1. Higher Order Visual Search for Information in Multidimensional Data Sets
Scalable Visual Analytics (SPP 1335)
Good morning, my namy name is Georgia Albuquerque and I am here with Dirk Lehman to present you some results of our project on… from the last year.
Holger Theisel, University of Magdeburg, Visual Computing Group
Marcus Magnor, TU Braunschweig, Computer Graphics Lab

2. Higher Order Visual Search: Team
University Magdeburg
TU Braunschweig
Holger Theisel,
Head of Visual Computing Group
Marcus Magnor,
Head of Computer Graphics Lab
Dirk J. Lehmann
Georgia Albuquerque
Martin Eisemann
First I want to introduce the complete team: From the University of Magdeburg in the Visual Computing Group we have Holger Theisel and Dirk Lehmann, and from the TU Braunschweig from the Institut für Computergraphik our group is composed by Marcus Magnor , Martin Eisemann and Me.
I want to apologize because I was not here in the last meeting, I was 1 year in parental vacations in a small private project :) . Let me Introduce you the little Isabella, now 15 months old.

3. Higher Order Visual Search: Team
University Magdeburg
TU Braunschweig
Holger Theisel,
Head of Visual Computing Group
Marcus Magnor,
Head of Computer Graphics Lab
Dirk J. Lehmann
Georgia Albuquerque
Martin Eisemann
Baby on board!

4. Higher Order Visual Search
Extend Exhaustive Visual Search for:
Higher order relations in multidimensional data sets
WP1: 2D hypothesis testing by user-drawn sketches
WP2: Relations only visible in continuous visualizations
WP3: Relations between more than 2 dimensions
WP4: Evaluation
So… the main Idea our project in this second phase was to search for higher order relations in multidimensional datasets.
Now I want to recap our work packages, I will begin with our first WP…

5. WP1: Sketch-based Structure Search
Best projections are selected by a quality metric
2D hypothesis testing by user-drawn sketches
Sketch-based structure search
That we call Sketch-based Structure Search, for this work package we consider that the best projections of a dataset were select by a quality metric and presented to the user.
Now the user have some interesting projections of the data and know better what he is searching for.
Our idea is then to allow this user to test two-dimensional Hypothesis using sketches, these sketches can be then used find an approximate model of structure in projection or to searcht for similar structures in the remaining projections.

6. WP1: Sketch-based Structure Search
Selecting Coherent and Relevant Plots in Large Scatterplot Matrices,
D. J. Lehmann, G. Albuquerque, M. Eisemann, M. Magnor, H. Theisel,
Computer Graphics Forum, 2012
Semi-Automatic Classification of Weather Maps
G. Albuquerque, D. J. Lehmann, T. Rodermund, M. Eisemann, T. Nocke, H. Theisel, M. Magnor,
Technical Report , TU Braunschweig, 2012
In the scope of this WP Dirk presente two of our publications in our last meeting: Selecting Coherent and Relevant Plots in Large Scatterplot Matrices and Semi-Automatic Classification of Weather Maps

7. WP2: Continuous Visualizations
Relations only visible in continuous visualizations
Quality metrics for continuous visualizations
Continuous data
New continuous visualizations for discrete data
Smooth density functions of point clouds with sharp structures
Structure identification
Reconstruction
Compression of continuous visualizations
In the second WP of our project the main idea was to search for relations that are only visible in continuous visualizations.
This continuous visualizations can be from original continuous data or
from new continuous visualizations for discrete data that can be obtained using a smoothing density function over a point cloud for example.
These quality metrics should be able to find important structures in the visualization and can be later used for
Reconstruction and compression of continuous visualizations.

