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

P c Π C ∇φ(C) = P c ∇φ(C) = r(t) d ′ (r(t)) (sin θ, − cos θ)P c P c Π C ∇φ(C) = −r(t) 2 d ′ (r(t)) (cos θ, sin θ)φ(C)D s C = d(r(t))(− sin θ, cos θ)P c (φ(C)D s C) = r(t)d(r(t)) (cos θ, sin θ)∇˜H1E(C) = Lavg c (∇φ(C)) − 1λL P ( ) 1CP C Π C ∇φ(C) +λL P (CΠ C φ(C)Ds C ) ==1 (d(r(t)) + r(t) d ′ (r(t)) ) (cos θ, sin θ) ;λ2πwhereas for the ˜H 0 gradient descent we use the formula (10.27)∇ H 0E = L∇φ(C) − LD s (φ(C)D s C) == 2π ( r(t)d ′ (r(t)) + d(r(t)) ) (cos θ, sin θ)We will show in the following theorems how to prove that ˜H 1 gradient flowsof common energies are well defined; to this end, we will provide a detailed prooffor the centroid energy E (2.9) discussed in the previous section, and for thegeodesic active contour model [7, 27] (that was presented in Section 2.2); thoseproofs illustrates methods that may be used for many other energies.10.6.1 Lemmas and inequalitiesLet us also prepare the proof by presenting some useful inequalities in threeLemma.Definition 10.24 We recall from 3.11 that C 0 = C 0 (S 1 → lR n ) is the space ofcontinuous functions, that is a Banach space with norm‖c‖ 0 :=sup |c(θ)| .θ∈[0,2π]Similarly, C 1 = C 1 (S 1 → lR n ) is the space of continuously differentiable functions,that is a Banach space with norm‖c‖ 1 := ‖c‖ 0 + ‖c ′ ‖ 0where c ′ (θ) = ∂c∂θ(θ) is the usual parametric derivative of c.Lemma 10.25 • We will use repeatedly the two following inequalities. Ifa 1 , a 2 , b 1 , b 2 ∈ lR then|a 1 b 1 − a 2 b 2 | ≤ |b1|+|b2|2|a 1 − a 2 | + |a1|+|a2|2|b 1 − b 2 || a1b 1− a2b 2| ≤ |b1|+|b2|2|b |a 1b 2| 1 − a 2 | + |a1|+|a2|2|b 1b 2||b 1 − b 2 |(i)73