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

10.1 Sobolev-type metricsRecently in [53–58, 35] Yezzi–M.–Sundaramoorthi studied a family of Sobolevtypemetrics. Let D s := 1|c ′ | ∂ θ be the derivative with respect to arc parameter.Let j ≥ 1 be an integer 14 and∫〈h 1 , h 2 〉 Hj := 〈Dsh j 1 , Dsh j 2 〉 ds (10.1)0cwhere ∫ · · · ds was defined in 1.9. Let λ > 0 a fixed constant; we define thecSobolev-type metrics∫〈h 1 , h 2 〉 H j := 〈h 1 , h 2 〉 ds + λL 2j 〈h 1 , h 2 〉 Hj(10.2)〈h 1 , h 2 〉 ˜Hj:=c〈 ∫ h 1 ds,c∫c〉h 2 ds + λL 2j 〈h 1 , h 2 〉 Hj0where L = len(c). Notice that these metrics are geometric:0(10.3)• they are easily seen to be invariant w.r.to rotations and translations, (in astronger sense than in page 39, indeed in this case〈h 1 , h 2 〉 Ac = 〈h 1 , h 2 〉 c , 〈Rh 1 , Rh 2 〉 c = 〈h 1 , h 2 〉 cfor any Euclidean transformation A and rotation R);• they are reparameterization invariant due to the usage of the arc parameterin derivatives and integrals;• they are scale invariant, since the normalizing factors L 2j make them0-homogeneous w.r.to rescaling of the curve.For this last reason, we redefine the H 0 metric to be∫〈h 1 , h 2〉H := h 1 · h 2 ds (10.4)0so that it is again 0-homogeneous.This is a conformal version of the H 0 metricdefined in eqn. (2.4), so a gradient descent flow is a time reparameterization ofthe flow for the original H 0 metric; and the induced distance is again degenerate.When we will present a shape optimization energy E and we will minimizeE using a gradient descent flow driven by the H j or ˜H j gradient of E, we willcall the resulting algorithm a Sobolev active contour method (abbreviatedas SAC in the following).14 It is though possible to define Sobolev metrics for any j ∈ lR, j > 0; see Prop. 3.1 in [57].c61