Research update (by Ronald)

We have decided how to deal with the submission of the "Beltrami Holomorphic Flow" paper. We are going to separate this long paper into several small papers. The reason is to show the motivation and contribution clearly in separate papers, instead of putting them all in one paper.
Here is what we have discussed:
1. Paper will be divided into 3 short papers:

• a. Optimization of surface diffeomorphism;
• b. Solving Beltrami equation;
• c. A novel method for computing Riemann maps.

2. Ronald will be responsible for paper a. Alvin will be responsible for paper c. We will later discuss paper b further.

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As for the Alzhemier disease analysis, we have gathered all the data from LONI. Basically, we have: set of NC, MCI and ADNI hippocampus surfaces measured at year 0 (T1) and year 1 (T2). We have decided to do the following experiments:
1. Determine the structural difference between NC, MCI and ADNI using conformal factor and mean curvature. We will use ICA for systematic data analysis.
2. Using Beltrami coefficient to measure the deformation from T1 to T2. The goal is to detect region of deformation due to Alzhemier disease. And hopefully we can determine which MCI surfaces would develop into AD.

Wenlu (harvard medical school collaborator) is processing the medical data from Harvard medical school. Once the hippocampus are all segmented out, we will have 1000 NC, MCI and ADNI measured at four different times. We can make use of these data for more convincing experiements.

Alvin is working on the converter.

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The numerical Yamabe flow paper is almost done. The numerical proof is ready (but need careful checking). We need numerical experiments to validate. We will have to set deadlines for this project soon. Basically, we need to do the following numerical experiments:
1. Synthetic surface 1a (with known conformal parameter) onto rectangle;
2. Synthetic surface 2a (with known conformal parameter) onto rectangle;
3. Synthetic surface 1b (with known conformal parameter) onto disk;
4. Synthetic surface 2b (with known conformal parameter) onto disk;
5. Real brain surface 1a (with known conformal parameter) onto rectangle;
6. Real human face surface 2a (with known conformal parameter) onto rectangle;
7. Real brain surface 1a (with known conformal parameter) onto disk;
8. Real human face surface 2a (with known conformal parameter) onto disk;

*We have to check the convergence of \lambda, g_ij and angle and determine its order of convergence.

Alvin is working on the numerical Quasiconformal Yamabe flow

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The Teichmuller extremal map project is still at its preliminary stage. We are hoping to finish a paper draft before the summer. We are preparing a ppt illustrating the idea right now. We will have to set a calander for this project soon.

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Hyperbolic Ricci flow
We are designing experiment for this new method for shape analysis. The basic idea is to cut the surface along important landmarks so that it becomes multiply-connected surface. We can then decompose the surface into pants and determine the unique hyperbolic geodesic for shape descriptor. We might consider applying this method to WS brain and AD hippocampus.

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The all level set representation for branching curve is still at a very preliminary stage. The basic idea is to allow the gradient of the level set function equal to 0 at some location (with measure 0).
I am preparing a ppt illustrating the basic idea. We will have a plan for this project soon.