Computational Topology

This page is hopelessly outdated. Proceed to the front of the new site: Computer Vision Primer. Or you can start here: Computational Topology.

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 Introduction

In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.-- Hermann Weyl 

If it's just turning the crank it's algebra, but if it's got an idea in it, it's topology. -- Solomon Lefschetz

…Я алгеброй гармонию поверил. – А. С. Пушкин, «Моцарт и Сальери»

You wouldn't recognize mathematics if a calculus textbook fell on your head! -- Frustrated Mathematician

Fomenko

Fomenko explains: the picture represents a fibration. The vertical figures are the fibers, the horizontal shape is the base of the fibration.

How many tunnels does this protein on the right have? How many voids? What about that piece of an alloy?  How do you teach a computer to distinguish between letters C and O, O and P, P and B? These and related questions are properly addressed via the new discipline called Computational Topology. It is the result of recent attempts to apply such a theoretical discipline as algebraic topology in the areas of practical importance.

Topics of possible students' projects 

  • Image analysis and recognition;
  • Visual image search;
  • Structure of proteins;
  • Topology of data structures;
  • Topology of networks;
  • Computational topology and geometry.

See also  Computer Vision Primer.

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Take a look at a couple of patent applications published recently by the Patent and Trademark Office. Both apply pure mathematics in imaging. The first is US Patent Application No. 20050232511 “Image model based on n-pixels and defined in algebraic topology, and applications thereof” published last October. The inventors are three Canadian topologists. The second is US Patent Application No. 20060013505 “Analysis of geometric surfaces by conformal structure”  published in January. One of the inventors is a Field medalist from Harvard. The patents have not been awarded yet, but it is clear that things are happening… 

And this is my patent: Topology based method of partition, analysis, and simplification of dynamical images and its applications. Abstract: A method of partition, analysis, and simplification of images has been developed. Still images are partitioned into collections of connected regions. Then, the image is simplified by removing regions that are small in terms of their sizes  or contrasts. The applications are image enhancement, computer vision, surface and curve reconstruction, scientific image analysis, image recognition and matching. The algorithm is also designed to conduct motion tracking and analysis of sequences of images.  The result is real time identification of objects in each frame, following each object from frame to frame, computation of locations and velocities of these objects, as well as their measurements (areas, volumes, moments, etc). A part of PASS has been implemented as Pixcavator (there is also a Software Development Kit).

There is a number of overlapping names and areas of research related to this discipline:

  • computational topology and geometry,
  • digital topology and geometry,
  • finite topology,
  • algorithmic topology,
  • image analysis,
  • intrinsic image analysis,
  • combinatorial image analysis,
  • automatic image analysis,
  • machine vision,
  • computer vision,
  • low level vision,
  • infoimaging,
  • pattern recognition,
  • ...

None is quite appropriate for what I have in mind...

It's not too much of a stretch to translate "topology" as "image analysis". As long as you study the image, not what's being imaged...

 

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