ChapterZero

One of the reasons I’m so fascinated with image processing is the fact that you easily get direct feedback about the usefulness of your analysis. Among other things, this means that papers in this area are fun for browsing: even if you can’t follow the analysis, you can look at the pictures and get a sense of how worthwhile the results are. And if the papers are written well, you can duplicate the results. Here are links to three particularly interesting papers: Isophote-based Interpolation, Image Processing via the Beltrami operator, A General Framework for Low Level Vision.

The first presents the idea of interpolating using curvature driven evolution of isophotes (curves of constant image intensity), to preserve the geometry of the image; the results for the given test images– given as comparisons with bicubic, nearest neighbor, and linear interpolation– look encouraging. An interesting question is whether there is some analogous idea for multichannel images.

The remaining two papers present a geometric approach to scale-space image processing based on the Polyakov action from high energy physics. The idea is to represent images as surfaces and the processing of them as a minimization of the surface area; this approach has several advantages. In particular, it applies as directly to vector images as it does to scalar images. The second paper has two beautiful examples of this approach used on color images, but doesn’t justify the use of the Polyakov action and surface area minimization approach. The third paper goes into the details of the mathematics.

I came across this problem from a Putnam exam: Let be a finite group of matrices in such that . Show that .

I haven’t been able to solve it in three days of sporadic efforts, but I’m slowly getting a handle on this: I’ve convinced myself that the are diagonalizable with only in their spectrum.

Solution Sweet! Lieven from Neverendingbooks came up with this neat proof: Note that is a projection onto its eigenspace– this follows from the basic group theoretical fact that . We know that the dimension of the range space of a projection operator is the trace of the operator, so voila!

I already have my two front teeth, but an OC-255 connection would make for a very special Christmas. A T3 connection is acceptable also.

You get … uh … something good, if you can identify the book from which I culled these quotes:

Why, I can smile, and murder whiles I smile
And cry ‘Content’ to that which grieves my heart,
And wet my cheeks with artificial tears,
And frame my face to all occasions.

I can add colours to the chameleon,
Change shapes with Proteus for advantages,
And set the murderous Machiavel to school.

Hints: it was published in the last two years, and the first chapter is titled “The boy who stole too much.”

It bothers me that getting a random orthonormal basis in Matlab is as simple as onb = orth(rand(n)). Why does this work? Equivalently, why, if you construct a matrix in with entries chosen i.i.d uniformly in the interval , is this matrix with high probability invertible? Slightly more general: if vectors are constructed in the same manner, what is the probability that the resulting collection of vectors is independent?

via The Art of Problem Solving, let’s play a (two-person) game.

Start with a 3×3 zero matrix, and at their turn each player changes one of the zero entries to a number between 1-9. The first player’s goal is to make the determinant positive, while the second’s is to make it negative. Numbers may be used only once.

Play it a couple of times. Now analyze it!

According to my personal classification system– either analytic-flavored or algebraic-flavored– most math blogging is algebraic in nature. Is that because those flavors of math are more accessible, or interesting, or are analysts just not the type of people to blog? Perhaps they’re too busy

I’m planning a series of posts on optimization, a rather practical analytic-flavored topic. As I’ve discovered this term, efficient, and general convex optimization algorithms are surprisingly easy to understand, so I’m going to cover the basics of convexity and convex optimization, duality theory, unconstrained optimization, and then interior point methods.

Via Ars Mathematica, I came across a paper describing TeX’s math layout algorithm and offering an implementation of it in Standard ML. Sweet! I only wish I had come across this paper (it is about 10 years old) when I was daydreaming about a WYSIWYG gui interface that worked for multiple CASes. If I ever get back around to it, this will be really useful.