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

can be solved for k (this is not so easy to prove: see Michor and Mumford [37],or 4.4.7 and 4.6.6 in Hamilton [24]).11.2 Reparameterization to normal motionIn the preceding proposition we decided to use “orthogonal motion” as distinguishedchart for the manifold B. This choice leads also to a “lifting”.Lemma 11.3 Given any smooth homotopy C of immersed curves, there existsa reparameterization given by a parameterized family of diffeomorphisms Φ :[0, 1] × S 1 → S 1 , so that setting˜C(t, θ) := C(t, Φ(t, θ)) ;we have that ∂ t ˜C is orthogonal to ∂θ ˜C at all points; more precisely,π ˜T∂ t ˜C = 0 , πÑ∂ t ˜C = πN ∂ t C(where the last equality is up to the reparameterization Φ).For the proof, see Thm. 3.10 in [67] or §2.5 in [37].This explains what is commonly done in the level set method, where thetangent part of the flow is discarded.It is unclear that this choice is actually the best possible choice for computervision application. Consider the following example.Example 11.4 Suppose that c(θ) = (cos(θ), sin(θ)) is a planar circle. LetC(t, θ) = c(θ) + vt be an uniform translation of c. Let ˜C be as in the previousproposition; the Figure 17 shows the two motions. The two motions C and˜C coincide in B but are represented differently in M. Which one is best? Both“translation” and “orthogonal motions” seem a natural idea at first glance.C C ~The dotted line represents the trajectory of a point on the curve.Figure 17: Translation of a circle as a normal-only motion, by Prop. 11.3.90