8. WP2: Continuous Visualizations
Automatic Detection and Visualization of Qualitative Hemodynamic Characteristics in Cerebral Aneurysms, R. Gasteiger, D. J. Lehmann, R. van Pelt, G. Janiga, O. Beuing,
A. Vilanova, H. Theisel, B. Preim,
IEEE Transactions on Visualization and Computer Graphics (Proc. IEEE Visualization ), 2012
awarded by MedVis-Award 2012 (Karl Heinz Höhne Award)
Automating Transfer Function Design with Valley Cell-based Clustering of 2D Density Plots,
Y. Wang, J. Zhang, D. J. Lehmann, H. Theisel, X. Chi,
Computer Graphics Forum (Proc. EuroVis), 2012
Reflected Vector Fields for Finding FTLE Ridges,
M. Schulze, C. Roessl, D. J. Lehmann and H. Theisel,
Technical Report FIN , Otto-von-Guericke-University, Magdeburg, 2013
In the scope of this WP we have three publications. I am not going into detail on these today but they are available in our project page.

9. WP3: Higher Order relations
Relations between more than 2 dimensions
Search for multidimensional structures
Quality metrics based on
iconized visualizations
Scatterplot cubes
3D Extension of scatterplot matrix
Theisel 1998
In the last WP that we call Higher order relations, our idea is to search for relations between more than two dimensions.
As for example search for existing multidimensional structures or quality metrics for iconized visualization
Another idea for this WP are scatterplot cubes, that is basically a 3d extension of scatterplot matrix.
Doka 2006

10. WP3: Higher Order relations
D. J. Lehmann and H. Theisel
Orthographic Star Coordinates
IEEE Transactions on Visualization and Computer Graphics
(Proc. IEEE Information Visualization), 2013
In the scope of this WP, we have the Orthographic Star Coordinates that was presented in the InfoVis this year, and know I will hand over to Dirk that will explain it in more detail.

11. nD Data Space
2D Visualization Space x y d i ) x y (
Dimension
Axes ) x 2 y ( d 1 2 3
High-Dimensional Data
First of all, we start with a short introduction to Star Coordinates:
let's consider a high-dimensional data set given over an nD Data Space.
Each axis $d$ of it is accociated with a 2D axis representations $(x and y)$ within a 2D Visualization Space. x 1 y ( ) ) x 3 y (
Motivation
Orthographic
Conditions
Standard Configuration
Interaction
Morphing
Conclusion

12. nD Data Space
2D Visualization Space x y 1 ( ) 2 3 ) x 2 y ( d 1 2 3 p m
m=p
Now, a data point $m$ is projected on a point $p$ in the visualizations space by using a matrix multiplication with a Matrix $A$.
This Matix $A$ is constructed by the (x and y) axis of the 2D visualization space. x 1 y ( ) ) x 3 y (

13. nD Data Space
2D Visualization Space x y 1 ( ) 2 3 d 1 2 3 p m
Appling Matrix $A$ to all data points yield the representation of the data in the 2D visualization space.
This kind of multivariate projection has been introduced 2000 by Kandogan, denoted as Star Coordinates.

14. nD Data Space
2D Visualization Space
causes Affine Projection
causes Projective Projection
Star Class
Ma & Teoh
2003
Normalized RadViz
2012
Daniels et al.
Clusters in RadViz
Nováková & Stepánková
2006
3D Star Coordinate System
Shaik & Yeasin
Star Coordinates
Kandogan
2000
Hoffman et al.
1997
RadViz
Trends Using RadViz
2011 x y 1 ( ) 2 3 d 1 2 3
Unfortunately, projective as well as affine projections have a serious drawback. They aren't able to preserve distance properties of the data within
the visualization space. What does this means?
To answer this question, let's consider 4 data records, placed on the periphery of a hyper-spher….

15. nD Data Space
2D Visualization Space
Projective Projection
causes Affine Projection
causes Projective Projection
Star Class
Ma & Teoh
2003
Normalized RadViz
2012
Daniels et al.
Clusters in RadViz
Nováková & Stepánková
2006
3D Star Coordinate System
Shaik & Yeasin
Star Coordinates
Kandogan
2000
Hoffman et al.
1997
RadViz
Trends Using RadViz
2011 x y 1 ( ) 2 3
Affine Projection d 1 2 3
Unfortunately, projective as well as affine projections have a serious drawback. They aren't able to preserve distance properties of the data within
the visualization space. What does this means?
To answer this question, let's consider 4 data records, placed on the periphery of a hyper-spher….

