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

We already presented all the calculus needed to compute the H 0 and ˜H 1gradients of E 1 (see Section 2.4.4 and Example 10.21); let us summarize all theresults.• In the case of H 0 gradient descent flow, the second term is ill posed , sinceL φ /L 2 > 0 and ∇ H 0L is the driving term in the geometric heat flow.This is also clear from the explicit formula∇ H 0E 1 (c) = [∇φ · N − κ(φ − avg c (φ))]N (10.41)since avg c (φ) is the average value of φ, therefore φ − avg c (φ) is negativeon roughly half of the curve; so the second term tries to increase length onhalf of curve using a geometric heat flow (and this is ill posed); whereasthe first term ∇φ does not stabilize the flow.• In the case of the Sobolev gradient, we saw in the proof of Theorem 10.31that the gradients ∇˜H1L φ and ∇˜H1L are locally Lipschitz in C 1 , so, by usingLemma 10.25 we conclude that the gradient ∇˜H1E 1 is locally Lipschitz aswell, hence the gradient flow is well defined.10.8.2 New edge-based active contour modelsThe average weighted length idea can be used to build new energies with interestingapplications in tracking.Let us call∫E old (c) = φ(c(s)) dsthe traditional edge-based active contour energy.When using a Sobolev–type norm, we can consider non-shrinking edge-basedmodels, as in the following examples.Example 10.36 Let us consider the energy∫E new (c) = φ(c(s)) ( L −1 + αLκ 2 (s) ) dswherecc• the first term is edge-detection without shrinking bias (nor regularity),• and the second term is a kind of elastic regularization (that will bediscussed also in the next section) that is moreover relaxed near edges;the length terms render the whole energy scale invariant.The frames in Figure 14 on the following page show initialization and finalresults obtained with different norms and energies.86