… |SABCD|
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OL-HeatMap: Final visualization
Coloring the boxesEach common region S should be colored only once
based on their intersection cardinality. We skip
drawing of rectangles which are completely covered
by another.
Currently ~30% less overlaps are colored
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Experimental Evaluation
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⦁ Accuracy performance
⦁ Runtime performance
⦁ OL-HeatMap versatility (extension to 1D
objects)
⦁ OL-HeatMap flexibility (real world use-cases)
⦁ OL-HeatMap scalability
Experiment overview
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1-D intervals
Randomly generated objects
2-D rectangles –gaussian distribution
2-D rectangles –uniform distribution
2-D rectangles – bi-modal distribution
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Accuracy
Measurement of accuracy for different grid sizes
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Accuracy
Accuracy performance of OL-HeatMap vs. grid-based
OL-HeatMap is 100% accurate. However, a finer grid can
achieve similar accuracy
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Runtime cost
Comparison of time for different data-set sizes
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Runtime cost
Comparison of time for different data distributions
Finer grid sizes takes a lot of time to compute in order
to achieve similar accuracy that of OL-HeatMap
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Scalability
Execution Time vs OL-HeatMap Scalability
OL-HeatMap can scale up-to a million regions
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Real World Use Cases
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The Data
⦁ US Airline Carrier Data (1987-present)
⦁ We used John Wayne Airport, Orange County, California
⦁ 1D intervals created by time aircraft spent on runway
Visualization Goal
⦁ Find highest density of runway traffic
⦁ Finding least used time slot for a runway
⦁ Overview of airport usage in a single day (February 1st, 2019)
⦁ Providing aid in Air Traffic Management
Real-world use cases (1D)
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Airline carrier data
Overview of the
February 1st, 2019
Time Left to Right –
0000 – 2359 Hours
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Airline carrier data
100 Grid. Time - 0000-2359 Hours [24 Minute Intervals]
50 Grid. Time - 0000-2359 Hours [48 Minute Intervals]
OL-HeatMap. Time - 0000-2359 Hours
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The Data
⦁ US Storm Events Database, NOAA (1953-present).
⦁ Relevant information regarding significant weather event.
⦁ Begin Long., Lat., and End Long., Lat. Used to create bounding
boxes
Visualization Goal
⦁ Determining storm hot-spots in US during 2017-2019
⦁ Finding states with less severe weather incidents
⦁ Finding the borders of “Tornado Alley”
⦁ Visualize using OL-HeatMap to show the sizes, density and severity
of these events
⦁ Finding all hurricanes in Florida from 1953-2018 {Using a subset of
the entire dataset}
Real-world use cases (2D)
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US storm events database
Grid-based visualization
OL-HeatMap
Storms in US [2017-2019]
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US storm events database
Overview of Florida [1953-2018]
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US storm events database
Grid-based visualization OL-HeatMap
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Proof-of-Concept Demo System
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System overview
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User interface
Input Data UI
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User interface
Visualization UI
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Faster visualization rendering
Finding multiple axis-aligned object intersections
OL-HeatMap – a powerful sweep-line based algorithm for finding density
OL-HeatMap properties:
- fast
- exact
- versatile
Take-away message
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Thank you!
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Questions?