Algorithm 1 Unbiased Nonlinear Registration

Initialize t = 0 and u(x, 0) = 0. Given u(x, t), calculate the force field f(x, u(x, t)) using equation (18). Note that the fluid model, obtains the force field using equation (18) with λ = 0. Solve (21) for the instantaneous velocity v(x, t). Steps 4-6 describe the procedure for solving equation (20), advancing u(x, t) in time. Calculate the perturbation of the displacement field R(x) = v(x, t) − v(x, t) · ∇u(x, t). Time step Δt is calculated adaptively so that Δt·max(‖R‖2) = δu, where δu is the maximal displacement allowed in one iteration. Results in this work are obtained with δu = 0.1. Advance equation (20), i.e. ∂u(x, t)/∂t = R(x), in time, with time step from step 4, solving for u(x, t). If the cost functional, defined in either one of (22) through (25), decreases in the last fifty iterations by less than 1% of the total decrease in energy, then stop. Let t := t + Δt and go to step 2.