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

H 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . time → . . . . . . . . . . . . . . . . . . . . . . . . . . . . .“Rigidified”(Translationfavored). . . . . . . . . . . . . . . . . . . . . . . . . . . . . time → . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . time → . . . . . . . . . . . . . . . . . . . . . . . . . . . . .SobolevActiveContoursFigure 13: Comparison of minimization flows when initialized near a localminimum; see Example 10.35. (From [57] c○ 2008 IEEE. Reproduced with permission).10.8 New shape optimization energiesMost of what follows was originally presented in [56, 58].There are many useful active contour energies that cannot be optimized usingH 0 gradient flow. There are mainly two reasons why this happens.• Some energies result in ill-posed H 0 flows.• Some energies result in high-order PDE and are difficult to implementnumerically.In many cases, we can optimize these energies using Sobolev active contoursand avoid ill-posed problems and reduce order of PDE.We remark that other gradients (e.g. the “rigidified” active contours ofCharpiat et al. [10]) cannot be used in the examples we will present: theminimization flows still are ill posed or high order. Similarly global methods(e.g. graph cuts) cannot deal with these types of energies.10.8.1 Average weighted lengthA simple example that falls in the first category is the average weightedlength energy that we presented in equation (2.6) in the introduction:∫E 1 (c) = φ(c(s)) ds = avg c (φ) ;cthis energy was introduced since it does not present the short length bias (thatwas discussed in §2.4.2).Let L φ = ∫ c φ ds and L = len(c) so that E 1(c) = L φ /L, then the gradient is∇E 1 = 1 L ∇L φ − L φL 2 ∇L .85