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

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

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

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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**• Introduction & Motivation • Multidimensional Visualization • Hyper-Space Diagonal Counting (HSDC) • Results • Conclusions • Future Work**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**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 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**• 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**Glyphs Scatterplot Matrix**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**• 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**• Similarly, we can map points from an n-dimensional space to unique points on a line • Hyper-Space Diagonal Counting (HSDC) in nD**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**• 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**• 245 Pareto points were generated**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 Technique**• Representation of Pareto frontier using binning technique**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**• Spanning the grid • To what diagonal to count – to span the entire grid?**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**Straight line – HSDC (10 bins/axis)**Y=500X Y=3X Y=0 Y=(-500)X**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**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 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 at (0,0), edge = 2 units, inclination with major axis=0 HSDC plot of the adjacent square**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)**Inclination with major axis=10º Inclination with major axis=45º**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**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 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 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 = 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 = 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)**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 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**• 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