Archive for November, 2007

Notes from the underbelly

Monday, November 26th, 2007

I somtimes kvetch about TAing because of the demands — grading, reviewing the material, and preparing for recitations generally makes a sick mess of my Sundays.

The silver lining is that sometimes I see wonderful work. One of the problems on the set I’m grading now is to assume the solutions to the differential equation y^{\prime\prime} + \left(1+ \frac{1}{x^2}\right) y = 0 have asymptotic approximations of the form y = e^{\pm i x} \left(a_0 + \frac{a_1}{x} + \frac{a_2}{x^2} + \cdots \right) and find the coefficients a_k.

One student’s clever idea was rather simple: instead of applying the chain rule brute force to \left( e^{\pm i x} \right) \cdot \left(a_0 + \frac{a_1}{x} + \frac{a_2}{x^2} + \cdots \right) , he called the negative power series r(x), let \alpha = \pm i and rewrote the differential equation in terms of e^{\alpha x} r(x) , getting r^{\prime\prime}(x) + 2\alpha r^\prime(x) + \frac{1}{x^2} r(x) = 0 ; in addition to being rather fun, this preprocessing step removes as much messy error-prone algebra as possible. As a broader observation, I like the way this guy does/thinks about mathematics in general … we seem to be interested in the same type of math, also.

The second student consistently surprises me with the depth of his answers — maybe it’s the way they teach math in Poland?. Whereas the furthest the other students and myself took the expression for the coefficients a_k is a_0 \frac{(-\alpha)^k}{2^k k!} \prod_{j=1}^k (j^2 - j +1) , he asserted that in fact this implies

a_k = \frac{a_0 (-\alpha)^k}{2^k k ! \pi} \cosh \frac{\sqrt{3} \pi}{2} \Gamma(\omega + k) \Gamma(\overline{\omega} + k)

where \omega = \frac{1+ i 3}{2} . Alas without proof, but I’ll look at it later and figure it out, or ask him tomorrow.

Posted in Mathematics | 2 Comments »

Analysis of non-rigid shapes

Thursday, November 15th, 2007

I attended a talk given by Michael Bronstein yesterday, on the analysis of non-rigid shapes using metric geometry. That may sound a bit abstract, but in fact, it delved into just enough theory to whet the appetite, while emphasizing the fascinating and almost miraculous practial fruits of their techniques.

The Bronsteins (twin brothers) and their advisor Kimmel have been doing some fascinating work in — I guess their research falls under this rubric — computer vision and pattern recognition. Classical techniques for shape analysis: comparing and registering them, for example, assume that they are rigid; there are a few– snakes and some stochastic methods come to mind– which don’t make this assumption, but these tend to not be very robust. The novel approach that this group has come up with identifies an object with its intrinsic geometry rather than its extrinsic– i.e. the geodesic distances between points on a surface is considered more important than the distance in \R^n. The idea is that inelastic deformations will not change these geodesic distances, so they make a more precise and robust fingerprint for object identification.

The problem is how to compare geodesic distances on two candidate objects: if you did this naively, you’d have to look at each possible correspondence of the objects, find one which minimizes the overall difference in geodesic distances, and then use the ’stress’ of this mapping as a measure of the similarity between the objects– this operation is exponential in the number of points considered. Instead, they propose using multidimensional scaling techniques to find a minimally metric distorting embedding of the object into a Euclidean space; the image of the object under this embedding is called the canonical form. To compare two objects, one then compares their canonical forms using classical techniques for rigid shape analysis. Clever trick, and judging from some of their results, incredibly fruitful.

The math involved is a beautiful combination of optimization (for finding the minimally metric distorting embedding) and (metric) geometry. I’ll definitely be reading his book when it comes out.

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Unrealistic, but inspiring

Friday, November 9th, 2007 
       If
       by Rudyard Kipling

       If you can keep your head when all about you
       Are losing theirs and blaming it on you;
       If you can trust yourself when all men doubt you,
       But make allowance for their doubting too;
       If you can wait and not be tired by waiting,
       Or, being lied about, don't deal in lies,
       Or, being hated, don't give way to hating,
       And yet don't look too good, nor talk too wise; 

       If you can dream - and not make dreams your master;
       If you can think - and not make thoughts your aim;
       If you can meet with triumph and disaster
       And treat those two imposters just the same;
       If you can bear to hear the truth you've spoken
       Twisted by knaves to make a trap for fools,
       Or watch the things you gave your life to broken,
       And stoop and build 'em up with worn out tools; 

       If you can make one heap of all your winnings
       And risk it on one turn of pitch-and-toss,
       And lose, and start again at your beginnings
       And never breath a word about your loss;
       If you can force your heart and nerve and sinew
       To serve your turn long after they are gone,
       And so hold on when there is nothing in you
       Except the Will which says to them: "Hold on"; 

       If you can talk with crowds and keep your virtue,
       Or walk with kings - nor lose the common touch;
       If neither foes nor loving friends can hurt you;
       If all men count with you, but none too much;
       If you can fill the unforgiving minute
       With sixty seconds' worth of distance run -
       Yours is the Earth and everything that's in it,
       And - which is more - you'll be a Man, my son.

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