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

κ > 0HNNHNHκ < 0Figure 1: Example of a regular curve and its curvature.2 Shapes in applicationsA number of methods have been proposed in shape analysis to define distancesbetween shapes, averages of shapes and statistical models of shapes. At thesame time, there has been much previous work in shape optimization, for exampleimage segmentation via active contours, 3D stereo reconstruction viadeformable surfaces; in these later methods, many authors have defined energyfunctionals F (c) on curves (or on surfaces), whose minima represent the desiredsegmentation/reconstruction; and then utilized the calculus of variations to derivecurve evolutions to search minima of F (c), often referring to these evolutionsas gradient flows. The reference to these flows as gradient flows implies a certainRiemannian metric on the space of curves; but this fact has been largelyoverlooked. We call this metric H 0 , and properly define it in eqn. (2.4).2.1 Shape analysisMany method and tools comprise the shape analysis. We may list• distances between shapes,• averages for shapes,• principle component analysis for shapes and• probabilistic models of shapes.We will present a short overview of the above, in theory and in applications. Webegin by defining the distance function and signed distance function, two toolsthat we will use often in this theory.Definition 2.1 Let A, B ⊂ lR n be compact sets.7