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

It is interesting to notice that the H 1 and ˜H 1 gradient flows are well-posedfor both ascent and descent while the H 0 gradient flow is only well-posed fordescent. This is related to the fact that the H 0 gradient descent flow smoothsthe curve, whereas the ˜H 1 gradient descent (or ascent) has neither a beneficialnor a detrimental effect on the regularity of the curve.10.7 Regularization of energy vs regularization of flow/metricImagine an energy E minimized on curves (numerically sampled to p points).Suppose it is not satisfactory in applications (it may be ill posed, or not robustto noise). We have two solutions available.• Add a regularization term R(c) and minimize E(c)+εR(c) by H 0 gradientdescent. The numerical complexity of computing ∇ H 0R is of order O(p).• Minimize E(c) by ˜H 1 gradient descent. The numerical complexity ofcomputing ∇˜H1E using the equations in Prop. 10.10 is of order O(p).The numerical complexity of the two approaches is similar; but in the secondcase we are minimizing the original energy. This can bring evident benefits, asshown in the following example (originally presented in the 2006 IMA conferenceon shape spaces).Example 10.33 We consider a region-based segmentation of a synthetic noisyimage, where the energy is the Chan-Vese energyE(c) =∫(I − u) 2 dA +c in∫(I − v) 2 dAc outplus the regularization terms α len(c) for H 0 flows.When flowing according to −∇ H 0E + ακN and a small value of α > 0, weobtain the following evolution of the curve, where the curve gets trapped in alocal minima due to noise.For a larger value of α, the regularization term forces the curve away fromthe square shape.When evolving using −∇˜H1E, the curve captures the correct shape on coarsescale, and then segments the noisy boundary of the square further.83