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

10.7.1 Robustness w.r.to local minima due to noiseThe concept of local depends on the norm.Definition 10.34 A curve c 0 is a local minimum of an energy E iff ∃ɛ > 0such that if ‖h‖ c0 < ɛ then E(c 0 ) ≤ E(c 0 + h).Note that Sobolev-type norms dominate H 0 -type norm:‖h‖ H 0 ≤ ‖h‖ Sobolev,and the norms are not equivalent. As a result the neighborhood of critical pointsin Sobolev-type space is different than in H 0 space.We present an experimental demonstration of what “local” means (originallypresented in [57]).Example 10.351. We initialize a contour in a noisy image.2. We run the H 0 gradient flow on energyE(c) = E cv (c) + α len(c),where E cv is the Chan-Vese energy. Let’s call the convergedcontour c 0 ; it is shown in the picture on the right.3. We adjust c 0 at one sample point by one pixel, and call the modificationĉ 0 . Note: ĉ 0 is an H 0 (but also “rigidified”) local perturbation of c 0 .4. We run and compare H 0 , “rigidified,” and Sobolev gradient flows initializedwith ĉ 0 .The results are in Figure 13 on the next page. As we can see, the SACevolution can escape from the local minimum that is induced by noise. Surprisingly,in the numerical experiments, the same result holds for SAC even withoutperturbing the local minimum: this is due to the effect of numerical noise.Many other examples, on synthetic and on real images, are present in [55,57, 58].84