16. ( ) y x nD Data Space 2D Visualization Space
Note that a transla-
tion of the data influences location size and shape of the projections:
the closer the spheres come to the projection center, the larger is their
projection
Projective projections map straight lines to straight lines and preserve the cross ratio of four collinear points
map straight lines to straight lines and preserve the ratio of three collinear points
nD Data Space
2D Visualization Space
Projective Projection x y 1 ( ) 2 3
Affine Projection d 1 2 3
…and lets further consider its related projection in visualization space.
For projective and affine projections apply that two points close to each other in the projection can be far away from each other in data space.
Also the opposite is true: two points close to each other in data space can be far away from each other in the projection.
Consequently, distortions occours, meaning that a hyper-sphere in data space is mapped onto an ellipse in visualization space.
clearly, two points close to each other in the projection can be far away
from each other in nD. Unfortunately, also the opposite is true: two
points close to each other in nD can be far away from each other in the
projection. Fig. 1(c) gives an illustration

17. nD Data Space
2D Visualization Space
Projective Projection x y
Affine Projection
Orthographic Projection ) x 2 y ( d 1 2 3
If the projection matrix $A$ is orthographic, then the distance between two points in visualization space is not larger than their distance in the data space.
Therefore, an orthographic projection avoids distortions, meaning that a hypher-sphere in data space is mapped onto a circle in visualization space with the same radius. x 1 y ( ) ) x 3 y (

18. ? ( ) y x nD Data Space 2D Visualization Space Affine Projection
However, a good projection should
only show properties of the data, not properties of the projection.
nD Data Space
2D Visualization Space ?
Affine Projection
Projective Projection
Orthographic Projection x y 1 ( ) 2 3 d 1 2 3
With this knowledge, Let's go back to the inital visualization example:
Here, we do not know -without more ado- which kind of projection applies for matrix A,
and consequently we do not know where in the projection distortions may occour, which makes a reliable interpretation a hard task. --
Thus, we introduce in the following the orthographic star coordinates that restricts star coordinates to orthographic projections.
Furthermore, we present related interaction techniques and data tours, that enable a fast exploration of unknown data w.r.t. to patterns, cluster, trends, an so on and so forth.

19. ? nD Data Space 2D Visualization Space Affine Projection
Conditions for being orthographic ?
Affine Projection
Projective Projection
Orthographic Projection d 3 d 1 2 3 d 2 d 1
Mutual orthogonal column vectors
Unit length of column vectors
With this knowledge, Let's go back to the inital visualization example:
Here, we do not know -without more ado- which kind of projection applies for matrix A,
and consequently we do not know where in the projection distortions may occour, which makes a reliable interpretation a hard task. --
Thus, we introduce in the following the orthographic star coordinates that restricts star coordinates to orthographic projections.
Furthermore, we present related interaction techniques and data tours, that enable a fast exploration of unknown data w.r.t. to patterns, cluster, trends, an so on and so forth.

20. Conditions for being orthographic1 2 3
Orthographic Condition:
Mutual orthogonal column vectors
Unit length column vectors
Unit length
Orthogonal
Thereto, please node that the x and y components of the projection Matrix $A$ forms the row vectors x and y.
Now, three "orthographic conditions" need to be fulfilled for an orthographic projection matrix $A$:
the Eucledian-norm of the row vectors $x$ and $y$ minus $1$, as well as, the inner product
of the row vectors $x$ and $y$ needs to be zero.
With this "orthographic conditions", a scalar "orthographic energy" $e$ can be definied, as the squared sum
of these three “orthographic conditions”.
If this "orthographic energy" $e$ of a matrix $A$ is zero, the Matrix $A$ being an orthographic projection, else it is not.

