So I was looking for an old video in numberphile (one about a particular kind of prime) and re-saw this
https://www.youtube.com/watch?v=yGsIw8LHXM8 (two square theorem)
or if you like the original video that one comes from
https://www.youtube.com/watch?v=SyJlRUBoVp0 The Prime Problem with a One Sentence Proof )
Anyway I found myself rewatching it. and decide to see if I can make a simpler version of the proof. not less space mind you, simpler as in more straight forward. I think I've done that here (it's probably not new, most things in math aren't) but regardless I submit it for your entertainment/utility.
Also it's worth saying that the you can then figure out exactly what form the X and Y (K or what have you) need to have by just unraveling all that. Here is that little bit of extra formula stuffs spelled out.
Well, weeks and weeks of working on the cancer contest have brought be back to where i started from. I want to use Dynamic time warping to match images. Once the images are matched, I think maybe look at a difference between the original and the target images to see what is left. This is probably your best place to start looking for cancer.
So why don't i do this? Because the run time is abhorrent. For the single comparison of two images I think Naively the Big O notation is like N^4 . N^2 is linear DTW but you can't just add a dimension and go up by 1 power. if i understand it right you have to add 2 to properly do 2d matching. Where N is the number of pixels in the image. So really it's like yeah. bad.
Maybe there is a way it can be done in N^3 and I'm missing something, but really it needs to be done in something like linear or at least N*log(N) time to really work. So that's where I'm leaving it.
There is a cervical cancer contest out there that is very similar except the photos are from some sort of normal optical camera and while maybe it could be done the same way, it has the same problem. I think if we solve this problem the world will have much much better analysis systems (in general).
It's worth mentioning I think most people do their analysis using deep neural networks. Quite honestly I'm not sure how they would do a good job processing 2-d image data but apparently it does work. I've got 3 weeks before the contest is over. if i can come up with a good way to do the DTW I will, otherwise I'm throwing in the towel on this one :( .