Monday, February 21, 2011

Bigfoot update: Skeleton semantics, Footplant WIP.

I took some time today to work on Bigfoot (my animation research testbed) a bit.
The first thing I added was not animation related, my UI library did not support multi-sampling when rendering 3D to textures, now it does, and all my skeletons look happier.

On the animation side, I added some skeleton semantic detection code. Before, the skeletons were only analyzed for branches, chains of bones with one child only. Now it also find symmetries between branches, in the beginning of the video you can see how it detects that the left and right limbs are symmetrical. The next step is to give it some human skeleton knowledge, so that it automatically figures out what is a head, a foot, a leg, etc... The point of this is that it would enable running the code on large mocap databases without the need for human annotation for purposes like machine learning.

The other new feature which is still very much work in progress is footplant detection. While seemingly innocent, it can be quite tricky to get this right. Mocap is noisy and I also want to support the more general case of 'support contact', where for an animation of an athlete hanging on a bar per example, the contact points with the bar would be detected, or for an unrealistic animation of a martial arts kick after taking a few steps on a vertical wall, the steps on the wall would be registered as well. This needs a different technique than simply foot height. I am researching this slowly when I find myself needing a break from Mathematics and want to do something instantly gratifying.

In the video, green spheres are generated when there is a local minimum in joint height, blue ones when there is a local minimum in joint velocity and white for both.
You can see lots of them firing during footplants. I tried to filter the signals and that did improve the detection, but this is only the beginning, it needs to get much better.

Saturday, February 19, 2011

Friday, February 18, 2011

Is this 'limit replacement' property trivially true?

I have lately been bumping calculus proofs where this property of limits would be really useful, but I am not sure it is true even though it seems trivially true. I have just formulated it but not tried to prove it yet, I will try using the formal definition of limits. Comments welcome.

Thursday, February 17, 2011

Microscopically intuitive FTC#1, take 2.

Last week, I added a small paragraph at the bottom of my Fundamental Theorem of Calculus proof attempt, trying to cast an intuitive view on the theorem,
based on my observation that, when focusing on one tiny interval, proving the FTC and understanding it is intuitive.
The paragraph was rushed and did not contain a much needed figure, therefore when I showed it to Tom he could not make sense out of it although I thought it was a really nice insight.
While analyzing some FTC proofs, I became even more aware of how useful this intuition was.
I also become aware that some proofs simply require one to work out each detail of their intuition rigorously and patiently and not much more.
I decided to take the time to polish my insight, and see if I can manage to explain it better.
One thing I realized is that it takes it quite an amount text to explain even the simplest ideas if ones wants to do it right...

Limit over an interval

We are analyzing several FTC proofs to gain some insights. For now it seems all of them need analysis to be stated with enough detail to be convincing.
In this proof, I stumbled upon an assumption that can be reduced to claiming that this statement is true:

As usual, this is intuitively very true, the interval vanishes, leaving the 'sup' to act on only 'one point' if f is continuous. But that is no proof.

I have tried to detail this a bit more to see if I can prove it, the main idea behind my proof is: Courage.
I have found courage to be an essential component across many proofs and bold inventions in mathematics.
I am not sure how good it is, it feels pretty convincing, but there are 2 spots where it needs more detail, and I suspect that for these spots, there is an inescapable need for analysis (luckily we will be tackling that in the foreseeable future).


Sunday, February 13, 2011

Nested limits technicality

The last two weeks, we have been dealing with the Fundamental theorem of calculus and it's proofs. Both me and Tom created proofs that hang on annoying technicalities and because of that they do not hold.
One of the problems would be solved if we could prove an innocent statement about nested limits.
I have proven it in the following document, but this proof only holds if f is continuous, which is not the case in our proofs, I will look some more.

The proof does seems trivial, but we are realizing more and more the importance of the tiniest details in the relationships between continuousness, differentiability, integrability, and it is not always clear, specially in elementary calculus books where proofs are given a flimsy and vague treatment, we seem to be heading straight into analysis whether we like it or not (and we do!!)

Here is the same tiny proof as a pdf:

Friday, February 4, 2011

Fundemental Theoreom of Calculus version 1

I managed a very 'weak' proof of FTC1, still it was fun and will come in handy when I see better proofs and how they solved the parts that I treated with too little formality.

I also included in a second part, yet another intuitive perspective on the theorem, using simple infinitesimal algebra.