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Implications of using the HyperSpace Diagonal Counting (HSDC) method for visually representing 3-D shapes By Mallepally Mithun K ReddyPowerPoint Presentation

Implications of using the HyperSpace Diagonal Counting (HSDC) method for visually representing 3-D shapes By Mallepally Mithun K Reddy

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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 (HSDC) method for visually representing 3-D shapes

- Introduction & Motivation
- Multidimensional Visualization
- Hyper-Space Diagonal Counting (HSDC)
- Results
- Conclusions
- Future Work

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

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 (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

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

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

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 (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

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

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

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

- 245 Pareto points were generated

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

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

- Representation of Pareto frontier using binning technique

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

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

- Spanning the grid
- To what diagonal to count – to span the entire grid?

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

- Inspect 2D shapes – observe trends

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 (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

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 (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

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

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

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

Inclination with major axis=10º

Inclination with major axis=45º

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 (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 (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

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

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

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 = 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 (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 (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

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

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