21. 2D Visualization Space y x Conditions for being orthographic
Construct an orthographic Matrix
Reconditioning
Orthographic Conditions
Orthographic Condition:
Unit length
Orthogonal y
2D Visualization Space x
Thereto, please node that the x and y components of the projection Matrix $A$ forms the row vectors x and y.
Now, three "orthographic conditions" need to be fulfilled for an orthographic projection matrix $A$:
the Eucledian-norm of the row vectors $x$ and $y$ minus $1$, as well as, the inner product
of the row vectors $x$ and $y$ needs to be zero.
With this "orthographic conditions", a scalar "orthographic energy" $e$ can be definied, as the squared sum
of these three “orthographic conditions”.
If this "orthographic energy" $e$ of a matrix $A$ is zero, the Matrix $A$ being an orthographic projection, else it is not.
Scalable Visual Analytics

22. 2D Visualization Space y 1 x 3 6 Construct an orthographic Matrix
Reconditioning
Orthographic Conditions
2D Visualization Space 1 y
The second approach is a geometrical reconditioning of matrix $A$, which is based on the Gram-Schmidt-Process.
Fist, the row vector $y$ is projected in a way that the resulting vector is orthogonal to the row vector $x$. This fulfills the fist condition.
Then, both vectors will be nomalized to unit length, which fullfill the remaining orthographic conditions. The outcome is an orthographic matrix $A$.
---
Based on the Energy Minimization and the Reconditioning, we define in the following interaction techniques for orthographic star coordinates. x
Scalable Visual Analytics
Orthographic Energy 6 3

23. X y 2D Visualization Space 1 x x
Orthography-preserving Axis Interaction
Construct an orthographic Matrix
Reconditioning
Orthographic Conditions X y
Axis Interaction
update
2D Visualization Space 1
An interaction is given by moving one axis x,y in the visualizations space and updating then the projection matrix $A$.
As you can see here in the video, this interaction is not orthography-preserving, wherefore the projected circle deforms. x x
-Axes interaction causes distortions
Scalable Visual Analytics

24. 2D Visualization Space y y y y x x
Orthography-preserving Axis Interaction
Orthographic Conditions
2D Visualization Space y y y y
Axis Interaction
By Reconditioning
update
Construct an
Orthographic
Therfore, an additional orthography-construction step is done, in order to maintain orthography:
for this, an "orthography energy-minimization" is applied.
In the vid, it can be seen that the circle preserves its shape now; because the remaining axis adjusted themselves this way that the orthographic conditions are
always fulfilled. x x
Scalable Visual Analytics

25. Using Restrictions during Axis interaction
Orthography-preserving Axis Interaction
Orthographic Conditions
Using Restrictions during Axis interaction
2D Visualization Space y y y y y
Axis Interaction
By Reconditioning
Direction
update
Fixed
Radial
Construct an
Orthographic
Restrictions:
Fixed
Obviously, a typical characteristic of orthographic star coordinates is that all axes move freely while the interaction. A user
might want to restrict this movement in certain situations, to further ease the visual search for instance.
Therfore, we provide a conditional interaction by restricting the degree of freedom for the axes.
Three restrictions are of interest:
Fist: Forbid any movement of the axis, which means to fix the axes.
Secound: Allow only the movement in the direction of the axis.
Third: Allow only the movement along the periphery of a circle with certain radius.
With the aid of the reconditioning approach, this conditions can be purposefully realized.
For this, after the reconditioning the conditions are insert into the resulting orthographic matrix, which give a non-orthographic matrix.
Therefore, the reconditioning and the insertion of the conditions are iteratively repeated until this process converges.
(vid aktivieren:) The result is an interaction scheme where the orthography is preserved and where the mentioned conditions are realized as well.
You can see, that the red axis really is fices, that the blue axis only moves along its direction, and that the yellow axis moves along a certain periphery.
Direction
Radial x x
Scalable Visual Analytics

