Topological Data Analysis

Like I mentioned in a previous post, I am taking a Computer Vision (CV) course. One of the things I am interested in, is trying to see if I have something to offer from my mathematics background that might be of use to CV problems. I realize there has been a lot of work / effort put in by many people on these problems, so I am not trying to make it seem as if someone with as little background in CV as myself could solve the issues. However, I am not afraid to put forth some thought on the subject.

One of the things that has attracted me is the idea of how to funnel the data within an image into categories that can be used for recognition / labeling. One of the thoughts that keeps coming into my head is the idea of invariants of immersions of objects from 3D into the plane. Essentially, I would like to look at edges, (hopefully curves (??)) in the 2D image we capture and attempt to find invariants that will help in categorization of items within the image as specific objects. I say invariant because what I mean is that objects in 3D, much of the time depending on angle of projection/immersion, have varied 2D depictions. So, I’m hoping that the use of invariants can shed light on how to better recognize 3D objects in a 2D world…. Are there invariants that are more useful? Are there semi-invariants that are useful to this?

sorry to hack this last paragraph together, but like I said in another blog post, some of these explanations will be short/rough as I lack time and I lack, much of the time, a complete background that is useful to go into long explanations!

SO… with my interest in topology, I came across this Wiki article:

http://en.wikipedia.org/wiki/Topological_data_analysis

Amazing Effort

So, I recently found a link to a supposed human solution to two amazing theorems that previously had computer-based proofs only.

Mathematical Proofs of Two Conjectures: The Four Color Problem and The Uniquely 4-colorable Planar Graph

I find it most interesting that the author, Jin Xu, had his first draft of the paper in 1991; however, due to some fatal flaws in the paper (issues with the proofs), it was clearly not ready. From that time until August of 2009 (yes, that’s 18 years), he worked on the problems and finally came to the “answer.” So, how long was he working on it prior to that 1991 first draft?? That is an amazing amount of effort put forth in the hopes of solving a mathematical “problem” that was supposedly solved via computer in 1976 (first big theorem to be proved with a computer). I wonder if his work would have stopped had the computer proof not come about until mid 1990s and thus impacting his thought process?

Quick one from Godel’s Lost Letters and P = NP blog

http://rjlipton.wordpress.com/2009/11/12/more-on-mathematical-diseases/

Not sure what to say, but go check it out… basically talk on problems; some that seem simple but in fact are extremely complex to solve. There’s quite a intro on John Conway; I’ve read tales on jhc from various places… fascinating person it seems.

BTW: I realize I spelled Godel’s name incorrectly, but I don’t know how to make wordpress like LaTeX… I keep having issues :-(

space debris

problem to solve: dealing with space debris…. how to make it a non-issue, essentially. I have zero background on this, but I would imagine that, based on my security background, we’d want to deal for the worst case scenario. I imagine this to be a cloud of small sized debris since it would be hard to navigate through as well as destroy particular bits with a specific weapon. So, I guess the idea is a blanket force field that is a protective layer? Too bad I have no background in this; seems like a neat thing to think about.

Along these lines of things I have zero background in, but I find fascinating and know there are maths behind the ideas that people are using to solve these problems, I share this:

http://www.ams.org/bull/2009-46-01/S0273-0979-08-01232-9/home.html

It is on the mathematics behind invisibility and was published in AMS bulletin in 2008… long, dense, but good read. I don’t have the background to grasp all of it, but I was able to get some bits out and it was cool. So, if you have a decent college math background, go for it.

Grothendieck

Considering I started down the applied mathematics approach, especially considering I feel I was more a computer scientist (joke?) for a long time than one who did anything math related, I have come to appreciate Alexander Grothendieck. His ability to think abstractly led to much great work in 20th century mathematics. From topology and algebraic geometry to category theory and his thoughts on society, I feel his mark on maths will be felt for a long time to come. Here are some URLs, albeit they’d be easy for you to find, that are pretty much mostly wiki, but are reasons for my interest and fascination with this brilliant man:

Grothendieck’s wiki page
Grothendieck Circle – they try to make published his writings
Grothendieck Universe – a construct to allow for math (set theory)
Grothendieck’s Relative Point of View

From the American Mathematical Society, a 2 part bio:
Part 1
Part 2

There is much much more out there on him; just search if interested…

Learning from One Example

I am currently taking a Computer Vision course and am finding a number of fascinating technological, philosophical, and psychological papers pertaining to all these areas.  One area of interest to me is the idea of how we, as humans, are capable of recognizing objects after seeing them just once.  This is also an area of interest to my professor (Erik Learned-Miller).  Here is a paper by him and others that is a good read.

http://www.cs.umass.edu/~elm/papers/cvpr2000.pdf

Example of Finding Game Theoretic Scenarios

Found via reddit, but this following blog post does some neat analysis of game theory in The Dark Knight (Batman movie). I feel there could be more meat to their blog post, but who the hell am I to say? I guess what I mean is that it could have been more formalized. Perhaps that’s a problem left to the reader.

http://www.brazencareerist.com/2008/08/19/game-theory-in-the-dark-knight-a-critical-review-of-the-opening-scene-spoilers/

Translating

Currently, I am working on translating “Extending Immersions in Codimension 1″ by Valentin Poenaru.  It’s in french and is in a Seminaire Bourbaki book, so we’ll see how it goes.  I am hoping it sheds some light on the work done by Samuel Blank (Northeastern University).

A reasonable introduction to Tensor math

http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/documents/Tensors_TM2002211716.pdf

Change on how this blog is run

Since I clearly have failed to post anything, I attempted to figure out why in order to remedy the situation. It seems that my schedule / life is too busy to be taking the time to write a blog post. The only way for me to blog is if I do quick posts; maybe only post a link with minimal associated information. Hopefully this will encourage longer posts in the future when I have more motivation to take the time to write them.