Teichmüller Shape Descriptor and Its Application to Alzheimer’s Disease Study

Abstract

We propose a novel method to apply Teichmüller space theory to study the signature of a family of nonintersecting closed 3D curves on a general genus zero closed surface. Our algorithm provides an efficient method to encode both global surface and local contour shape information. The signature—Teichmüller shape descriptor—is computed by surface Ricci flow method, which is equivalent to solving an elliptic partial differential equation on surfaces and is numerically stable. We propose to apply the new signature to analyze abnormalities in brain cortical morphometry. Experimental results with 3D MRI data from Alzheimer’s disease neuroimaging initiative (ADNI) dataset [152 healthy control subjects versus 169 Alzheimer’s disease (AD) patients] demonstrate the effectiveness of our method and illustrate its potential as a novel surface-based cortical morphometry measurement in AD research.

Appendix: Proof of Theorem 4

Proof

See Fig. 10. In the left frame, a family of planar smooth curves $\Gamma =\{\gamma _0, \ldots , \gamma _5\}$ divide the plane to segments $\{\Omega _0, \Omega _1, \ldots , \Omega _6\}$, where $\Omega _0$ contains the $\infty$ point. We represent the segments and the curves as a tree in the second frame, where each node represents a segment $\Omega _k$, each link represents a curve $\gamma _i$. If $\Omega _j$ is included by $\Omega _i$, and $\Omega _i$ and $\Omega _j$ shares a curve $\gamma _k$, then the link $\gamma _k$ in the tree connects $\Omega _j$ to $\Omega _i$, denoted as $\gamma _k:\Omega _i \rightarrow \Omega _j$. In the third frame, each segment $\Omega _k$ is mapped conformally to a circle domain $D_k$ by $\Phi _k$. The signature for each closed curve $\gamma _k$ is computed $f_{ij} = \Phi _i\circ \Phi _j^{-1}|_{\gamma _k}$, where $\gamma _k: \Omega _i\rightarrow \Omega _j$ in the tree. In the last frame, we construct a Riemann sphere by gluing circle domains $D_k$’s using $f_{ij}$’s in the following way. The gluing process is of bottom up. We first glue the leaf nodes to their fathers. Let $\gamma _k: D_i \rightarrow D_j, D_j$ be a leaf of the tree. For each point $z=re^{i\theta }$ in $D_j$, the extension map is

$$\begin{aligned} G_{ij} ( re^{i \theta } ) = r e^{f_{ij}(\theta )}. \end{aligned}$$

(15)

We denote the image of $D_j$ under $G_{ij}$ as $S_j$. Then we glue $S_j$ with $D_i$. By repeating this gluing procedure bottom up, we glue all leafs to their fathers. Then we prune all leaves from the tree, and glue all the leaves of the new tree, and prune again. By repeating this procedure, eventually, we get a tree with only the root node, then we get a Riemann sphere, denoted as $S$. Each circle domain $D_k$ is mapped to a segment $S_k$ in the last frame, by a sequence of extension maps. Suppose $D_k$ is a circle domain, a path from the root $D_0$ to $D_k$ is $\{i_0=0, i_1, i_2, \ldots , i_n=k\}$, then the map from $G_k: D_k\rightarrow S_k$ is given by:

$$\begin{aligned} G_k = G_{i_0i_1}\circ G_{i_1i_2}\circ \ldots \circ G_{i_{n-1}i_n}. \end{aligned}$$

(16)

Note that, $G_0$ is identity. Then the Beltrami coefficient of $G_k^{-1}: S_k \rightarrow D_k$ can be directly computed, denoted as $\mu _k: S_k \rightarrow \mathbb C$. The composition $\Phi _k \circ G_k^{-1} : S_k \rightarrow \Omega _k$ maps $S_k$ to $\Omega _k$, because $\Phi _k$ is conformal, therefore the Beltrami coefficient of $\Phi _k \circ G_k^{-1}$ equals to $\mu _k$.

Fig. 10

Proof for the main theorem, the signature uniquely determines the family of closed curves unique up to a Möbius transformation

We want to find a map from the Riemann sphere $S$ to the original Riemann sphere $\Omega , \Phi : S\rightarrow \Omega$. The Beltrami-coefficient $\mu : S \rightarrow \mathbb C$ is the union of $\mu _k$’s each segments: $\mu (z) = \mu _k(z), \forall z \in S_k$. The solution exists and is unique up to a Möbius transformation according to Quasi-conformal Mapping theorem (Gardiner and Lakic 2000). $\square$

Note that, the discrete computational method is more direct without explicitly solving the Beltrami equation. From the Beltrami coefficient $\mu$, one can deform the conformal structure of $S_k$ to that of $\Omega _k$, under the conformal structures of $\Omega _k, \Phi :S\rightarrow \Omega$ becomes a conformal mapping. The conformal structure of $\Omega _k$ is equivalent to that of $D_k$, therefore, one can use the conformal structure of $D_k$ directly. In discrete case, the conformal structure is represented as the angle structure. Therefore in our algorithm, we copy the angle structures of $D_k$’s to $S$, and compute the conformal map $\Phi$ directly.