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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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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 l.jpg
Overview (HSDC) method for visually representing 3-D shapes

  • Introduction & Motivation

  • Multidimensional Visualization

  • Hyper-Space Diagonal Counting (HSDC)

  • Results

  • Conclusions

  • Future Work


Introduction motivation l.jpg
Introduction & Motivation (HSDC) method for visually representing 3-D shapes

  • 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 l.jpg

Low order (HSDC) method for visually representing 3-D shapes

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 l.jpg
Dimension Reduction (HSDC) method for visually representing 3-D shapes

  • 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


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MDMV Techniques (HSDC) method for visually representing 3-D shapes

  • 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 l.jpg
MDMV Examples (HSDC) method for visually representing 3-D shapes

Glyphs

Scatterplot Matrix


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HSDC - Methodology Development (HSDC) method for visually representing 3-D shapes

  • 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


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Methodology Development (HSDC) method for visually representing 3-D shapes

  • 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 l.jpg
Methodology (HSDC) method for visually representing 3-D shapes

  • Similarly, we can map points from an n-dimensional space to unique points on a line

  • Hyper-Space Diagonal Counting (HSDC) in nD


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Relevance (HSDC) method for visually representing 3-D shapes

  • 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


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Binning technique - explained (HSDC) method for visually representing 3-D shapes

  • To be able to use HSDC for multiobjective problems

    • Need an index based approach

    • Binning technique – index based representation

  • Consider a bi-objective problem


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Traditional Pareto Frontier (HSDC) method for visually representing 3-D shapes

  • 245 Pareto points were generated


Binning technique l.jpg
Binning Technique (HSDC) method for visually representing 3-D shapes

  • 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


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Binning Technique (HSDC) method for visually representing 3-D shapes

  • Representation of Pareto frontier using binning technique


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Binning Technique (HSDC) method for visually representing 3-D shapes

  • 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


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Grid Spanning (HSDC) method for visually representing 3-D shapes

  • Spanning the grid

    • To what diagonal to count – to span the entire grid?


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Outline of research - results (HSDC) method for visually representing 3-D shapes

  • Idea – search for extensibility of trends

  • Procedure

    • Inspect 2D shapes – observe trends

      • Straight lines

      • Circles

      • Squares/rectangles

    • Inspect 3D shapes

      • Cube

      • Sphere


Straight line hsdc 10 bins axis l.jpg
Straight line – HSDC (10 bins/axis) (HSDC) method for visually representing 3-D shapes

Y=500X

Y=3X

Y=0

Y=(-500)X


Conclusions for a straight line l.jpg
Conclusions for a straight line (HSDC) method for visually representing 3-D shapes

  • 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


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Circle (HSDC) method for visually representing 3-D shapes

Radius = 1, Center (2,2)

Radius =1, Center (1,1)

Radius =2, Center (-2,2)

Radius =10, Center (2, -2)


Conclusions for a circle l.jpg
Conclusions for a circle (HSDC) method for visually representing 3-D shapes

  • 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


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Square (HSDC) method for visually representing 3-D shapes

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

HSDC plot of the adjacent square


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Square.. (HSDC) method for visually representing 3-D shapes

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


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Rectangle (length=5, breadth=1) (HSDC) method for visually representing 3-D shapes

Inclination with major axis=10º

Inclination with major axis=45º


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Conclusions about square/rectangle (HSDC) method for visually representing 3-D shapes

  • 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 l.jpg
Circle vs square (HSDC) method for visually representing 3-D shapes

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 l.jpg
3D shapes -motivation (HSDC) method for visually representing 3-D shapes

  • 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


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Cube (Edge 10 units; Inclination with all axes=0º) (HSDC) method for visually representing 3-D shapes

HSDC plot of cube

HSDC plot of square with inclination of 0º

- There are similarities in both the figures


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Cube (Edge 10 units; inclination with (HSDC) method for visually representing 3-D shapesX-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


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Cube (Edge 10 units; inclination with (HSDC) method for visually representing 3-D shapesX-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 l.jpg
Sphere (inclination with all axes = 0º) (HSDC) method for visually representing 3-D shapes

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 l.jpg
Conclusions (HSDC) method for visually representing 3-D shapes

  • 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 l.jpg
Future work (HSDC) method for visually representing 3-D shapes

  • 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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Thank You !!! (HSDC) method for visually representing 3-D shapes


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