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Implications of using the HyperSpace Diagonal Counting (HSDC) method for visually representing 3-D shapes By Mallepally Mithun K Reddy Department of Mechanical and Aerospace Engineering Project Defense December 29 th , 2006 Overview Introduction & Motivation Multidimensional Visualization

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slide1
Implications of using the HyperSpace Diagonal Counting (HSDC) method for visually representing 3-D shapes

By

Mallepally Mithun K Reddy

Department of Mechanical and Aerospace Engineering

Project Defense

December 29th, 2006

overview
Overview
  • Introduction & Motivation
  • Multidimensional Visualization
  • Hyper-Space Diagonal Counting (HSDC)
  • Results
  • Conclusions
  • Future Work
introduction motivation
Introduction & Motivation
  • Scientific Visualization
    • Allows visual representation of data
    • 2D or 3D graphs
    • Easy to understand
  • Multidimensional Data
    • Difficult to visualize
    • Not so easy to understand
    • Numerous methods – different applications
multidimensional visualization
Low order

Multidimensional

Data

High order

Multidimensional

Data

Dimension

Reduction

Multidimensional

Visualization

Multidimensional Visualization
  • Multidimensional Multivariate Visualization (MDMV)
    • Translate multidimensional data into visual representations
    • Reduce dimensionality
  • Dimension Reduction
    • Some variables can be correlated
    • Few variables may be irrelevant
dimension reduction
Dimension Reduction
  • Dimension Reduction Techniques
    • Clustering of variables
  • Drawbacks
    • Mostly suitable for linear structures
    • Computationally expensive
    • Loss of meaning
    • Loss of ability to understand the representation intuitively
mdmv techniques
MDMV Techniques
  • Techniques designed for a fixed number of variables
    • Use of color
    • Animation
  • Techniques designed for any number of variables
    • Scatterplots
    • Chernoff faces – Glyphs
    • Many others
mdmv examples
MDMV Examples

Glyphs

Scatterplot Matrix

hsdc methodology development
HSDC - Methodology Development
  • Cantor’s Theory
    • One-to-one correspondence of points on a line and points on a 2D surface
    • 2D array of points can be laid flat on a line

Array of points on a surface Path through all the points

Graphic Proof of Cantor’s Theory

methodology development
Methodology Development
  • Points from 3D space – mapped to points on a line
    • Make an array of points in 3D space
    • Create a path through the points
methodology
Methodology
  • Similarly, we can map points from an n-dimensional space to unique points on a line
  • Hyper-Space Diagonal Counting (HSDC) in nD
relevance
Relevance
  • What has any of this to do with visualization?
    • HSDC allows collapsing multiple dimensions on a single axis
    • Counting covers each point in a lossless fashion
  • HSDC – wide breadth of applications
    • Overarching relationship in variables
    • No overarching relationship – data already generated
    • May include exploration of databases to identify trends
binning technique explained
Binning technique - explained
  • To be able to use HSDC for multiobjective problems
    • Need an index based approach
    • Binning technique – index based representation
  • Consider a bi-objective problem
traditional pareto frontier
Traditional Pareto Frontier
  • 245 Pareto points were generated
binning technique
Binning Technique
  • Binning Technique – steps involved
    • Obtain Pareto points
    • Identify Max. and Min. for each objective to establish a range
    • Divide ranges into some finite number of bins. Example, objective F1 can be divided into 100 bins, 1 through 100.
    • Indices of these bins can be plotted along an axis, thus we can have indices of F1 on X-axis and F2 indices on Y-axis
    • Each Pareto point, previously generated, will fall under some combination of these bins
    • Represented as a unit cylinder along the third axis
    • Multiple points may fall under the same set of indices
binning technique15
Binning Technique
  • Representation of Pareto frontier using binning technique
binning technique16
Binning Technique
  • Index-based representation of Pareto frontier
    • Same as traditional Pareto frontier
    • Small changes in representation – due to discretization
    • Multiple Pareto points in bins – again, due to discretization
  • IMPORTANT
    • Axes enumerate indices
    • Not actual function values
  • We can use HSDC for mapping two or more objectives on one axis
grid spanning
Grid Spanning
  • Spanning the grid
    • To what diagonal to count – to span the entire grid?
outline of research results
Outline of research - results
  • Idea – search for extensibility of trends
  • Procedure
    • Inspect 2D shapes – observe trends
      • Straight lines
      • Circles
      • Squares/rectangles
    • Inspect 3D shapes
      • Cube
      • Sphere
conclusions for a straight line
Conclusions for a straight line
  • No. of bins needed – depends on the slope
  • Spread of bins occupied also depends on the slope
  • Looking at the HSDC plot doesn’t lead us to conclude anything
circle
Circle

Radius = 1, Center (2,2)

Radius =1, Center (1,1)

Radius =2, Center (-2,2)

Radius =10, Center (2, -2)

conclusions for a circle
Conclusions for a circle
  • HSDC plots are the same – independent of radius and center of the circle
  • If a HSDC plot resembles the one got above – it is that of a circle
square
Square

Square at (0,0), edge = 2 units, inclination with major axis=0

HSDC plot of the adjacent square

square24
Square..

Square at (0,0), edge = 2 units, inclination with major axis=10º

Square at (0,0), edge=2 units, inclination with major axis=20º

HSDC plot of the above square

HSDC plot of the above square

rectangle length 5 breadth 1
Rectangle (length=5, breadth=1)

Inclination with major axis=10º

Inclination with major axis=45º

conclusions about square rectangle
Conclusions about square/rectangle
  • HSDC plots of square/rectangle of all configurations are similar
  • The points occur in pairs (similar to a circle but has differences)
circle vs square
Circle vs square

HSDC plot of square

HSDC plot of circle

- Though there is coupling in both the shapes, there is a difference in the spreads

3d shapes motivation
3D shapes -motivation
  • 3D shapes are extensions of 2D
  • If similar trends are found, it would mean that there is extensibility and can be extended to n-D objects similarly.
  • Looking at the HSDC plot of an unknown dataset, one can intuitively visualize the shape by comparing the HSDC plot with that of the known shapes
cube edge 10 units inclination with all axes 0
Cube (Edge 10 units; Inclination with all axes=0º)

HSDC plot of cube

HSDC plot of square with inclination of 0º

- There are similarities in both the figures

cube edge 10 units inclination with x axis 30
Cube (Edge 10 units; inclination with X-axis = 30º)

HSDC plot of cube

Points are color coded

- Similar to the earlier figure

- End points in the HSDC space, correspond to end points on the cube

cube edge 10 units inclination with x axis 45
Cube (Edge 10 units; inclination with X-axis = 45º)

Points are color coded

HSDC plot of cube

- Similar to the earlier figure

- End points in the HSDC space, correspond to end points on the cube

sphere inclination with all axes 0
Sphere (inclination with all axes = 0º)

Points on the surface – color coded

HSDC plot of the sphere

- Similar to the earlier figure

- End points in the HSDC space, correspond to end points on the sphere as expected

conclusions
Conclusions
  • HSDC method explained
  • HSDC method applied on
    • 2D shapes – line, circle, square, rectangle
    • 3D shapes – cube, sphere
  • Trends seen in 2D are seen in 3D
    • Method seems to be extensible to higher dimensions
future work
Future work
  • Explore hyper-cube and hyper-sphere (more than 4 dimensions) to verify that similar trends are seen
  • Exploring more shapes will give more insight into the trends
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