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

and̂∇˜HjE(0) = ̂∇ H 0E(0),̂∇˜HjE(l) = ̂∇ H 0E(l)λ(2πl) 2j for l ∈ Z\{0}, (10.13)It is clear from the previous expressions that high frequency components of∇ H 0E(c) are increasingly less pronounced in the various forms of the H j gradients.So H j and ˜H j gradients are much smoother than H 0 gradients. Note that usinga SAC method smooths the gradients, so it induces smoother minimization flows,but it does not smooth the curves themselves.The above phenomenon may be also explained by the following argument.The gradient satisfies the following property.Proposition 10.8 If DE(c) ≠ 0, the gradient ∇E(c) is the vector v in T c Mthat provides the maximum inDE(c)(v)|DE(c)(h)|= sup. (10.14)‖v‖ c h∈T cM\{0} ‖h‖ cThus, the gradient is the most efficient perturbation, i.e., it maximizeschange in energy by moving in direction hcost of moving in direction hBy constructing ‖ · ‖ c to penalize “unfavorable” directions, we have controlover the path taken to minimize E without changing the energy E. 20Since higher frequencies are more penalized in H 1 than in H 0 , this explainswhy a SAC prefers to move the curves in a coarser scale w.r.to a traditionalactive contour method.10.3.2 Flow regularizationIt is also possible to show mathematically that the Sobolev–type gradientsregularize the flows of well known energies, by reducing the degree of the P.D.E.,as shown in this example.Example 10.9 (Elastica) In the case of the elastic energy∫ ∫E(c) = κ 2 ds = |Dsc| 2 2 ds ,ccthe H 0 gradient is 21 ∇ H 0E = LD s (2D 3 sc + 3|D 2 sc| 2 D s c)20 In a sense, the focus of research in active contours has been mostly on the numerator ineqn. (10.14) — whereas we now focus on the denominator.21 We use the definition (10.4) of H 0 ; the directional derivative was computed in 4.666