Classification Using Wasserstein Distance for Brain Morphometry Analysis

, which is not necessary in our method. The elastic shape metric assumes two shapes are isotopic. However, the proposed intrinsic method is applicable for general Riemannian manifolds. Our approach solely depends on Riemannian metrics and is invariant under rigid motions and scalings such that it intrinsically measures shape distance, and thus more effective and efficient for shape classification and brain morphology analysis.

3 Theory

This section briefly introduces the theoretic foundation, for thorough treatments, we refer readers to [9] for conformal geometry, and [6, 15] for optimal mass transport theory.

3.1 Conformal Mapping

Suppose S is a topological surface, a Riemannian metric $\mathbf {g}$ on S is a family of inner products on the tangent planes. Locally, a metric tensor is represented as a positive definite matrix $\left( g_{ij}\right)$ . Let $\varphi : (S_1,\mathbf {g}_1)\rightarrow (S_2,\mathbf {g}_2)$ be a diffeomorphic map between two Riemannian surfaces, the pull back metric on the source induced by $\varphi$ is $\varphi ^*\mathbf {g}_2 = J^T \mathbf {g}_2 J$ , where J is the Jacobian matrix of $\varphi$ . The mapping is conformal or angle-preserving, if the pull back metric and the original metric differ by a scalar function, $\varphi ^*\mathbf {g}_2 = e^{2\lambda }\mathbf {g}_1$ , where $\lambda : S_1\rightarrow \mathbb {R}$ is called the conformal factor.

Hamilton’s surface Ricci flow conformably deforms the Riemannian metric proprotional to the curvature, such that the curvature evolves according to a non-linear heat diffusion process, and becomes constant everywhere.

Definition 1

(Surface Ricci Flow). The normalized surface Ricci flow is defined as

$\begin{aligned} \frac{dg_{ij}(p,t)}{dt} = \left( \frac{4\pi \chi (S)}{A(0)} - 2K(p,t) \right) g_{ij}(p,t), \end{aligned}$

where $\chi (S)$ is the Euler characteristic number of the surface, A(0) is the total area at the initial time.

Theorem 1

(Uniformization). Suppose $(S,\mathbf {g})$ is a closed compact Riemannian surface with genus g, then there is a conformal factor function $\lambda : S\rightarrow \mathbb {R}$ , such that the conformal metric $e^{2\lambda }\mathbf {g}$ induces constant Gaussian curvature. Depending on the $\chi (S)$ is positive, zero or negative, the const is $$+1$$

, 0 or

In the current work, we apply surface Ricci flow to deform the human cortical surface to the unit sphere.

3.2 Optimal Mass Transport

The Optimal mass transportation problem was first raised by Monge [5] in the 18th century. Suppose $(S,\mathbf {g})$ is a Riemannian manifold with a metric $\mathbf {g}$ . Let $\mu$ and $\nu$ be two probability measures on S with the same total mass $\int _S d\mu = \int _S d\nu$ , $\varphi :S\rightarrow S$ be a diffeomorphism, the pull back measure induced by $\nu$ is $\varphi ^*\nu = det(J)\nu \circ \varphi$ . The mapping is called measure preserving, if the pull back measure equals to the initial measure, $\varphi ^*\nu = \mu$ . The transportation cost of $\varphi$ is defined as

$\begin{aligned} \mathcal {C}(\varphi ):=\int _S d_{\mathbf {g}}^2(p,\varphi (p)) d\mu (p). \end{aligned}$

(1)

The optimal mass transportation problem is to find the measure preserving mapping, which minimizes the transportation cost,

$\begin{aligned} \begin{array}{c} \mathop {\min }\nolimits _{\varphi } \mathcal {C}(\varphi )\\ s.t.~\varphi :S \rightarrow S, ~\varphi ^*\nu = \mu \end{array} \end{aligned}$

In the 1940s, Kantorovich introduced the relaxation of Monge’s problem and solved it using linear programming method [15].

Theorem 2

(Kantorovich). Suppose $(M,\mathbf {g})$ is a Riemannian manifold, probability measures $\mu$ and $\nu$ have the same total mass, $\mu$ is absolutely continuous, $\nu$ has finite second moment, the cost function is the squared geodesic distance, then the optimal mass transportation map exists and is unique.

If $S \subset \mathbb {R}^n$ is a convex domain in the Euclidean space, then Brenier proved the following theorem.

Theorem 3

(Brenier). There is a convex function $u:S\rightarrow \mathbb {R}$ , the optimal map is given by the gradient map $p\rightarrow \nabla u(p)$ .

Solving the optimal transportation problem is equivalent to solve the following Monge-Amperé equation,

$\begin{aligned} det \left( \frac{\partial ^2 u}{\partial x_i\partial x_j} \right) \nu \circ \nabla u(\mathbf {x}) = \mu (\mathbf {x}). \end{aligned}$

3.3 Wasserstein Metric Space

Suppose $(S,\mathbf {g})$ is a Riemannian manifold with a Riemannian metric $\mathbf {g}$ .

Definition 2

(Wasserstein Space). For $p\ge 1$ , let $\mathcal {P}_p(S)$ denote the space of all probability measures $\mu$ on M with finite $p^{th}$ moment, for some $x_0\in S$ , $\int _S d(x,x_0)^p d\mu (x) < +\infty ,$ where d is the geodesic distance induced by $\mathbf {g}$ .

Given two probability $\mu$ and $\nu$ in $\mathcal {P}_p$ , the Wasserstein distance between them is defined as the transportation cost induced by the optimal transportation map $\varphi :S\rightarrow S$ ,

$\begin{aligned} W_p(\mu ,\nu ) := \inf _{\varphi ^*\nu = \mu } \left( \int _M d_{\mathbf {g}}^p(x,\varphi (x)) d\mu (x) \right) ^{\frac{1}{p}}. \end{aligned}$

The following theorem plays a fundamental role for the current work

Theorem 4

The Wasserstein distance $$W_p$$

is a Riemannian metric of the Wasserstein space $\mathcal {P}_p(S)$ .

Detailed proof can be found in [28].

3.4 Discrete Optimal Mass Transport

Let $P=\{p_1,p_2,\dots ,p_k\}$ be a discrete point set on S, $\mathbf {h}=\{h_1,h_2,\dots ,h_k\}$ be the weight vector.

Definition 3

(Geodesic Power Voronoi Diagram). Given the point set P and the weight $\mathbf {h}$ , the geodesic power voronoi diagram induced by $(P,\mathbf {h})$ is a cell decomposition of the manifold $(S,\mathbf {g})$ , such that the cell associated with $$p_i$$

is given by

$\begin{aligned} W_i:=\{q\in S | d_\mathbf {g}^2(p_i,q) - h_i \le d_\mathbf {g}^2(p_j,q) - h_j \}. \end{aligned}$

Theorem 5

(Discrete Optimal Mass Transportation Map). Given a Riemannian manifold $(S,\mathbf {g})$ , two probability measures $\mu$ and $\nu$ are of the same total mass. $\nu$ is a Dirac measure, with discrete point set support $P=\{p_1,p_2,\cdots ,p_k\}$ , $\nu (p_i)=\nu _i$ . There exists a weight $\mathbf {h}=\{h_1,h_2,\cdots ,h_k\}$ , unique up to a constant, the geodesic power Voronoi diagram induced by $(P,\mathbf {h})$ gives the optimal mass transportation map,

$\begin{aligned} \varphi : W_i \rightarrow p_i, i = 1, 2,\cdots , k, \end{aligned}$