I’ve been all over the place, mathematically speaking, this past week. Not that that’s unusual.
The new book – bought used thru ABEbooks, but new to me – about computer vision came in last Saturday afternoon. I looked through the entire thing as soon as it arrived. It’s “introductory techniques for 3-D computer vision” by Trucco and Verri, Prentice-Hall, 1998. It is far less specialized than the previous 2 I talked about, and provides a nice overview – with supporting detail! – of the things we need to do or can do in the field.
I suspect that if I want to do computer vision, this is where I will look 1st. If I want, instead, to do projective geometry, then I will look at the projective geometry book… it’s significant that I think of it as that, rather than as computer vision (Hartley and Zisserman’s “Multiple View Geometry in computer vision, Cambridge, 2003).
My alter ego the kid has been looking at Donald Greenspan’s “Particle Modeling”, Birkhauser, 1997. The original approach to the numerical solution of partial differential equations was “finite differences”, in which we replace derivatives by approximations. The next approach was “the finite element method”, in which we combine dimensions and solve approximations on (usually, I think) triangular elements.
Nowadays, it is feasible to replace our continuum by a collection of particles, and solve for their motion. This book begins with several chapter-length much-simplified examples – and they do look fascinating. No, I haven’t even thought about programming them in Mathematica yet… but there is Fortran code in the back of the book so that I can see the details I’ll need to work out. (And that’s a job for Mr. Belvedere… the kid is chasing butterflies, not catching them.)
The kid has also been looking at Baez and Munian’s “Gauge Fields, Knots and Gravity”, world scientific press, 1994. I had heard that a 2nd edition was in the works… but a quick check on Amazon does not show one.
Anyway, he uses bundles and connections – advanced differential geometry – but since the book is introductory (well, from where I stand), it looks like a good place for me to start.
In addition, the other kids – my alter egos the undergraduate and the graduate student – have continued working on trusses and non-unique-factorization-domains. I think this Monday’s post will be
the Howe truss again, but a Pratt truss with a load of snow on the top. Then next week I’ll compare it to two other trusses.
(See what I mean about alter egos? The undergraduate knows what he’s supposed to do, even if Rip and The Diarist don’t.)
My busiest alter ego is the poor guy who is responsible for general time series analysis (including regression and wavelets), differential geometry, quantum mechanics, control theory, dynamical systems, and color. (Not that you’ve been keeping track, but the Big Five became the Big Six.) I haven’t figured out if he needs designated multiple personalities too, or just designated multiple hands. Or maybe I just need to limit him to two subjects max at any one time. In any case, working a little bit at a time, he has brought a couple of frequency analysis posts and the latest multi-collinearity post through stage IV – that is, the narrative on top of the mathematics is done, and all that remains is to implement them on the blog. (And all those count as one subject, time series.)
That I have made such good progress on multiple fronts is why I am delighted with my organization of work into multiple personalities. It’s seems to be an effective way of keeping track of things, and especially useful for making progress in small steps.
He has also been looking at the Higgs mechanism (a second subject) – which imparts mass to other fundamental particles, and which is why we care about whether the large hadron collider has found it. But, as with the other things, he’s been working a little bit at a time, so I haven’t yet worked out the calculations for the Higgs mechanism itself. I’m still stuck on a calculation that yields a massless Goldstone boson.
FYI, that poor guy is named Jimmie. I’ll tell you that he’s named for a TV character… but I’ll be stunned if anyone figures out which one.
Jimmie is also thinking about picking up control theory. Oh, that’s a third subject. But his interest in the Higgs particle will be over soon, one way or another. (Let’s see… a third eye instead of a third hand?)
But I am intimidated by control theory. Lord knows I’ve jumped into enough other things without knowing their place in the scheme of things… perhaps at least knowing what comes next, but not knowing what comes after that.
In statics, for example, I am quite happy to be working plane determinate truss problems, in which we assume the beams are completely undistorted… knowing that in the real world one must cope with distortions of the structure, and work in 3 dimensions, and investigate at the joints, and…. I’m just having fun solving the simplest possible models, knowing that that’s all I’m doing.
But for control theory… I hesitate to just have fun solving the simplest possible models. Well, I’m going to try to overcome that hesitation.
And with that, I need some lunch and I think the undergraduate needs to take this Monday’s post thru stage V, all put together and awaiting only final edits.