LDDMM, Hands On

Three pictures behind Beg, Miller, Trouvé & Younes (2005): why a warp is built by flowing a smooth velocity field, why the gradient is taken in a smoothed space, and why the optimal path has constant speed.

1. One big step versus a flow

Two control points push toward each other (the left one to the right, the right one to the left). The smooth velocity field v they create is the same in both panels. On the left, the warp is the single-step displacement map x + v(x) (the small-deformation model). On the right, the same field is flowed in N small steps. Red cells are folded: the warp has turned them inside out.

displacement x + v(x)
flow of v in N steps

Raise the strength until the left panel folds, then watch the right panel: the field is the same but the flow keeps every cell the right way round. With N=1 the two panels coincide. With very few steps the right panel can fold too: that comes from the coarse stepping, not from the flow itself, and is why the paper's numerical scheme uses many small steps.

2. Why the gradient is taken in the smoothed space

The operator is L = −α∇² + γ. On a wave of frequency k it multiplies by A(k) = γ + α(2πk)², and L†L multiplies by λ(k) = A². The ordinary (L²) gradient contains a term 2L†L·v, so each frequency of v is scaled by λ(k): high frequencies are amplified. The V-gradient (Hilbert gradient) instead scales the force term by 1/λ(k): high frequencies are damped.

Orange: the L² gradient's factor λ(k) on v (grows with frequency, so a gradient-descent step size must shrink to keep the high frequencies stable). Blue: the V-gradient's factor 1/λ(k) on the data force (falls with frequency, so rough force components are suppressed). Vertical axis is logarithmic. The paper's implementation uses a discrete version of the symbol on a periodic grid; this plot uses the continuous one.

3. Why constant speed

For a given path with speed ‖vt‖ at time t on [0,1], the energy is ∫‖vt‖² dt and the length is ∫‖vt‖ dt. The Cauchy–Schwarz inequality says energy ≥ length², with equality exactly when the speed is constant. Re-timing the same path changes the speed but not where it goes, so a path with uneven speed can always be improved without changing its end point.

Left: a speed profile over time (the shaded area is the length). Right: energy against length² before re-timing, and after (where speed is constant at the average). The paper does this re-timing every 10 gradient iterations.