0. Initialize variables: ξ, η, I.

for t = 1 to T do

1.1 Compute the horizontal lifts vξ and vη (Sect. 4.3).

1.2 Solve EPDiff until time δt = 1/T with initial condition v = vξ. Let ϕ0 be the diffeomorphism obtained at time δt and v0 the velocity field obtained at the same time.

1.3 Set I0 ← I ∘ (ϕ0)−1.

1.4 Set ξ0← ξv0 = −〈∇ I0, v0〉.

1.5 Computer w0 = B (1/T, vξ, vη) (Theorem 1).

1.6 Set η0 ← T ξw0 = −T 〈∇ I0, w0〉.

1.7 Set I ← I0, ξ ← ξ0 and η ← η0.

end for

2. Return the translated vector, η.