Visual%20data%20analysis%20a%20chance%20and%20a%20challenge%20for%20mathematicians

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Visual data analysis a chance and a challenge
for mathematicians

• Krzysztof S. Nowinski (ICM)

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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
• Problem extract information from data

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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

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Example cosmological simulations

• 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

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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)
• ?1(X), ?2(X), ?3(X)

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Example (cont.)

• Required
• Description of 3D landscape of
• E(?1,?2,?3)min(E(X) ?i(X)?i, i1,2,3)
• Local minima and their values (quasi-stationary
  states)
• Minimum energy paths joining these minima state
  transition tree

E
Transition state
Local equilibrium
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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?

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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

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Pictures now
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Example (cont.)

• Discover hidden symmetry
• Unexpected
• Clearly seen
• Impossible to be found by any form of numerical
  (statistical) analysis
• Unless known beforehand

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Examples

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

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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)

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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

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In 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

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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?

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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
• Michal Chlebiej
• Bartosz Borucki
• Hubert Orlik-Grzesik
• Michal Lyczek