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

Visual data analysis Visual data analysis a a chance and chance and a a challenge challenge

for mathematiciansfor mathematicians

Krzysztof S. Nowiński (ICM)

Visual data analysisVisual 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

• Problem: extract informationinformation from datadata

Mathematics for key technologies and innovation Warsaw, February 21-22, 2008

Statistics versus VisualizationStatistics 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

conjecturesMathematics for key technologies and innovation Warsaw,

February 21-22, 2008

Example: cosmological Example: cosmological simulationssimulations


• 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

ExampleExampleGiven:• 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) ω1(X), ω2(X), ω3(X)


Example (cont.)Example (cont.)

Required: Description of 3D

landscape of

E(ω1,ω2,ω3)=min(E(X): ωi(X)=ωi, i=1,2,3):

• Local minima and their values (quasi-stationary states)

• Minimum energy paths joining these minima – state transition tree


EE

Local equilibrium

Transition state


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?



• With some visualization system at hand – why not to look first at the raw data?

• Finding local minima – visually trivialJust look at isosurfaces corresponding to

small energy values• Finding transition paths – slightly

harder but still easyPick moments (threshold values) when

isosurfaces start to join


Pictures nowPictures now



Discover hidden symmetryDiscover hidden symmetry• Unexpected• Clearly seen• Impossible to be found by any

form of numerical (statistical) analysis

• Unless known beforehand


ExamplesExamples


• Vector field in plane –– from simplest

possible – to artistic– and formal –

singular points detection

Applicable in 3D• Question –

Tensor fields

Example – biomedical Example – biomedical applicationsapplications

• 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

Volume segmentationVolume segmentation

Segmented volume growing – Evolution of characteristic function

Well established numerical algorithms, but

large data to operate onvs.

Evolution of surfacefast, efficient,

butvariable topology


In need for algorithmic homotopyIn need for algorithmic homotopy


• 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

In need for algorithmic homotopyIn need for algorithmic homotopy

The problem: „Diagnose for surgery”,That is, find points orclosed curves (cycles) thatcan become singular innearest future.They must be small but essential (at

least locally)How to find them on the fly?


Final remarksFinal 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


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

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