For a sequence $x^k\in R^N$, if we know $x^k$ is uniformly bounded, and the difference $\|x^{k+1}-x^k\|\rightarrow 0$, can we have the convergence of the sequence $x^k$.

This problem arised when I want to prove the convergence of  an algorithm.  It is true that the sequence converges along a subsequence, but the total converge can not be estalished.  The reason is that there might be inifinite number of accumulation points, and the accumulation point doesn’t converge to each other. Thomas gave a very nice example to generate such a sequence. For example, for a first cycle,  we can take points between an interval for example $[0,1]$ in a sample interval $a$, and when we hit the bound, come back in a sample intervale $0.5a$, and repeat. Such a sequene is obviously bounded, and the distance between two sequential points go to zero, while the accumalation points will cover the whole line. And the sequence does not converge!

This is true if we take a sequence like $||x^{k+1}-x^k||=1/k$ along a closed curve,  thus the difference of two sequential points go to zero, but the point doesn’t converge.

Ok, it is good to figure out this !!

http://www.youtube.com/watch?v=fhAFzmnFVqU

http://www.amara.com/current/wavesoft.html

http://taco.poly.edu/WaveletSoftware/index.html

Image sequences  f_ij, i=1:n, j=1:3

After the registration:

(u_1,u_2,u_3) classfications to 4 kinds of dye.

No training, unfortunate.

Clustering?

http://www.cs.technion.ac.il/~elad/publications/journals/2008/SuperResPaper_IEEE_TIP.pdf