Gyrification analysis using conformal module (by Ronald)

After doing some literature review, we found that researchers are commonly interested in understanding the gyrification pattern at different portion. There are commonly accepted partition of the brain surface as below:














We have 40 healthy and 40 unhealthy (Williams Symdrome) subjects. Gyrification usually happens for WS patients. We would like to study its pattern for each partition. Landmarks has been labeled. We have identified the right landmarks corresponding to the boundaries of the partition. They are labeled with landmark label # as shown in the figure.

We test the partition on L1608.m (Control):












We parameterize each partition conformally onto rectangles using Yamabe flow. Results show that each portion is mapped to different rectangles with different dimensions. We can use the conformal module to study the gyrification pattern:


















To compare the shape distance for each portion between different subjects, we can use the Teichmuller shape distance:



















Experimental results is as follow:

Research Update (by Alvin)

Found 2 resources for solving Poisson equation on unstructured mesh:

DistMesh - A Simple Mesh Generator in MATLAB
Poisson's Equation by the FEM using a MATLAB Mesh Generator

We will look into details of these tools and use them in our paper (to be finished in August).

Research discussion summary (by Ronald)

David, Alvin and I have had several fruitful research discussions through msn last few days. Here is a brief summary.

We're looking for good practical applications for the tools we've developed (Most reviewers commented that our works should be supported by real applications). We've proposed a possible practical application of our "Beltrami Holomorphic flow" method and "Teichmuller extremal map" method.

Basically, the problems we want to solve are:
1. Given two simply-connected open surface S1 and S2. Suppose we know the boundary correspondence between the two surfaces (but we know nothing about the interior: i.e. no registration is done in the interior). How can we define a measure to describe their shape difference?

2. In real situation, we might not know the correspondence for every points on the boundaries. Suppose we only know the correspondence for 2n points on each boundaries of S1 and S2. Again, we do not know the registration in the interior. How can we define a measure to describe their shape difference?

3. Now, we consider a more general situation. Suppose we have two genus zero closed surfaces M1 and M2. Given no information about the correspondence between M1 and M2. How can we define the best measure to describe their shape difference?

We've proposed the following methods to solve each problem:

For 1:
We map S1 and S2 conformally to unit disks D, using Yamabe flow. The boundary correspondence between S1 and S2 gives us a map from circle to circle. Denote it by f: S^1 -> S^1. Extend f to a harmonic map F: D -> D. F can be regarded as the best optimized conformal map that agree with boundary correspondence. F is conformal if S1 and S2 are conformally equivalent. We define the shape index as:
Shape(S1, S2) = \int_D |\mu_F| + a\int_D |H1 - H2(F)| + b\int_D |K1 - K2(F)|
where: \mu_F is the Beltrami coefficient of F; Hi = mean curvature; Ki = Gaussian curvature; a and b are the rescaling parameters.
We can prove that (S1,S2) has the same geometry (same shape up to rigid motion) if and only if Shape(S1,S2) = 0.

For 2:
Again, we map S1 and S2 conformally to unit disks D, using Yamabe flow. The 2n points correspondence between S1 and S2 give us the 2n points correspondence between the circle. We can apply our Teichmuller extremal map method to find an extremal map F:D -> D that matches the 2n points. Define:
Shape(S1, S2) = k + a\int_D |H1 - H2(F)| + b\int_D |K1 - K2(F)|
where: k is the Beltrami coefficient of F; Hi = mean curvature; Ki = Gaussian curvature; a and b are the rescaling parameters.
We can prove that (S1,S2) has the same geometry (same shape up to rigid motion) if and only if Shape(S1,S2) = 0.

For 3:
We map M1 and M2 conformally to unit sphere. We consider the following minimization problem:
E(\mu, T) = \int_S^2 |\mu| + a\int_S^2 |H1(T) - H2(F_mu)| + b\int_S^2 |K1(T) - K2(F_mu)|
where F_mu = quasiconformal map associated with mu; T is a Mobius transformation
Define the shape index as: Shape(M1, M2) = Min E(\mu, T).
We will need to prove that:
(S1,S2) has the same geometry (same shape up to rigid motion) if and only if Shape(S1,S2) = 0.

First Post (by Alvin)

Hello, this is Alvin. This is my first post here. Research has been slow recently while the preparation of the ATC slides is ongoing, but I will be sure to keep up the efficiency, as the time is running tight now! A very nice beamer template has been found, so the only and major part remains is to fill in the details and figures of our projects. The ATC Plan should make a nice framework for the presentation.

Update: Hippocampal Shape analysis (by Ronald)

I have been doing literature review on the structure of hippocampus. I've found that the most common approach for shape analysis on hippocampal surface for Alzhemier disease is to partition the surface into several parts, namely, LZ, SZ, IMZ. See figure.
I am trying to partition the hippocampus surface accordingly on the template. Using surface registration, I am going to partition all the hippocampus surfaces. The goal is to study the atrophy at different parts due to Alzhemier Disease.

Our group has developed a nice algorithm to do shape analysis using hyperbolic ricci flow and pant decomposition. The partition may possibly help us to modify the topology of the surface so that we can use hyperbolic geometry to study the shape. I am thinking a consistent algorithm for topological modification.

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.

------------------------------------------
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.

-----------------------------------------
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

----------------------------------------
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.

---------------------------------------
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.

-------------------------------------
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.

First Blog (by Ronald)

HiHi. This is our first research blog. Our research team consists of several people: Tony Chan, S-T Yau, Paul Thompson, David Gu, Ronald Lui, Alvin Wong and Wei Zeng.

We are going to keep a blog of our research progress daily. Hopefully we can be more productive! The blog is mainly maintained by Alvin and me!

Hey hello Alvin!! Why don't you write your first blog and say Hi!!