Algorithm 1.

Riemannian Optimal Mass Transport Map

Input: A triangle mesh M, measure μ and Dirac measure {(p1, ν1), (p2, ν2), ⋯ , (pk, νk)},  ${\int }_{M}u\left(p\right)\mathit{dp}={\sum }_{i=1}^k{\nu }_i$; a threshold ϵ.

Output: The unique discrete Optimal Mass Transport Map φ : (M, μ) → (P, ν).

Subdivide M for several levels, until each triangle size is small enough.

for all pi ∈ P do

Compute the geodesic from pi to every other vertex on M,

end for

h ← (0, 0, ⋯ , 0).

repeat

for all vertex vj on M do

Find the minimum weighted squared geodesic distance, decide which Voronoi cell vi    belongs to, vi ∈ Wt(h)

$t={\mathit{argmin}}_k{d}_{\mathrm{g}}^2(v_j,p_k)+h_k$

end for

for all pi ∈ P do

Compute the current cell area wi = ∫Wi(h) dμ,

end for

for all hi ∈ h do

Update hi, hi = hi + δ(νi − wi)

end for

until ∣νi − wi∣ < ϵ, ∀i.

return Power geodesic Voronoi diagram.