1. Visual data analysis a chance and a challenge for mathematicians Krzysztof S. Nowiński (ICM)

2. Visual data analysis Motivation: Mathematical model of any piece of reality – Verifiable – Operative - applicable Verification and application usually done by computational implementation Verification by comparison to observation or experiment informationdata Problem: extract information from data Mathematics for key technologies and innovation Warsaw, February 21-22, 2008

3. Statistics versus Visualization Statistics – Provides easily comparable, simple, compressed information – Provides answers to questions Visualization – Provides often complicated, hard to describe images or movies – Difficult to compare and compress but – Shows the unexpected – Allows to pose questions and state conjectures Mathematics for key technologies and innovation Warsaw, February 21-22, 2008

4. Example: cosmological simulations Mathematics for key technologies and innovation Warsaw, February 21-22, 2008 Universe evolution model with gravity as the only driving force Does it correctly reproduce the current state – Voids – Walls – Strings The picture confirms this conjecture

5. Example Given: A mathematical (classical) model of internal energy E(X) of a molecule – balls and springs flavor Three principal geometric variables (dihedral angles in a large ring) ω 1 (X), ω 2 (X), ω 3 (X) Mathematics for key technologies and innovation Warsaw, February 21-22, 2008

6. Example (cont.) Required: Description of 3D landscape of E(ω 1,ω 2,ω 3 )=min(E(X): ω i (X)=ω i, i=1,2,3): Local minima and their values (quasi-stationary states) Minimum energy paths joining these minima – state transition tree Mathematics for key technologies and innovation Warsaw, February 21-22, 2008 E Local equilibrium Transition state

7. Example (cont.) Numerical implementation provided 30x30x30 matrix of energy values Finding local minima – numerically trivial Finding transition paths – slightly harder but possible With some visualization system at hand – why not to look first at the raw data? Mathematics for key technologies and innovation Warsaw, February 21-22, 2008

8. Example (cont.) With some visualization system at hand – why not to look first at the raw data? Finding local minima – visually trivial Just look at isosurfaces corresponding to small energy values Finding transition paths – slightly harder but still easy Pick moments (threshold values) when isosurfaces start to join Mathematics for key technologies and innovation Warsaw, February 21-22, 2008

9. Pictures now Mathematics for key technologies and innovation Warsaw, February 21-22, 2008

10. Example (cont.) Discover hidden symmetry Unexpected Clearly seen Impossible to be found by any form of numerical (statistical) analysis Unless known beforehand Mathematics for key technologies and innovation Warsaw, February 21-22, 2008

11. Examples Vector field in plane – – from simplest possible – to artistic – and formal – singular points detection Applicable in 3D Question – Tensor fields

12. Example – biomedical applications Volume segmentation – essential for diagnosis and therapy planning Preceeded by volume preprocessing and tissue classification Lots of techniques – Freehand – Semi-interactive – volume growing – Automatic (atlas deformation) Mathematics for key technologies and innovation Warsaw, February 21-22, 2008

13. Volume segmentation Segmented volume growing – Evolution of characteristic function Well established numerical algorithms, but large data to operate on vs. Evolution of surface fast, efficient, but variable topology Mathematics for key technologies and innovation Warsaw, February 21-22, 2008

14. In need for algorithmic homotopy Mathematics for key technologies and innovation Warsaw, February 21-22, 2008 Surfaces usually evolve in a smooth way (elementary stability theory) However, ocassionally they pass through singularities (slightly more advanced stability theory) Passing through a singularity almost always (again, stability theory) means undergoing a surgery – cutting away a small ring and filling holes with two disks or reverse operation

15. In need for algorithmic homotopy The problem: „Diagnose for surgery”, That is, find points or closed curves (cycles) that can become singular in nearest future. They must be small but essential (at least locally) How to find them on the fly? Mathematics for key technologies and innovation Warsaw, February 21-22, 2008

16. Final remarks Majority of images made with VisNow – Open sourced, Java based visualization system – currently targeted at biomedical application – and its derivatives http://visnow.icm.edu.pl/ Thanks to my collaborators – Michał Chlebiej – Bartosz Borucki – Hubert Orlik-Grzesik – Michał Łyczek Mathematics for key technologies and innovation Warsaw, February 21-22, 2008