Algorithm 2.

Computing Wasserstein Distance

Input: Two topological spherical surfaces (S1, g1), (S2, g2).

Output: The Wasserstein distance between S1 and S2.

1. Scale and normalize S1 and S2 such that the total area of each surface is 4π.

2. Compute the conformal maps by [21] ${\varphi }_1:S_1\to {\mathbb{S}}^2$ and ${\varphi }_2:S_2\to {\mathbb{S}}^2$, where ${\mathbb{S}}^2$ is the unit  sphere, and ϕ1 and ϕ2 are with normalization conditions: the mass center of the image points  are at the sphere center.

3. Compute the conformal factors λ1 and λ2 by [8]. Construct the measure μ ← e2λ1dA.

4. Discretize ${\mathbb{S}}^2$ into a discrete point set with measure (P, ν), where ν is computed by Eqn. 3.

5. With $({\mathbb{S}}^2,\mu )$ and (P, ν) as inputs of Algorithm 1, we compute the Riemannian Optimal  Mass Transport map.

6. Wasserstein distance between S1 and S2 can be computed by Eqn. 4.