After acquiring some knowledge that you didn't have before, it is amazing how you can see new things within something you have seen many times before.
For example, during my first semester at MIT, I was reading the book chapter "Regularization in Image Restoration and Reconstruction" by Clem Karl, when it hit me that the cameraman image was taken at MIT. I had seen that image several times before, but before coming to MIT, I hadn't paid attention to the background.
Something similar happened last week. I talked with Nir Sochen, who was visiting LIDS last Monday and Tuesday, about his and his colleagues' work on curve evolution on manifolds. Then on Wednesday, Biz gave a talk about open surfaces, something which I had heard about from him several times. This time, however, it hit me that his work is essentially curve evolution on manifolds. A shape is to be found that minimizes some functional with the shape as its argument. Curve evolution is basically gradient descent minimization starting from some initial shape. When the shape is restricted to lie on a manifold, then the problem becomes curve evolution on a manifold.
In my meeting with Dr. Sochen, I also told him about the work I will be presenting in a couple of weeks at the IEEE Workshop on Machine Learning for Signal Processing in Cancun. The basic idea is to take a level set/curve evolution formulation usually applied in image segmentation and to apply it in supervised classification problems of machine learning.
The topic he discussed in the LIDS Seminar was not related to curve evolution, but to designing overcomplete dictionaries. For discrete-time signals of length p, a very large dictionary of order p3 atoms is constructed. The mutual coherence of the dictionary is low: on the order of p-1/2. (See slide 16 of this for a definition of mutual coherence.) The atoms of the dictionary are related to the eigenvectors of the harmonic oscillator. Two versions of the paper that give the details have appeared recently; they may be found here and here.
