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

• Sundaramoorthi et al. [53] Sobolev type metrics∫∫〈h 1 , h 2 〉 H n = 〈h 1 , h 2 〉 ds + len(c) 2n 〈∂s n h 1 , ∂s n h 2 〉 ds〈h 1 , h 2 〉 ˜Hn=c〈 ∫ h 1 ds,c∫cc〉 ∫h 2 ds + len(c) 2n 〈∂s n h 1 , ∂s n h 2 〉 ds .cWe will now present a quick overview of all metrics (but for the latter, thatis discussed in the next section).9.1 H 0The H 0 inner product was defined in eqn. (2.4) as∫〈 〉h1 , h 2 := hH 0 1 (s) · h 2 (s) ds ; (9.1)cit is possibly the simplest geometric inner product that we may imagine toapply to curves. We already noted that the minimizing flows implemented intraditional active contour methods are “gradient flows” only if we use this H 0inner product.We will show in Sec. 11.3 that the H 0 -induced distance is identically zero. In[67] there is a result that shows that the distance is non degenerate, and minimalgeodesics exists, when the shape space is restricted to curves with uniformlybounded curvature.9.2 H AMichor and Mumford [37] propose the metric H A〈h 1 , h 2 〉 H A =|c∫ L0(1 + A|κ c | 2 )〈h 1 , h 2 〉 dswhere κ c is the curvature of c, and A > 0 is fixed.Properties 9.1• The induced distance is non degenerate.• The completion M (intended in the metric sense) isBV 2 ⊂ M ⊂ Lipwhere BV 2 are the curves that admit curvature as a measure and Lip arethe rectifiable curves.• There are compactness bounds.59