Metrics of curves in shape optimization and analysis - Andrea Carlo ...

and we simply integrate twice! Moreover, exploiting the identity∫ s(∫ t) ∫ sh(r) dr dt = (s − r) h(r) dr ,00and with some extra computations, we obtain the result.In the end we obtain that the ˜H 1 gradient need not be computed as a convolution;so the ˜H 1 gradient enjoys nearly the same computational speed as the H 0gradient; moreover the resulting gradient flow is more stable, so it works fine innumerical implementations for a larger choice of time step discretization. Forthese reasons, ˜H1 Sobolev active contours are very fast.10.5 Analysis and calculus of ˜H 1 gradientsWe write h ∈ T c M as h = avg c (h) + ˜h; this decomposes0T c M = lR n ⊕ D c M (10.16)withD c M :={}h : S 1 → lR n | avg c (h) = 0 .If we assign to lR n its usual Euclidean norm, and to D c M the H j 0 norm22 ,then‖h‖ 2˜Hj = |avg c (h)| 2 lR + n λL2j ‖˜h‖ 2 .H j 0This means that the two spaces lR n and D c M are orthogonal w.r.to ˜H j .A remark on this decomposition is due.Remark 10.12 In the above, lR n is akin to be the space of translations andD c M the space of non-translating deformations.That labeling is not rigorous, though! since the subspace of T c M of deformationsthat do not move the center of mass avg c (c) is not D c M, but rather{ ∫h : h + ( c − avg c (c) ) }〈D s h · T 〉 ds = 0 ,as is easily deduced from equation (2.8).cThe decomposition (10.16) is quite useful in the calculus when analyticallycomputing ˜H 1 gradients and proving existence of flows. Indeed, let f = ∇ H 0E,g = ∇˜H1E; decomposeTo solvewe have to solve two equations:f = avg c (f) + ˜f , g = avg c (g) + ˜g .f = avg c (g) − λL 2 D 2 sg22 H j 0 was defined in (10.1). Note that Hj 0 is a seminorm on TcM, since its value is zero onconstant fields; but H j 0 is a norm on DcM, by Poincaré inequality (10.5).68