26. + - - + Using Restrictions during Axis interaction Star Coordinates
Orthography-preserving Axis Interaction y x
Using Restrictions during Axis interaction
Construct an
Orthographic
Axis Interaction
update
By Reconditioning
Fixed
Direction
Radial
Restrictions:
2D Visualization Space
Orthographic Conditions
Final Comparison
Star Coordinates
[E. Kandogan 2000]
Orthographic Star Coordinates
I'd like to close this talk with a comparision between trational star coordinates and our orthographic star coordinates.
(vid starten) Here you can see a snippet of an interacvtive visual search of the "wine data set" for both traditional star coordinates and our orthographic star coordinates.
One can see that the influence of a single axis can be better investigated with star coordinates compared to our approach, since only one axis move at time there.
However, projection based distortions do not occour in our orthographic star coordinates, thus, a visual search for global pattern
can be well done with the orthographic start coordinates. Exspecially the three clusters in the data can be better seen with our approach compares
to trationsal star coordinates.
In conclusion, the lession to learn is that both techniques complement each other and both are necessary with respect to a complete visual search.
So far, thats it. I thank you for your attantion and I'm open for your quations. Thank You. + -
Influence of single axis is clear
Influence of single axis is unclear - +
Distortions negatively influence visual search
Absence of distortions ease visual search
=Visual Analytics Tool

27. Presentation of …. Fin Orthographic Star Coordinates Contributions
D. J. Lehmann and H. Theisel
Orthographic Star Coordinates
IEEE Transactions on Visualization and Computer Graphics
(Proc. IEEE Information Visualization), 2013
Contributions
Orthographic Conditions
Regarding this, since the last SPP-meeting in Stuttgart, we were able to turn out 5 results, as you can see here:
two approaches are dealing with structure identification within 3D volume data, one is dealing with structure identification in nD data,
and one is dealing with hyphothesis testing from specific visualizations. >>>
Orthographic Configurations
Orthographic Interactions
Orthographic Morphing
Orthographic Data Tours

28. Publications: Second Project Stage
Selecting Coherent and Relevant Plots in Large Scatterplot Matrices,
D. J. Lehmann, G. Albuquerque, M. Eisemann, M. Magnor, H. Theisel,
Computer Graphics Forum, 2012
Automatic Detection and Visualization of Qualitative Hemodynamic Characteristics in Cerebral Aneurysms, R. Gasteiger, D. J. Lehmann, R. van Pelt, G. Janiga, O. Beuing,
A. Vilanova, H. Theisel, B. Preim,
IEEE Transactions on Visualization and Computer Graphics (Proc. IEEE Visualization ), 2012
awarded by MedVis-Award 2012 (Karl Heinz Höhne Award)
Semi-Automatic Classification of Weather Maps
G. Albuquerque, D. J. Lehmann, T. Rodermund, M. Eisemann, T. Nocke, H. Theisel, M. Magnor,
Technical Report , TU Braunschweig, 2012
Automating Transfer Function Design with Valley Cell-based Clustering of 2D Density Plots,
Y. Wang, J. Zhang, D. J. Lehmann, H. Theisel, X. Chi,
Computer Graphics Forum (Proc. EuroVis), 2012
Novel Methods and Applications for the Feature Extraction from Visualizations of Multi-Parameter Data, D. J. Lehmann, Ph.D. thesis , University Magdeburg, 2012
Here we have our list of publications in this second project phase.
Orthographic Star Coordinates,
D. J. Lehmann, H. Theisel,
IEEE Transactions on Visualization and Computer Graphics (Proc. IEEE Info .Visualization), 2013
Reflected Vector Fields for Finding FTLE Ridges,
M. Schulze, C. Roessl, D. J. Lehmann and H. Theisel,
Technical Report FIN , Otto-von-Guericke-University, Magdeburg, 2013

29. Outlook: New Ideas Quality metrics for continuous visualizations
New visualization methods to better visualize relations in high-dimensional space
Evaluation
And now I want to give a short overview about some future and ongoing work for the next year of the project.
In this next year we plan to develop other quality metrics for continuous visualizations as aforementioned and
to investigate new visualization methods to better visualize relations in high-dimensional space.
And finally was plan to develop new test cases to evaluate the methods we are working on.

30. Thank You Higher Order Visual Search 6 Results, including
This work was supported by the
German Science Foundation (DFG),
within the priority program on
Scalable Visual Analytics (SPP 1335)
6 Results, including
1 x IEEE Visualization
1 x IEEE Eurovis
1 x Computer Graphics Forum
2 x Technical Report
1 x IEEE Infovis
Holger Theisel, University of Magdeburg, Visual Computing Group
Marcus Magnor, TU Braunschweig, Computer Graphics Lab