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

10.9 New regularization methodsTypically, in active contour energies a length penalty is added to obtain regularityof the evolving contour:E(c) = E data (c) + α len(c) .It is important to note that this length penalty will regularize the curve onlyif the curve evolution is derived as a H 0 gradient descent flow. If another curvemetric is used to derive the gradient descent flow, then a priori the length penaltymay have no regularizing effect on the curve regularity (cf. 10.32).10.9.1 Elastic regularizationWe can now consider an alternative approach for regularity of curve:∫E(c) = E data (c) + α len(c) κ 2 (s) dswhere α > 0 is a fixed constant. This energy favors regularity of curve, but thisregularization does not rely on properties of the metric; and the regularizationis scale invariant. Note that the H 0 gradient flow of this energy is ill posed.The numerical results (originally presented in [56]) are in Fig. 16. In allframes, the final limit of the gradient descent flow is shown; between differentframes, the value of α is increased.c. . . . . . . . . . . .minima for α increasing → . . . . . . . . . . . .E data (c) + α len(c)(H 0 gradient descent)E data (c) +α len(c) ∫ c κ2 (s) ds(Sobolev gradient descent)Figure 16: Elastic regularization. (From [58] c○ 2008 IEEE. Reproduced with permission).88