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

1. 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

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

3. 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

4. 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

5. 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

6. 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

7. MDMV Examples Glyphs Scatterplot Matrix

8. 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

9. 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

10. Methodology • Similarly, we can map points from an n-dimensional space to unique points on a line • Hyper-Space Diagonal Counting (HSDC) in nD

11. 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

12. 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

13. Traditional Pareto Frontier • 245 Pareto points were generated

14. 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

15. Binning Technique • Representation of Pareto frontier using binning technique

16. 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

17. Grid Spanning • Spanning the grid • To what diagonal to count – to span the entire grid?

18. 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

19. Straight line – HSDC (10 bins/axis) Y=500X Y=3X Y=0 Y=(-500)X

20. 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

21. Circle Radius = 1, Center (2,2) Radius =1, Center (1,1) Radius =2, Center (-2,2) Radius =10, Center (2, -2)

22. 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

23. Square Square at (0,0), edge = 2 units, inclination with major axis=0 HSDC plot of the adjacent square

24. 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

25. Rectangle (length=5, breadth=1) Inclination with major axis=10º Inclination with major axis=45º

26. 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)

27. 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

28. 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

29. 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

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

31. 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

32. 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

33. 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

34. 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

35. Thank You !!!